LMFDB - Embedded newform 8020.2.a.f.1.12

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:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-1.55757 q^{3} +1.00000 q^{5} +3.26992 q^{7} -0.573979 q^{9} +O(q^{10})$	$q-1.55757 q^{3} +1.00000 q^{5} +3.26992 q^{7} -0.573979 q^{9} +3.84499 q^{11} +4.55328 q^{13} -1.55757 q^{15} +3.75593 q^{17} +4.33352 q^{19} -5.09312 q^{21} -6.95144 q^{23} +1.00000 q^{25} +5.56672 q^{27} +8.89661 q^{29} +4.65700 q^{31} -5.98884 q^{33} +3.26992 q^{35} +9.51805 q^{37} -7.09204 q^{39} -4.95057 q^{41} +0.695724 q^{43} -0.573979 q^{45} -1.13212 q^{47} +3.69236 q^{49} -5.85012 q^{51} +4.06053 q^{53} +3.84499 q^{55} -6.74975 q^{57} +12.8466 q^{59} -10.8622 q^{61} -1.87686 q^{63} +4.55328 q^{65} +12.0433 q^{67} +10.8274 q^{69} -4.14991 q^{71} -4.89555 q^{73} -1.55757 q^{75} +12.5728 q^{77} -7.64922 q^{79} -6.94861 q^{81} -6.72530 q^{83} +3.75593 q^{85} -13.8571 q^{87} +4.16805 q^{89} +14.8888 q^{91} -7.25359 q^{93} +4.33352 q^{95} -10.4983 q^{97} -2.20695 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * 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
$3$	−1.55757	−0.899263	−0.449631	−	0.893214i	$-0.648445\pi$
$3$	−1.55757	−0.899263	−0.449631	+	0.893214i	$0.648445\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.26992	1.23591	0.617956	−	0.786212i	$-0.287961\pi$
$7$	3.26992	1.23591	0.617956	+	0.786212i	$0.287961\pi$
$8$	0	0
$9$	−0.573979	−0.191326
$10$	0	0
$11$	3.84499	1.15931	0.579655	−	0.814862i	$-0.303187\pi$
$11$	3.84499	1.15931	0.579655	+	0.814862i	$0.303187\pi$
$12$	0	0
$13$	4.55328	1.26285	0.631426	−	0.775436i	$-0.282470\pi$
$13$	4.55328	1.26285	0.631426	+	0.775436i	$0.282470\pi$
$14$	0	0
$15$	−1.55757	−0.402163
$16$	0	0
$17$	3.75593	0.910947	0.455473	−	0.890249i	$-0.349470\pi$
$17$	3.75593	0.910947	0.455473	+	0.890249i	$0.349470\pi$
$18$	0	0
$19$	4.33352	0.994177	0.497089	−	0.867700i	$-0.334402\pi$
$19$	4.33352	0.994177	0.497089	+	0.867700i	$0.334402\pi$
$20$	0	0
$21$	−5.09312	−1.11141
$22$	0	0
$23$	−6.95144	−1.44948	−0.724738	−	0.689025i	$-0.758039\pi$
$23$	−6.95144	−1.44948	−0.724738	+	0.689025i	$0.758039\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.56672	1.07132
$28$	0	0
$29$	8.89661	1.65206	0.826030	−	0.563627i	$-0.190594\pi$
$29$	8.89661	1.65206	0.826030	+	0.563627i	$0.190594\pi$
$30$	0	0
$31$	4.65700	0.836421	0.418211	−	0.908350i	$-0.362657\pi$
$31$	4.65700	0.836421	0.418211	+	0.908350i	$0.362657\pi$
$32$	0	0
$33$	−5.98884	−1.04252
$34$	0	0
$35$	3.26992	0.552717
$36$	0	0
$37$	9.51805	1.56476	0.782379	−	0.622803i	$-0.214006\pi$
$37$	9.51805	1.56476	0.782379	+	0.622803i	$0.214006\pi$
$38$	0	0
$39$	−7.09204	−1.13564
$40$	0	0
$41$	−4.95057	−0.773148	−0.386574	−	0.922258i	$-0.626342\pi$
$41$	−4.95057	−0.773148	−0.386574	+	0.922258i	$0.626342\pi$
$42$	0	0
$43$	0.695724	0.106097	0.0530485	−	0.998592i	$-0.483106\pi$
$43$	0.695724	0.106097	0.0530485	+	0.998592i	$0.483106\pi$
$44$	0	0
$45$	−0.573979	−0.0855637
$46$	0	0
$47$	−1.13212	−0.165137	−0.0825683	−	0.996585i	$-0.526312\pi$
$47$	−1.13212	−0.165137	−0.0825683	+	0.996585i	$0.526312\pi$
$48$	0	0
$49$	3.69236	0.527481
$50$	0	0
$51$	−5.85012	−0.819180
$52$	0	0
$53$	4.06053	0.557756	0.278878	−	0.960326i	$-0.410037\pi$
$53$	4.06053	0.557756	0.278878	+	0.960326i	$0.410037\pi$
$54$	0	0
$55$	3.84499	0.518459
$56$	0	0
$57$	−6.74975	−0.894027
$58$	0	0
$59$	12.8466	1.67248	0.836241	−	0.548362i	$-0.184749\pi$
$59$	12.8466	1.67248	0.836241	+	0.548362i	$0.184749\pi$
$60$	0	0
$61$	−10.8622	−1.39076	−0.695379	−	0.718643i	$-0.744764\pi$
$61$	−10.8622	−1.39076	−0.695379	+	0.718643i	$0.744764\pi$
$62$	0	0
$63$	−1.87686	−0.236463
$64$	0	0
$65$	4.55328	0.564764
$66$	0	0
$67$	12.0433	1.47132	0.735661	−	0.677350i	$-0.236872\pi$
$67$	12.0433	1.47132	0.735661	+	0.677350i	$0.236872\pi$
$68$	0	0
$69$	10.8274	1.30346
$70$	0	0
$71$	−4.14991	−0.492503	−0.246252	−	0.969206i	$-0.579199\pi$
$71$	−4.14991	−0.492503	−0.246252	+	0.969206i	$0.579199\pi$
$72$	0	0
$73$	−4.89555	−0.572981	−0.286490	−	0.958083i	$-0.592489\pi$
$73$	−4.89555	−0.572981	−0.286490	+	0.958083i	$0.592489\pi$
$74$	0	0
$75$	−1.55757	−0.179853
$76$	0	0
$77$	12.5728	1.43281
$78$	0	0
$79$	−7.64922	−0.860604	−0.430302	−	0.902685i	$-0.641593\pi$
$79$	−7.64922	−0.860604	−0.430302	+	0.902685i	$0.641593\pi$
$80$	0	0
$81$	−6.94861	−0.772068
$82$	0	0
$83$	−6.72530	−0.738198	−0.369099	−	0.929390i	$-0.620334\pi$
$83$	−6.72530	−0.738198	−0.369099	+	0.929390i	$0.620334\pi$
