LMFDB - Embedded newform 8020.2.a.c.1.22

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:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
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)

$f(q)$	$=$	$q+1.95384 q^{3} -1.00000 q^{5} +2.17117 q^{7} +0.817489 q^{9} +O(q^{10})$	$q+1.95384 q^{3} -1.00000 q^{5} +2.17117 q^{7} +0.817489 q^{9} +1.00263 q^{11} -0.272452 q^{13} -1.95384 q^{15} -5.86725 q^{17} -8.21631 q^{19} +4.24213 q^{21} +4.96981 q^{23} +1.00000 q^{25} -4.26428 q^{27} -6.90963 q^{29} +8.98865 q^{31} +1.95898 q^{33} -2.17117 q^{35} -5.23167 q^{37} -0.532328 q^{39} +1.37318 q^{41} +2.99770 q^{43} -0.817489 q^{45} -2.43702 q^{47} -2.28600 q^{49} -11.4637 q^{51} +6.76735 q^{53} -1.00263 q^{55} -16.0534 q^{57} +2.44495 q^{59} +4.43871 q^{61} +1.77491 q^{63} +0.272452 q^{65} -12.9828 q^{67} +9.71022 q^{69} -6.91721 q^{71} +3.98738 q^{73} +1.95384 q^{75} +2.17689 q^{77} -9.61707 q^{79} -10.7842 q^{81} -11.8176 q^{83} +5.86725 q^{85} -13.5003 q^{87} -11.9940 q^{89} -0.591541 q^{91} +17.5624 q^{93} +8.21631 q^{95} +10.3855 q^{97} +0.819641 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * 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
$3$	1.95384	1.12805	0.564025	−	0.825758i	$-0.309252\pi$
$3$	1.95384	1.12805	0.564025	+	0.825758i	$0.309252\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.17117	0.820627	0.410313	−	0.911945i	$-0.365419\pi$
$7$	2.17117	0.820627	0.410313	+	0.911945i	$0.365419\pi$
$8$	0	0
$9$	0.817489	0.272496
$10$	0	0
$11$	1.00263	0.302305	0.151153	−	0.988510i	$-0.451702\pi$
$11$	1.00263	0.302305	0.151153	+	0.988510i	$0.451702\pi$
$12$	0	0
$13$	−0.272452	−0.0755646	−0.0377823	−	0.999286i	$-0.512029\pi$
$13$	−0.272452	−0.0755646	−0.0377823	+	0.999286i	$0.512029\pi$
$14$	0	0
$15$	−1.95384	−0.504479
$16$	0	0
$17$	−5.86725	−1.42302	−0.711509	−	0.702677i	$-0.751988\pi$
$17$	−5.86725	−1.42302	−0.711509	+	0.702677i	$0.751988\pi$
$18$	0	0
$19$	−8.21631	−1.88495	−0.942476	−	0.334275i	$-0.891509\pi$
$19$	−8.21631	−1.88495	−0.942476	+	0.334275i	$0.891509\pi$
$20$	0	0
$21$	4.24213	0.925708
$22$	0	0
$23$	4.96981	1.03628	0.518139	−	0.855297i	$-0.326625\pi$
$23$	4.96981	1.03628	0.518139	+	0.855297i	$0.326625\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.26428	−0.820660
$28$	0	0
$29$	−6.90963	−1.28309	−0.641543	−	0.767087i	$-0.721705\pi$
$29$	−6.90963	−1.28309	−0.641543	+	0.767087i	$0.721705\pi$
$30$	0	0
$31$	8.98865	1.61441	0.807204	−	0.590272i	$-0.200980\pi$
$31$	8.98865	1.61441	0.807204	+	0.590272i	$0.200980\pi$
$32$	0	0
$33$	1.95898	0.341015
$34$	0	0
$35$	−2.17117	−0.366995