$84$	0	0
$85$	3.75593	0.407388
$86$	0	0
$87$	−13.8571	−1.48564
$88$	0	0
$89$	4.16805	0.441812	0.220906	−	0.975295i	$-0.429099\pi$
$89$	4.16805	0.441812	0.220906	+	0.975295i	$0.429099\pi$
$90$	0	0
$91$	14.8888	1.56077
$92$	0	0
$93$	−7.25359	−0.752162
$94$	0	0
$95$	4.33352	0.444610
$96$	0	0
$97$	−10.4983	−1.06594	−0.532970	−	0.846134i	$-0.678924\pi$
$97$	−10.4983	−1.06594	−0.532970	+	0.846134i	$0.678924\pi$
$98$	0	0
$99$	−2.20695	−0.221806
$100$	0	0
$101$	8.80349	0.875980	0.437990	−	0.898980i	$-0.355691\pi$
$101$	8.80349	0.875980	0.437990	+	0.898980i	$0.355691\pi$
$102$	0	0
$103$	−12.2902	−1.21099	−0.605495	−	0.795849i	$-0.707025\pi$
$103$	−12.2902	−1.21099	−0.605495	+	0.795849i	$0.707025\pi$
$104$	0	0
$105$	−5.09312	−0.497038
$106$	0	0
$107$	15.3777	1.48662	0.743310	−	0.668947i	$-0.233255\pi$
$107$	15.3777	1.48662	0.743310	+	0.668947i	$0.233255\pi$
$108$	0	0
$109$	1.00714	0.0964661	0.0482330	−	0.998836i	$-0.484641\pi$
$109$	1.00714	0.0964661	0.0482330	+	0.998836i	$0.484641\pi$
$110$	0	0
$111$	−14.8250	−1.40713
$112$	0	0
$113$	3.94815	0.371411	0.185705	−	0.982605i	$-0.440543\pi$
$113$	3.94815	0.371411	0.185705	+	0.982605i	$0.440543\pi$
$114$	0	0
$115$	−6.95144	−0.648225
$116$	0	0
$117$	−2.61348	−0.241617
$118$	0	0
$119$	12.2816	1.12585
$120$	0	0
$121$	3.78398	0.343998
$122$	0	0
$123$	7.71085	0.695264
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−15.3379	−1.36102	−0.680508	−	0.732741i	$-0.738241\pi$
$127$	−15.3379	−1.36102	−0.680508	+	0.732741i	$0.738241\pi$
$128$	0	0
$129$	−1.08364	−0.0954091
$130$	0	0
$131$	−1.59053	−0.138965	−0.0694825	−	0.997583i	$-0.522135\pi$
$131$	−1.59053	−0.138965	−0.0694825	+	0.997583i	$0.522135\pi$
$132$	0	0
$133$	14.1702	1.22872
$134$	0	0
$135$	5.56672	0.479107
$136$	0	0
$137$	−21.5615	−1.84212	−0.921060	−	0.389420i	$-0.872676\pi$
$137$	−21.5615	−1.84212	−0.921060	+	0.389420i	$0.872676\pi$
$138$	0	0
$139$	−20.8402	−1.76764	−0.883820	−	0.467828i	$-0.845037\pi$
$139$	−20.8402	−1.76764	−0.883820	+	0.467828i	$0.845037\pi$
$140$	0	0
$141$	1.76335	0.148501
$142$	0	0
$143$	17.5073	1.46404
$144$	0	0
$145$	8.89661	0.738823
$146$	0	0
$147$	−5.75111	−0.474344
$148$	0	0
$149$	5.46074	0.447362	0.223681	−	0.974662i	$-0.428193\pi$
$149$	5.46074	0.447362	0.223681	+	0.974662i	$0.428193\pi$
$150$	0	0
$151$	7.27391	0.591943	0.295971	−	0.955197i	$-0.404357\pi$
$151$	7.27391	0.591943	0.295971	+	0.955197i	$0.404357\pi$
$152$	0	0
$153$	−2.15582	−0.174288
$154$	0	0
$155$	4.65700	0.374059
$156$	0	0
$157$	−9.68925	−0.773287	−0.386643	−	0.922229i	$-0.626366\pi$
$157$	−9.68925	−0.773287	−0.386643	+	0.922229i	$0.626366\pi$
$158$	0	0
$159$	−6.32455	−0.501570
$160$	0	0
$161$	−22.7306	−1.79143
$162$	0	0
$163$	−18.0106	−1.41070	−0.705351	−	0.708858i	$-0.749211\pi$
$163$	−18.0106	−1.41070	−0.705351	+	0.708858i	$0.749211\pi$
$164$	0	0
$165$	−5.98884	−0.466231
$166$	0	0
$167$	18.9322	1.46502	0.732510	−	0.680756i	$-0.238349\pi$
$167$	18.9322	1.46502	0.732510	+	0.680756i	$0.238349\pi$
$168$	0	0
$169$	7.73233	0.594794
$170$	0	0
$171$	−2.48735	−0.190212
$172$	0	0
$173$	19.9273	1.51505	0.757523	−	0.652809i	$-0.226410\pi$
$173$	19.9273	1.51505	0.757523	+	0.652809i	$0.226410\pi$
$174$	0	0
$175$	3.26992	0.247183
$176$	0	0
$177$	−20.0094	−1.50400
$178$	0	0
$179$	−12.1178	−0.905725	−0.452863	−	0.891580i	$-0.649597\pi$
$179$	−12.1178	−0.905725	−0.452863	+	0.891580i	$0.649597\pi$
$180$	0	0
$181$	6.56162	0.487722	0.243861	−	0.969810i	$-0.421586\pi$
$181$	6.56162	0.487722	0.243861	+	0.969810i	$0.421586\pi$
$182$	0	0
$183$	16.9186	1.25066
$184$	0	0
$185$	9.51805	0.699781
$186$	0	0
$187$	14.4415	1.05607
$188$	0	0
$189$	18.2027	1.32405
$190$	0	0
$191$	−11.1325	−0.805517	−0.402758	−	0.915306i	$-0.631948\pi$
$191$	−11.1325	−0.805517	−0.402758	+	0.915306i	$0.631948\pi$
$192$	0	0
$193$	−0.224492	−0.0161593	−0.00807963	−	0.999967i	$-0.502572\pi$
$193$	−0.224492	−0.0161593	−0.00807963	+	0.999967i	$0.502572\pi$
$194$	0	0
$195$	−7.09204	−0.507872
$196$	0	0
$197$	−14.2467	−1.01504	−0.507519	−	0.861641i	$-0.669437\pi$
$197$	−14.2467	−1.01504	−0.507519	+	0.861641i	$0.669437\pi$
$198$	0	0
$199$	−14.0278	−0.994405	−0.497202	−	0.867635i	$-0.665639\pi$
$199$	−14.0278	−0.994405	−0.497202	+	0.867635i	$0.665639\pi$
$200$	0	0
$201$	−18.7583	−1.32310
$202$	0	0
$203$	29.0912	2.04180
$204$	0	0
$205$	−4.95057	−0.345762
$206$	0	0
$207$	3.98998	0.277323
$208$	0	0
$209$	16.6624	1.15256
$210$	0	0
$211$	−21.9377	−1.51025	−0.755127	−	0.655579i	$-0.772425\pi$
$211$	−21.9377	−1.51025	−0.755127	+	0.655579i	$0.772425\pi$
$212$	0	0
$213$	6.46377	0.442890
$214$	0	0
$215$	0.695724	0.0474480
$216$	0	0
$217$	15.2280	1.03374
$218$	0	0
$219$	7.62516	0.515260
$220$	0	0
$221$	17.1018	1.15039
$222$	0	0
$223$	−20.1518	−1.34946	−0.674732	−	0.738063i	$-0.735741\pi$
$223$	−20.1518	−1.34946	−0.674732	+	0.738063i	$0.735741\pi$
$224$	0	0
$225$	−0.573979	−0.0382653
$226$	0	0
$227$	−21.5549	−1.43065	−0.715325	−	0.698792i	$-0.753721\pi$
$227$	−21.5549	−1.43065	−0.715325	+	0.698792i	$0.753721\pi$
$228$	0	0
$229$	−11.8018	−0.779884	−0.389942	−	0.920839i	$-0.627505\pi$
$229$	−11.8018	−0.779884	−0.389942	+	0.920839i	$0.627505\pi$
$230$	0	0
$231$	−19.5830	−1.28847
$232$	0	0
$233$	−3.33851	−0.218713	−0.109357	−	0.994003i	$-0.534879\pi$
$233$	−3.33851	−0.218713	−0.109357	+	0.994003i	$0.534879\pi$
$234$	0	0
$235$	−1.13212	−0.0738513
$236$	0	0
$237$	11.9142	0.773909
$238$	0	0
$239$	−13.9926	−0.905107	−0.452553	−	0.891737i	$-0.649487\pi$
$239$	−13.9926	−0.905107	−0.452553	+	0.891737i	$0.649487\pi$
$240$	0	0
$241$	9.08798	0.585408	0.292704	−	0.956203i	$-0.405445\pi$
$241$	9.08798	0.585408	0.292704	+	0.956203i	$0.405445\pi$
$242$	0	0
$243$	−5.87721	−0.377023
$244$	0	0
$245$	3.69236	0.235897
$246$	0	0
$247$	19.7317	1.25550
$248$	0	0
$249$	10.4751	0.663834
$250$	0	0
$251$	−11.6958	−0.738230	−0.369115	−	0.929384i	$-0.620339\pi$
$251$	−11.6958	−0.738230	−0.369115	+	0.929384i	$0.620339\pi$
$252$	0	0
$253$	−26.7283	−1.68039
$254$	0	0
$255$	−5.85012	−0.366349
$256$	0	0
$257$	0.0674009	0.00420435	0.00210218	−	0.999998i	$-0.499331\pi$
$257$	0.0674009	0.00420435	0.00210218	+	0.999998i	$0.499331\pi$
$258$	0	0
$259$	31.1232	1.93390
$260$	0	0
$261$	−5.10647	−0.316082
$262$	0	0
$263$	15.5088	0.956312	0.478156	−	0.878275i	$-0.341305\pi$
$263$	15.5088	0.956312	0.478156	+	0.878275i	$0.341305\pi$
$264$	0	0
$265$	4.06053	0.249436
$266$	0	0
$267$	−6.49202	−0.397305
$268$	0	0
$269$	−6.58321	−0.401386	−0.200693	−	0.979654i	$-0.564319\pi$
$269$	−6.58321	−0.401386	−0.200693	+	0.979654i	$0.564319\pi$
$270$	0	0
$271$	−14.4108	−0.875393	−0.437696	−	0.899123i	$-0.644206\pi$
$271$	−14.4108	−0.875393	−0.437696	+	0.899123i	$0.644206\pi$
$272$	0	0
$273$	−23.1904	−1.40355
$274$	0	0
$275$	3.84499	0.231862
$276$	0	0
$277$	−15.1229	−0.908647	−0.454323	−	0.890837i	$-0.650119\pi$
$277$	−15.1229	−0.908647	−0.454323	+	0.890837i	$0.650119\pi$
$278$	0	0
$279$	−2.67302	−0.160029
$280$	0	0
$281$	6.89587	0.411373	0.205686	−	0.978618i	$-0.434057\pi$
$281$	6.89587	0.411373	0.205686	+	0.978618i	$0.434057\pi$
$282$	0	0
$283$	20.9807	1.24718	0.623588	−	0.781753i	$-0.285674\pi$
$283$	20.9807	1.24718	0.623588	+	0.781753i	$0.285674\pi$
$284$	0	0
$285$	−6.74975	−0.399821
$286$	0	0
$287$	−16.1879	−0.955544
$288$	0	0
$289$	−2.89300	−0.170176
$290$	0	0
$291$	16.3518	0.958561
$292$	0	0
$293$	6.92334	0.404466	0.202233	−	0.979337i	$-0.435180\pi$
$293$	6.92334	0.404466	0.202233	+	0.979337i	$0.435180\pi$
$294$	0	0
$295$	12.8466	0.747957
$296$	0	0
$297$	21.4040	1.24199
$298$	0	0
$299$	−31.6518	−1.83047
$300$	0	0
$301$	2.27496	0.131127
$302$	0	0
$303$	−13.7120	−0.787736
$304$	0	0
$305$	−10.8622	−0.621966
$306$	0	0
$307$	27.9265	1.59385	0.796924	−	0.604080i	$-0.206459\pi$
$307$	27.9265	1.59385	0.796924	+	0.604080i	$0.206459\pi$
$308$	0	0
$309$	19.1428	1.08900
$310$	0	0
$311$	30.2395	1.71473	0.857363	−	0.514713i	$-0.172101\pi$
$311$	30.2395	1.71473	0.857363	+	0.514713i	$0.172101\pi$
$312$	0	0
$313$	21.7424	1.22895	0.614477	−	0.788935i	$-0.289367\pi$
$313$	21.7424	1.22895	0.614477	+	0.788935i	$0.289367\pi$
$314$	0	0
$315$	−1.87686	−0.105749
$316$	0	0
$317$	−9.68366	−0.543889	−0.271944	−	0.962313i	$-0.587667\pi$
$317$	−9.68366	−0.543889	−0.271944	+	0.962313i	$0.587667\pi$
$318$	0	0
$319$	34.2074	1.91525
$320$	0	0
$321$	−23.9519	−1.33686
$322$	0	0
$323$	16.2764	0.905642
$324$	0	0
$325$	4.55328	0.252570
$326$	0	0
$327$	−1.56868	−0.0867484
$328$	0	0
$329$	−3.70194	−0.204094
$330$	0	0
$331$	−0.0431570	−0.00237212	−0.00118606	−	0.999999i	$-0.500378\pi$
$331$	−0.0431570	−0.00237212	−0.00118606	+	0.999999i	$0.500378\pi$
$332$	0	0
$333$	−5.46316	−0.299379
$334$	0	0
$335$	12.0433	0.657995
$336$	0	0
$337$	−11.6482	−0.634519	−0.317259	−	0.948339i	$-0.602763\pi$
$337$	−11.6482	−0.634519	−0.317259	+	0.948339i	$0.602763\pi$
$338$	0	0
$339$	−6.14951	−0.333996
$340$	0	0
$341$	17.9061	0.969671