$36$	0	0
$37$	−5.23167	−0.860080	−0.430040	−	0.902810i	$-0.641501\pi$
$37$	−5.23167	−0.860080	−0.430040	+	0.902810i	$0.641501\pi$
$38$	0	0
$39$	−0.532328	−0.0852406
$40$	0	0
$41$	1.37318	0.214455	0.107227	−	0.994235i	$-0.465803\pi$
$41$	1.37318	0.214455	0.107227	+	0.994235i	$0.465803\pi$
$42$	0	0
$43$	2.99770	0.457146	0.228573	−	0.973527i	$-0.426594\pi$
$43$	2.99770	0.457146	0.228573	+	0.973527i	$0.426594\pi$
$44$	0	0
$45$	−0.817489	−0.121864
$46$	0	0
$47$	−2.43702	−0.355475	−0.177738	−	0.984078i	$-0.556878\pi$
$47$	−2.43702	−0.355475	−0.177738	+	0.984078i	$0.556878\pi$
$48$	0	0
$49$	−2.28600	−0.326572
$50$	0	0
$51$	−11.4637	−1.60523
$52$	0	0
$53$	6.76735	0.929567	0.464784	−	0.885424i	$-0.346132\pi$
$53$	6.76735	0.929567	0.464784	+	0.885424i	$0.346132\pi$
$54$	0	0
$55$	−1.00263	−0.135195
$56$	0	0
$57$	−16.0534	−2.12632
$58$	0	0
$59$	2.44495	0.318306	0.159153	−	0.987254i	$-0.449124\pi$
$59$	2.44495	0.318306	0.159153	+	0.987254i	$0.449124\pi$
$60$	0	0
$61$	4.43871	0.568318	0.284159	−	0.958777i	$-0.408286\pi$
$61$	4.43871	0.568318	0.284159	+	0.958777i	$0.408286\pi$
$62$	0	0
$63$	1.77491	0.223618
$64$	0	0
$65$	0.272452	0.0337935
$66$	0	0
$67$	−12.9828	−1.58610	−0.793051	−	0.609155i	$-0.791509\pi$
$67$	−12.9828	−1.58610	−0.793051	+	0.609155i	$0.791509\pi$
$68$	0	0
$69$	9.71022	1.16897
$70$	0	0
$71$	−6.91721	−0.820922	−0.410461	−	0.911878i	$-0.634632\pi$
$71$	−6.91721	−0.820922	−0.410461	+	0.911878i	$0.634632\pi$
$72$	0	0
$73$	3.98738	0.466688	0.233344	−	0.972394i	$-0.425033\pi$
$73$	3.98738	0.466688	0.233344	+	0.972394i	$0.425033\pi$
$74$	0	0
$75$	1.95384	0.225610
$76$	0	0
$77$	2.17689	0.248080
$78$	0	0
$79$	−9.61707	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$79$	−9.61707	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$80$	0	0
$81$	−10.7842	−1.19824
$82$	0	0
$83$	−11.8176	−1.29715	−0.648573	−	0.761152i	$-0.724634\pi$
$83$	−11.8176	−1.29715	−0.648573	+	0.761152i	$0.724634\pi$
$84$	0	0
$85$	5.86725	0.636393
$86$	0	0
$87$	−13.5003	−1.44739
$88$	0	0
$89$	−11.9940	−1.27137	−0.635683	−	0.771950i	$-0.719281\pi$
$89$	−11.9940	−1.27137	−0.635683	+	0.771950i	$0.719281\pi$
$90$	0	0
$91$	−0.591541	−0.0620103
$92$	0	0
$93$	17.5624	1.82113
$94$	0	0
$95$	8.21631	0.842976
$96$	0	0
$97$	10.3855	1.05449	0.527243	−	0.849715i	$-0.323226\pi$
$97$	10.3855	1.05449	0.527243	+	0.849715i	$0.323226\pi$
$98$	0	0
$99$	0.819641	0.0823770
$100$	0	0
$101$	−4.61598	−0.459307	−0.229653	−	0.973272i	$-0.573759\pi$
$101$	−4.61598	−0.459307	−0.229653	+	0.973272i	$0.573759\pi$
$102$	0	0
$103$	−0.873853	−0.0861033	−0.0430516	−	0.999073i	$-0.513708\pi$