$342$	0	0
$343$	−10.8157	−0.583993
$344$	0	0
$345$	10.8274	0.582925
$346$	0	0
$347$	−5.48953	−0.294694	−0.147347	−	0.989085i	$-0.547073\pi$
$347$	−5.48953	−0.294694	−0.147347	+	0.989085i	$0.547073\pi$
$348$	0	0
$349$	34.5372	1.84873	0.924366	−	0.381507i	$-0.124595\pi$
$349$	34.5372	1.84873	0.924366	+	0.381507i	$0.124595\pi$
$350$	0	0
$351$	25.3468	1.35291
$352$	0	0
$353$	26.0609	1.38708	0.693541	−	0.720417i	$-0.256049\pi$
$353$	26.0609	1.38708	0.693541	+	0.720417i	$0.256049\pi$
$354$	0	0
$355$	−4.14991	−0.220254
$356$	0	0
$357$	−19.1294	−1.01244
$358$	0	0
$359$	35.1321	1.85420	0.927102	−	0.374809i	$-0.122292\pi$
$359$	35.1321	1.85420	0.927102	+	0.374809i	$0.122292\pi$
$360$	0	0
$361$	−0.220620	−0.0116116
$362$	0	0
$363$	−5.89381	−0.309345
$364$	0	0
$365$	−4.89555	−0.256245
$366$	0	0
$367$	−6.99093	−0.364924	−0.182462	−	0.983213i	$-0.558407\pi$
$367$	−6.99093	−0.364924	−0.182462	+	0.983213i	$0.558407\pi$
$368$	0	0
$369$	2.84152	0.147924
$370$	0	0
$371$	13.2776	0.689338
$372$	0	0
$373$	17.8511	0.924296	0.462148	−	0.886803i	$-0.347079\pi$
$373$	17.8511	0.924296	0.462148	+	0.886803i	$0.347079\pi$
$374$	0	0
$375$	−1.55757	−0.0804325
$376$	0	0
$377$	40.5087	2.08631
$378$	0	0
$379$	−18.7515	−0.963202	−0.481601	−	0.876391i	$-0.659945\pi$
$379$	−18.7515	−0.963202	−0.481601	+	0.876391i	$0.659945\pi$
$380$	0	0
$381$	23.8898	1.22391
$382$	0	0
$383$	−14.7326	−0.752799	−0.376399	−	0.926458i	$-0.622838\pi$
$383$	−14.7326	−0.752799	−0.376399	+	0.926458i	$0.622838\pi$
$384$	0	0
$385$	12.5728	0.640770
$386$	0	0
$387$	−0.399331	−0.0202991
$388$	0	0
$389$	−22.6498	−1.14839	−0.574195	−	0.818719i	$-0.694685\pi$
$389$	−22.6498	−1.14839	−0.574195	+	0.818719i	$0.694685\pi$
$390$	0	0
$391$	−26.1091	−1.32040
$392$	0	0
$393$	2.47736	0.124966
$394$	0	0
$395$	−7.64922	−0.384874
$396$	0	0
$397$	9.26595	0.465045	0.232522	−	0.972591i	$-0.425302\pi$
$397$	9.26595	0.465045	0.232522	+	0.972591i	$0.425302\pi$
$398$	0	0
$399$	−22.0711	−1.10494
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	21.2046	1.05628
$404$	0	0
$405$	−6.94861	−0.345279
$406$	0	0
$407$	36.5968	1.81404
$408$	0	0
$409$	8.91652	0.440893	0.220447	−	0.975399i	$-0.429248\pi$
$409$	8.91652	0.440893	0.220447	+	0.975399i	$0.429248\pi$
$410$	0	0
$411$	33.5835	1.65655
$412$	0	0
$413$	42.0073	2.06704
$414$	0	0
$415$	−6.72530	−0.330132
$416$	0	0
$417$	32.4600	1.58957
$418$	0	0
$419$	−18.8117	−0.919014	−0.459507	−	0.888174i	$-0.651974\pi$
$419$	−18.8117	−0.919014	−0.459507	+	0.888174i	$0.651974\pi$
$420$	0	0
$421$	24.0227	1.17079	0.585397	−	0.810747i	$-0.300939\pi$
$421$	24.0227	1.17079	0.585397	+	0.810747i	$0.300939\pi$
$422$	0	0
$423$	0.649813	0.0315950
$424$	0	0
$425$	3.75593	0.182189
$426$	0	0
$427$	−35.5184	−1.71886
$428$	0	0
$429$	−27.2689	−1.31655
$430$	0	0
$431$	10.3763	0.499809	0.249904	−	0.968271i	$-0.419601\pi$
$431$	10.3763	0.499809	0.249904	+	0.968271i	$0.419601\pi$
$432$	0	0
$433$	−10.2200	−0.491142	−0.245571	−	0.969379i	$-0.578975\pi$
$433$	−10.2200	−0.491142	−0.245571	+	0.969379i	$0.578975\pi$
$434$	0	0
$435$	−13.8571	−0.664396
$436$	0	0
$437$	−30.1242	−1.44104
$438$	0	0
$439$	18.1916	0.868236	0.434118	−	0.900856i	$-0.357060\pi$
$439$	18.1916	0.868236	0.434118	+	0.900856i	$0.357060\pi$
$440$	0	0
$441$	−2.11934	−0.100921
$442$	0	0
$443$	27.4568	1.30451	0.652257	−	0.757998i	$-0.273822\pi$
$443$	27.4568	1.30451	0.652257	+	0.757998i	$0.273822\pi$
$444$	0	0
$445$	4.16805	0.197584
$446$	0	0
$447$	−8.50548	−0.402296
$448$	0	0
$449$	37.7373	1.78093	0.890467	−	0.455047i	$-0.150378\pi$
$449$	37.7373	1.78093	0.890467	+	0.455047i	$0.150378\pi$
$450$	0	0
$451$	−19.0349	−0.896318
$452$	0	0
$453$	−11.3296	−0.532312
$454$	0	0
$455$	14.8888	0.698000
$456$	0	0
$457$	11.9472	0.558865	0.279432	−	0.960165i	$-0.409854\pi$
$457$	11.9472	0.558865	0.279432	+	0.960165i	$0.409854\pi$
$458$	0	0
$459$	20.9082	0.975911
$460$	0	0
$461$	−40.0303	−1.86440	−0.932200	−	0.361945i	$-0.882113\pi$
$461$	−40.0303	−1.86440	−0.932200	+	0.361945i	$0.882113\pi$
$462$	0	0
$463$	−18.3345	−0.852077	−0.426039	−	0.904705i	$-0.640091\pi$
$463$	−18.3345	−0.852077	−0.426039	+	0.904705i	$0.640091\pi$
$464$	0	0
$465$	−7.25359	−0.336377
$466$	0	0
$467$	6.49626	0.300611	0.150306	−	0.988640i	$-0.451974\pi$
$467$	6.49626	0.300611	0.150306	+	0.988640i	$0.451974\pi$
$468$	0	0
$469$	39.3806	1.81843
$470$	0	0
$471$	15.0917	0.695388
$472$	0	0
$473$	2.67506	0.122999
$474$	0	0
$475$	4.33352	0.198835
$476$	0	0
$477$	−2.33066	−0.106713
$478$	0	0
$479$	−15.7667	−0.720399	−0.360199	−	0.932875i	$-0.617291\pi$
$479$	−15.7667	−0.720399	−0.360199	+	0.932875i	$0.617291\pi$
$480$	0	0
$481$	43.3383	1.97606
$482$	0	0
$483$	35.4045	1.61096
$484$	0	0
$485$	−10.4983	−0.476703
$486$	0	0
$487$	−36.5514	−1.65630	−0.828150	−	0.560506i	$-0.810607\pi$
$487$	−36.5514	−1.65630	−0.828150	+	0.560506i	$0.810607\pi$
$488$	0	0
$489$	28.0528	1.26859
$490$	0	0
$491$	−42.1752	−1.90334	−0.951669	−	0.307125i	$-0.900633\pi$
$491$	−42.1752	−1.90334	−0.951669	+	0.307125i	$0.900633\pi$
$492$	0	0
$493$	33.4150	1.50494
$494$	0	0
$495$	−2.20695	−0.0991948
$496$	0	0
$497$	−13.5699	−0.608691
$498$	0	0
$499$	−20.0670	−0.898321	−0.449161	−	0.893451i	$-0.648277\pi$
$499$	−20.0670	−0.898321	−0.449161	+	0.893451i	$0.648277\pi$
$500$	0	0
$501$	−29.4882	−1.31744
$502$	0	0
$503$	−18.4027	−0.820537	−0.410269	−	0.911965i	$-0.634565\pi$
$503$	−18.4027	−0.820537	−0.410269	+	0.911965i	$0.634565\pi$
$504$	0	0
$505$	8.80349	0.391750
$506$	0	0
$507$	−12.0436	−0.534876
$508$	0	0
$509$	−10.6366	−0.471461	−0.235730	−	0.971818i	$-0.575748\pi$
$509$	−10.6366	−0.471461	−0.235730	+	0.971818i	$0.575748\pi$
$510$	0	0
$511$	−16.0081	−0.708154
$512$	0	0
$513$	24.1235	1.06508
$514$	0	0
$515$	−12.2902	−0.541571
$516$	0	0
$517$	−4.35299	−0.191444
$518$	0	0
$519$	−31.0382	−1.36242
$520$	0	0
$521$	−37.8556	−1.65848	−0.829242	−	0.558890i	$-0.811227\pi$
$521$	−37.8556	−1.65848	−0.829242	+	0.558890i	$0.811227\pi$
$522$	0	0
$523$	7.33615	0.320787	0.160394	−	0.987053i	$-0.448724\pi$
$523$	7.33615	0.320787	0.160394	+	0.987053i	$0.448724\pi$
$524$	0	0
$525$	−5.09312	−0.222282
$526$	0	0
$527$	17.4913	0.761935
$528$	0	0
$529$	25.3225	1.10098
$530$	0	0
$531$	−7.37366	−0.319990
$532$	0	0
$533$	−22.5413	−0.976372
$534$	0	0
$535$	15.3777	0.664837
$536$	0	0
$537$	18.8743	0.814485
$538$	0	0
$539$	14.1971	0.611513
$540$	0	0
$541$	−7.27432	−0.312748	−0.156374	−	0.987698i	$-0.549980\pi$
$541$	−7.27432	−0.312748	−0.156374	+	0.987698i	$0.549980\pi$
$542$	0	0
$543$	−10.2202	−0.438590
$544$	0	0
$545$	1.00714	0.0431409
$546$	0	0
$547$	30.4680	1.30272	0.651358	−	0.758771i	$-0.274200\pi$
$547$	30.4680	1.30272	0.651358	+	0.758771i	$0.274200\pi$
$548$	0	0
$549$	6.23466	0.266089
$550$	0	0
$551$	38.5536	1.64244
$552$	0	0
$553$	−25.0123	−1.06363
$554$	0	0
$555$	−14.8250	−0.629287
$556$	0	0
$557$	15.4428	0.654333	0.327166	−	0.944967i	$-0.393906\pi$
$557$	15.4428	0.654333	0.327166	+	0.944967i	$0.393906\pi$
$558$	0	0
$559$	3.16782	0.133985
$560$	0	0
$561$	−22.4937	−0.949684
$562$	0	0
$563$	20.9805	0.884221	0.442111	−	0.896961i	$-0.354230\pi$
$563$	20.9805	0.884221	0.442111	+	0.896961i	$0.354230\pi$
$564$	0	0
$565$	3.94815	0.166100
$566$	0	0
$567$	−22.7214	−0.954209
$568$	0	0
$569$	7.49042	0.314015	0.157007	−	0.987597i	$-0.449815\pi$
$569$	7.49042	0.314015	0.157007	+	0.987597i	$0.449815\pi$
$570$	0	0
$571$	−20.4171	−0.854428	−0.427214	−	0.904150i	$-0.640505\pi$
$571$	−20.4171	−0.854428	−0.427214	+	0.904150i	$0.640505\pi$
$572$	0	0
$573$	17.3396	0.724371
$574$	0	0
$575$	−6.95144	−0.289895
$576$	0	0
$577$	4.56445	0.190021	0.0950103	−	0.995476i	$-0.469712\pi$
$577$	4.56445	0.190021	0.0950103	+	0.995476i	$0.469712\pi$
$578$	0	0
$579$	0.349661	0.0145314
$580$	0	0
$581$	−21.9912	−0.912348
$582$	0	0
$583$	15.6127	0.646612
$584$	0	0
$585$	−2.61348	−0.108054
$586$	0	0
$587$	31.3083	1.29223	0.646116	−	0.763239i	$-0.276392\pi$
$587$	31.3083	1.29223	0.646116	+	0.763239i	$0.276392\pi$
$588$	0	0
$589$	20.1812	0.831551
$590$	0	0
$591$	22.1903	0.912785
$592$	0	0
$593$	−33.2958	−1.36729	−0.683647	−	0.729813i	$-0.739607\pi$
$593$	−33.2958	−1.36729	−0.683647	+	0.729813i	$0.739607\pi$
$594$	0	0
$595$	12.2816	0.503496
$596$	0	0
$597$	21.8493	0.894231
$598$	0	0
$599$	6.25374	0.255521	0.127761	−	0.991805i	$-0.459221\pi$
$599$	6.25374	0.255521	0.127761	+	0.991805i	$0.459221\pi$
$600$	0	0
$601$	−11.0868	−0.452241	−0.226121	−	0.974099i	$-0.572604\pi$
$601$	−11.0868	−0.452241	−0.226121	+	0.974099i	$0.572604\pi$
$602$	0	0
$603$	−6.91259	−0.281503
$604$	0	0
$605$	3.78398	0.153841
$606$	0	0
$607$	−7.01275	−0.284639	−0.142319	−	0.989821i	$-0.545456\pi$
$607$	−7.01275	−0.284639	−0.142319	+	0.989821i	$0.545456\pi$
$608$	0	0
$609$	−45.3115	−1.83612
$610$	0	0
$611$	−5.15485	−0.208543
$612$	0	0
$613$	3.29562	0.133109	0.0665544	−	0.997783i	$-0.478799\pi$
$613$	3.29562	0.133109	0.0665544	+	0.997783i	$0.478799\pi$
$614$	0	0
$615$	7.71085	0.310931
$616$	0	0
$617$	10.9327	0.440132	0.220066	−	0.975485i	$-0.429373\pi$
$617$	10.9327	0.440132	0.220066	+	0.975485i	$0.429373\pi$
$618$	0	0
$619$	−23.4399	−0.942131	−0.471065	−	0.882098i	$-0.656130\pi$
$619$	−23.4399	−0.942131	−0.471065	+	0.882098i	$0.656130\pi$
$620$	0	0
$621$	−38.6967	−1.55285
$622$	0	0
$623$	13.6292	0.546041
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−25.9528	−1.03645
$628$	0	0
$629$	35.7491	1.42541
$630$	0	0
$631$	−12.0109	−0.478144	−0.239072	−	0.971002i	$-0.576843\pi$
$631$	−12.0109	−0.478144	−0.239072	+	0.971002i	$0.576843\pi$
$632$	0	0
$633$	34.1695	1.35812
$634$	0	0
$635$	−15.3379	−0.608665
$636$	0	0
$637$	16.8124	0.666130
$638$	0	0
$639$	2.38196	0.0942289
$640$	0	0
$641$	21.1559	0.835606	0.417803	−	0.908538i	$-0.362800\pi$
$641$	21.1559	0.835606	0.417803	+	0.908538i	$0.362800\pi$
$642$	0	0
$643$	33.9969	1.34071	0.670354	−	0.742041i	$-0.266142\pi$
$643$	33.9969	1.34071	0.670354	+	0.742041i	$0.266142\pi$
$644$	0	0
$645$	−1.08364	−0.0426682
$646$	0	0
$647$	24.8546	0.977135	0.488567	−	0.872526i	$-0.337520\pi$
$647$	24.8546	0.977135	0.488567	+	0.872526i	$0.337520\pi$
$648$	0	0
$649$	49.3950	1.93892
$650$	0	0
$651$	−23.7186	−0.929607
$652$	0	0
$653$	18.5584	0.726245	0.363123	−	0.931741i	$-0.381711\pi$
$653$	18.5584	0.726245	0.363123	+	0.931741i	$0.381711\pi$
$654$	0	0
$655$	−1.59053	−0.0621471
$656$	0	0
$657$	2.80994	0.109626
$658$	0	0
$659$	−38.7550	−1.50968	−0.754841	−	0.655908i	$-0.772286\pi$
$659$	−38.7550	−1.50968	−0.754841	+	0.655908i	$0.772286\pi$
$660$	0	0
$661$	20.0010	0.777949	0.388974	−	0.921249i	$-0.372829\pi$
$661$	20.0010	0.777949	0.388974	+	0.921249i	$0.372829\pi$
$662$	0	0
$663$	−26.6372	−1.03450
$664$	0	0
$665$	14.1702	0.549499
$666$	0	0
$667$	−61.8443	−2.39462
$668$	0	0
$669$	31.3878	1.21352
$670$	0	0
$671$	−41.7650	−1.61232
$672$	0	0
$673$	6.30936	0.243208	0.121604	−	0.992579i	$-0.461196\pi$
$673$	6.30936	0.243208	0.121604	+	0.992579i	$0.461196\pi$
$674$	0	0
$675$	5.56672	0.214263
$676$	0	0
$677$	25.7904	0.991205	0.495603	−	0.868549i	$-0.334947\pi$
$677$	25.7904	0.991205	0.495603	+	0.868549i	$0.334947\pi$
$678$	0	0
$679$	−34.3286	−1.31741
$680$	0	0
$681$	33.5733	1.28653
$682$	0	0
$683$	43.5232	1.66537	0.832685	−	0.553747i	$-0.186803\pi$
$683$	43.5232	1.66537	0.832685	+	0.553747i	$0.186803\pi$
$684$	0	0
$685$	−21.5615	−0.823822
$686$	0	0
$687$	18.3821	0.701321
$688$	0	0
$689$	18.4887	0.704364
$690$	0	0
$691$	−43.0621	−1.63816	−0.819081	−	0.573678i	$-0.805516\pi$
$691$	−43.0621	−1.63816	−0.819081	+	0.573678i	$0.805516\pi$
$692$	0	0
$693$	−7.21653	−0.274133
$694$	0	0
$695$	−20.8402	−0.790512
$696$	0	0
$697$	−18.5940	−0.704297
$698$	0	0
$699$	5.19996	0.196681
$700$	0	0
$701$	−15.2553	−0.576184	−0.288092	−	0.957603i	$-0.593021\pi$
$701$	−15.2553	−0.576184	−0.288092	+	0.957603i	$0.593021\pi$
$702$	0	0
$703$	41.2466	1.55565
$704$	0	0
$705$	1.76335	0.0664118
$706$	0	0
$707$	28.7867	1.08264
$708$	0	0
$709$	8.97006	0.336878	0.168439	−	0.985712i	$-0.446127\pi$
$709$	8.97006	0.336878	0.168439	+	0.985712i	$0.446127\pi$
$710$	0	0
$711$	4.39049	0.164656
$712$	0	0
$713$	−32.3728	−1.21237
$714$	0	0
$715$	17.5073	0.654737
$716$	0	0
$717$	21.7945	0.813929
$718$	0	0
$719$	16.7855	0.625995	0.312997	−	0.949754i	$-0.398667\pi$
$719$	16.7855	0.625995	0.312997	+	0.949754i	$0.398667\pi$
$720$	0	0
$721$	−40.1880	−1.49668
$722$	0	0
$723$	−14.1552	−0.526436
$724$	0	0
$725$	8.89661	0.330412
$726$	0	0
$727$	−5.85260	−0.217061	−0.108530	−	0.994093i	$-0.534615\pi$
$727$	−5.85260	−0.217061	−0.108530	+	0.994093i	$0.534615\pi$
$728$	0	0
$729$	30.0000	1.11111
$730$	0	0
$731$	2.61309	0.0966487
$732$	0	0
$733$	−1.64199	−0.0606484	−0.0303242	−	0.999540i	$-0.509654\pi$
$733$	−1.64199	−0.0606484	−0.0303242	+	0.999540i	$0.509654\pi$
$734$	0	0
$735$	−5.75111	−0.212133
$736$	0	0
$737$	46.3064	1.70572
$738$	0	0
$739$	50.5500	1.85951	0.929756	−	0.368176i	$-0.120018\pi$
$739$	50.5500	1.85951	0.929756	+	0.368176i	$0.120018\pi$
$740$	0	0
$741$	−30.7335	−1.12902
$742$	0	0
$743$	−28.2484	−1.03633	−0.518166	−	0.855280i	$-0.673385\pi$
$743$	−28.2484	−1.03633	−0.518166	+	0.855280i	$0.673385\pi$
$744$	0	0
$745$	5.46074	0.200066
$746$	0	0
$747$	3.86018	0.141237
$748$	0	0
$749$	50.2839	1.83733
$750$	0	0
$751$	−28.4136	−1.03683	−0.518413	−	0.855131i	$-0.673477\pi$
$751$	−28.4136	−1.03683	−0.518413	+	0.855131i	$0.673477\pi$
$752$	0	0
$753$	18.2169	0.663863
$754$	0	0
$755$	7.27391	0.264725
$756$	0	0
$757$	1.67671	0.0609410	0.0304705	−	0.999536i	$-0.490299\pi$
$757$	1.67671	0.0609410	0.0304705	+	0.999536i	$0.490299\pi$
$758$	0	0
$759$	41.6311	1.51111
$760$	0	0
$761$	−9.75471	−0.353608	−0.176804	−	0.984246i	$-0.556576\pi$
$761$	−9.75471	−0.353608	−0.176804	+	0.984246i	$0.556576\pi$
$762$	0	0
$763$	3.29325	0.119224
$764$	0	0
$765$	−2.15582	−0.0779440
$766$	0	0
$767$	58.4940	2.11210
$768$	0	0
$769$	−14.9137	−0.537803	−0.268902	−	0.963168i	$-0.586661\pi$
$769$	−14.9137	−0.537803	−0.268902	+	0.963168i	$0.586661\pi$
$770$	0	0
$771$	−0.104982	−0.00378082
$772$	0	0
$773$	27.0502	0.972928	0.486464	−	0.873701i	$-0.338287\pi$
$773$	27.0502	0.972928	0.486464	+	0.873701i	$0.338287\pi$
$774$	0	0
$775$	4.65700	0.167284
$776$	0	0
$777$	−48.4766	−1.73909
$778$	0	0
$779$	−21.4534	−0.768647
$780$	0	0
$781$	−15.9564	−0.570964
$782$	0	0
$783$	49.5249	1.76988
$784$	0	0
$785$	−9.68925	−0.345824
$786$	0	0
$787$	27.0493	0.964203	0.482101	−	0.876115i	$-0.339874\pi$
$787$	27.0493	0.964203	0.482101	+	0.876115i	$0.339874\pi$
$788$	0	0
$789$	−24.1560	−0.859976
$790$	0	0
$791$	12.9101	0.459031
$792$	0	0
$793$	−49.4585	−1.75632
$794$	0	0
$795$	−6.32455	−0.224309
$796$	0	0
$797$	46.3701	1.64251	0.821257	−	0.570559i	$-0.193273\pi$
$797$	46.3701	1.64251	0.821257	+	0.570559i	$0.193273\pi$
$798$	0	0
$799$	−4.25216	−0.150431
$800$	0	0
$801$	−2.39237	−0.0845303
$802$	0	0
$803$	−18.8234	−0.664262
$804$	0	0
$805$	−22.7306	−0.801150
$806$	0	0
$807$	10.2538	0.360951
$808$	0	0
$809$	15.0837	0.530316	0.265158	−	0.964205i	$-0.414576\pi$
$809$	15.0837	0.530316	0.265158	+	0.964205i	$0.414576\pi$
$810$	0	0
$811$	−22.1040	−0.776178	−0.388089	−	0.921622i	$-0.626865\pi$
$811$	−22.1040	−0.776178	−0.388089	+	0.921622i	$0.626865\pi$
$812$	0	0
$813$	22.4458	0.787208
$814$	0	0
$815$	−18.0106	−0.630885
$816$	0	0
$817$	3.01493	0.105479
$818$	0	0
$819$	−8.54588	−0.298617
$820$	0	0
$821$	14.2049	0.495754	0.247877	−	0.968791i	$-0.420267\pi$
$821$	14.2049	0.495754	0.247877	+	0.968791i	$0.420267\pi$
$822$	0	0
$823$	21.3652	0.744745	0.372372	−	0.928083i	$-0.378544\pi$
$823$	21.3652	0.744745	0.372372	+	0.928083i	$0.378544\pi$
$824$	0	0
$825$	−5.98884	−0.208505
$826$	0	0
$827$	−21.6614	−0.753241	−0.376621	−	0.926368i	$-0.622914\pi$
$827$	−21.6614	−0.753241	−0.376621	+	0.926368i	$0.622914\pi$
$828$	0	0
$829$	41.6118	1.44524	0.722618	−	0.691247i	$-0.242939\pi$
$829$	41.6118	1.44524	0.722618	+	0.691247i	$0.242939\pi$
$830$	0	0
$831$	23.5549	0.817112
$832$	0	0
$833$	13.8683	0.480507
$834$	0	0
$835$	18.9322	0.655177
$836$	0	0
$837$	25.9242	0.896071
$838$	0	0
$839$	−48.5228	−1.67519	−0.837596	−	0.546290i	$-0.816040\pi$
$839$	−48.5228	−1.67519	−0.837596	+	0.546290i	$0.816040\pi$
$840$	0	0
$841$	50.1497	1.72930
$842$	0	0
$843$	−10.7408	−0.369932
$844$	0	0
$845$	7.73233	0.266000
$846$	0	0
$847$	12.3733	0.425152
$848$	0	0
$849$	−32.6789	−1.12154
$850$	0	0
$851$	−66.1642	−2.26808
$852$	0	0
$853$	6.44837	0.220788	0.110394	−	0.993888i	$-0.464789\pi$
$853$	6.44837	0.220788	0.110394	+	0.993888i	$0.464789\pi$
$854$	0	0
$855$	−2.48735	−0.0850655
$856$	0	0
$857$	40.1346	1.37097	0.685485	−	0.728086i	$-0.259590\pi$
$857$	40.1346	1.37097	0.685485	+	0.728086i	$0.259590\pi$
$858$	0	0
$859$	12.8608	0.438805	0.219402	−	0.975634i	$-0.429589\pi$
$859$	12.8608	0.438805	0.219402	+	0.975634i	$0.429589\pi$
$860$	0	0
$861$	25.2138	0.859285
$862$	0	0
$863$	47.6821	1.62312	0.811558	−	0.584271i	$-0.198620\pi$
$863$	47.6821	1.62312	0.811558	+	0.584271i	$0.198620\pi$
$864$	0	0
$865$	19.9273	0.677549
$866$	0	0
$867$	4.50605	0.153033
$868$	0	0
$869$	−29.4112	−0.997707
$870$	0	0
$871$	54.8364	1.85806
$872$	0	0
$873$	6.02580	0.203942
$874$	0	0
$875$	3.26992	0.110543
$876$	0	0
$877$	28.8836	0.975331	0.487665	−	0.873031i	$-0.337849\pi$
$877$	28.8836	0.975331	0.487665	+	0.873031i	$0.337849\pi$
$878$	0	0
$879$	−10.7836	−0.363721
$880$	0	0
$881$	21.1602	0.712904	0.356452	−	0.934314i	$-0.383986\pi$
$881$	21.1602	0.712904	0.356452	+	0.934314i	$0.383986\pi$
$882$	0	0
$883$	−35.6320	−1.19911	−0.599556	−	0.800333i	$-0.704656\pi$
$883$	−35.6320	−1.19911	−0.599556	+	0.800333i	$0.704656\pi$
$884$	0	0
$885$	−20.0094	−0.672610
$886$	0	0
$887$	−12.2836	−0.412443	−0.206222	−	0.978505i	$-0.566117\pi$
$887$	−12.2836	−0.412443	−0.206222	+	0.978505i	$0.566117\pi$
$888$	0	0
$889$	−50.1536	−1.68210
$890$	0	0
$891$	−26.7174	−0.895066
$892$	0	0
$893$	−4.90606	−0.164175
$894$	0	0
$895$	−12.1178	−0.405053
$896$	0	0
$897$	49.2999	1.64608
$898$	0	0
$899$	41.4315	1.38182
$900$	0	0
$901$	15.2511	0.508086
$902$	0	0
$903$	−3.54341	−0.117917
$904$	0	0
$905$	6.56162	0.218116
$906$	0	0
$907$	−0.500076	−0.0166048	−0.00830238	−	0.999966i	$-0.502643\pi$
$907$	−0.500076	−0.0166048	−0.00830238	+	0.999966i	$0.502643\pi$
$908$	0	0
$909$	−5.05302	−0.167598
$910$	0	0
$911$	−28.9438	−0.958952	−0.479476	−	0.877555i	$-0.659173\pi$
$911$	−28.9438	−0.958952	−0.479476	+	0.877555i	$0.659173\pi$
$912$	0	0
$913$	−25.8587	−0.855800
$914$	0	0
$915$	16.9186	0.559311
$916$	0	0
$917$	−5.20090	−0.171749
$918$	0	0
$919$	−21.8562	−0.720969	−0.360484	−	0.932765i	$-0.617389\pi$
$919$	−21.8562	−0.720969	−0.360484	+	0.932765i	$0.617389\pi$
$920$	0	0
$921$	−43.4974	−1.43329
$922$	0	0
$923$	−18.8957	−0.621959
$924$	0	0
$925$	9.51805	0.312952
$926$	0	0
$927$	7.05432	0.231694
$928$	0	0
$929$	−0.912054	−0.0299235	−0.0149618	−	0.999888i	$-0.504763\pi$
$929$	−0.912054	−0.0299235	−0.0149618	+	0.999888i	$0.504763\pi$
$930$	0	0
$931$	16.0009	0.524409
$932$	0	0
$933$	−47.1001	−1.54199
$934$	0	0
$935$	14.4415	0.472288
$936$	0	0
$937$	4.84133	0.158159	0.0790796	−	0.996868i	$-0.474802\pi$
$937$	4.84133	0.158159	0.0790796	+	0.996868i	$0.474802\pi$
$938$	0	0
$939$	−33.8653	−1.10515
$940$	0	0
$941$	−1.20097	−0.0391505	−0.0195753	−	0.999808i	$-0.506231\pi$
$941$	−1.20097	−0.0391505	−0.0195753	+	0.999808i	$0.506231\pi$
$942$	0	0
$943$	34.4136	1.12066
$944$	0	0
$945$	18.2027	0.592134
$946$	0	0
$947$	−15.3907	−0.500131	−0.250066	−	0.968229i	$-0.580452\pi$
$947$	−15.3907	−0.500131	−0.250066	+	0.968229i	$0.580452\pi$
$948$	0	0
$949$	−22.2908	−0.723590
$950$	0	0
$951$	15.0830	0.489099
$952$	0	0
$953$	23.7839	0.770436	0.385218	−	0.922826i	$-0.374126\pi$
$953$	23.7839	0.770436	0.385218	+	0.922826i	$0.374126\pi$
$954$	0	0
$955$	−11.1325	−0.360238
$956$	0	0
$957$	−53.2804	−1.72231
$958$	0	0
$959$	−70.5043	−2.27670
$960$	0	0
$961$	−9.31239	−0.300400
$962$	0	0
$963$	−8.82649	−0.284430
$964$	0	0
$965$	−0.224492	−0.00722664
$966$	0	0
$967$	4.67405	0.150307	0.0751536	−	0.997172i	$-0.476055\pi$
$967$	4.67405	0.150307	0.0751536	+	0.997172i	$0.476055\pi$
$968$	0	0
$969$	−25.3516	−0.814410
$970$	0	0
$971$	5.24722	0.168391	0.0841956	−	0.996449i	$-0.473168\pi$
$971$	5.24722	0.168391	0.0841956	+	0.996449i	$0.473168\pi$
$972$	0	0
$973$	−68.1456	−2.18465
$974$	0	0
$975$	−7.09204	−0.227127
$976$	0	0
$977$	15.9161	0.509200	0.254600	−	0.967046i	$-0.418056\pi$
$977$	15.9161	0.509200	0.254600	+	0.967046i	$0.418056\pi$
$978$	0	0
$979$	16.0261	0.512197
$980$	0	0
$981$	−0.578074	−0.0184565
$982$	0	0
$983$	−56.8967	−1.81472	−0.907362	−	0.420351i	$-0.861907\pi$
$983$	−56.8967	−1.81472	−0.907362	+	0.420351i	$0.861907\pi$
$984$	0	0
$985$	−14.2467	−0.453939
$986$	0	0
$987$	5.76602	0.183535
$988$	0	0
$989$	−4.83629	−0.153785
$990$	0	0
$991$	−15.2355	−0.483971	−0.241985	−	0.970280i	$-0.577799\pi$
$991$	−15.2355	−0.483971	−0.241985	+	0.970280i	$0.577799\pi$
$992$	0	0
$993$	0.0672199	0.00213316
$994$	0	0
$995$	−14.0278	−0.444711
$996$	0	0
$997$	60.8550	1.92730	0.963649	−	0.267173i	$-0.0860895\pi$
$997$	60.8550	1.92730	0.963649	+	0.267173i	$0.0860895\pi$
$998$	0	0
$999$	52.9843	1.67635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.f.1.12	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.f.1.12	✓	37	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)

\(f(q)\)	\(=\)	\(q-1.55757 q^{3} +1.00000 q^{5} +3.26992 q^{7} -0.573979 q^{9} +O(q^{10})\)	\(q-1.55757 q^{3} +1.00000 q^{5} +3.26992 q^{7} -0.573979 q^{9} +3.84499 q^{11} +4.55328 q^{13} -1.55757 q^{15} +3.75593 q^{17} +4.33352 q^{19} -5.09312 q^{21} -6.95144 q^{23} +1.00000 q^{25} +5.56672 q^{27} +8.89661 q^{29} +4.65700 q^{31} -5.98884 q^{33} +3.26992 q^{35} +9.51805 q^{37} -7.09204 q^{39} -4.95057 q^{41} +0.695724 q^{43} -0.573979 q^{45} -1.13212 q^{47} +3.69236 q^{49} -5.85012 q^{51} +4.06053 q^{53} +3.84499 q^{55} -6.74975 q^{57} +12.8466 q^{59} -10.8622 q^{61} -1.87686 q^{63} +4.55328 q^{65} +12.0433 q^{67} +10.8274 q^{69} -4.14991 q^{71} -4.89555 q^{73} -1.55757 q^{75} +12.5728 q^{77} -7.64922 q^{79} -6.94861 q^{81} -6.72530 q^{83} +3.75593 q^{85} -13.8571 q^{87} +4.16805 q^{89} +14.8888 q^{91} -7.25359 q^{93} +4.33352 q^{95} -10.4983 q^{97} -2.20695 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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