$103$	−0.873853	−0.0861033	−0.0430516	+	0.999073i	$0.513708\pi$
$104$	0	0
$105$	−4.24213	−0.413989
$106$	0	0
$107$	−9.17487	−0.886968	−0.443484	−	0.896282i	$-0.646258\pi$
$107$	−9.17487	−0.886968	−0.443484	+	0.896282i	$0.646258\pi$
$108$	0	0
$109$	−2.64702	−0.253538	−0.126769	−	0.991932i	$-0.540461\pi$
$109$	−2.64702	−0.253538	−0.126769	+	0.991932i	$0.540461\pi$
$110$	0	0
$111$	−10.2218	−0.970214
$112$	0	0
$113$	−6.83601	−0.643078	−0.321539	−	0.946896i	$-0.604200\pi$
$113$	−6.83601	−0.643078	−0.321539	+	0.946896i	$0.604200\pi$
$114$	0	0
$115$	−4.96981	−0.463437
$116$	0	0
$117$	−0.222727	−0.0205911
$118$	0	0
$119$	−12.7388	−1.16777
$120$	0	0
$121$	−9.99473	−0.908612
$122$	0	0
$123$	2.68297	0.241916
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0.696354	0.0617914	0.0308957	−	0.999523i	$-0.490164\pi$
$127$	0.696354	0.0617914	0.0308957	+	0.999523i	$0.490164\pi$
$128$	0	0
$129$	5.85703	0.515683
$130$	0	0
$131$	0.720727	0.0629702	0.0314851	−	0.999504i	$-0.489976\pi$
$131$	0.720727	0.0629702	0.0314851	+	0.999504i	$0.489976\pi$
$132$	0	0
$133$	−17.8390	−1.54684
$134$	0	0
$135$	4.26428	0.367010
$136$	0	0
$137$	−11.8434	−1.01185	−0.505924	−	0.862578i	$-0.668848\pi$
$137$	−11.8434	−1.01185	−0.505924	+	0.862578i	$0.668848\pi$
$138$	0	0
$139$	5.31971	0.451212	0.225606	−	0.974219i	$-0.427564\pi$
$139$	5.31971	0.451212	0.225606	+	0.974219i	$0.427564\pi$
$140$	0	0
$141$	−4.76154	−0.400994
$142$	0	0
$143$	−0.273169	−0.0228436
$144$	0	0
$145$	6.90963	0.573814
$146$	0	0
$147$	−4.46649	−0.368390
$148$	0	0
$149$	9.30295	0.762128	0.381064	−	0.924549i	$-0.375558\pi$
$149$	9.30295	0.762128	0.381064	+	0.924549i	$0.375558\pi$
$150$	0	0
$151$	2.32280	0.189027	0.0945135	−	0.995524i	$-0.469870\pi$
$151$	2.32280	0.189027	0.0945135	+	0.995524i	$0.469870\pi$
$152$	0	0
$153$	−4.79641	−0.387767
$154$	0	0
$155$	−8.98865	−0.721985
$156$	0	0
$157$	−14.5276	−1.15943	−0.579715	−	0.814819i	$-0.696836\pi$
$157$	−14.5276	−1.15943	−0.579715	+	0.814819i	$0.696836\pi$
$158$	0	0
$159$	13.2223	1.04860
$160$	0	0
$161$	10.7903	0.850397
$162$	0	0
$163$	−0.478693	−0.0374941	−0.0187470	−	0.999824i	$-0.505968\pi$
$163$	−0.478693	−0.0374941	−0.0187470	+	0.999824i	$0.505968\pi$
$164$	0	0
$165$	−1.95898	−0.152507
$166$	0	0
$167$	−20.9805	−1.62352	−0.811759	−	0.583993i	$-0.801489\pi$
$167$	−20.9805	−1.62352	−0.811759	+	0.583993i	$0.801489\pi$
$168$	0	0
$169$	−12.9258	−0.994290
$170$	0	0
$171$	−6.71674	−0.513642
$172$	0	0
$173$	10.0599	0.764838	0.382419	−	0.923989i	$-0.375091\pi$
$173$	10.0599	0.764838	0.382419	+	0.923989i	$0.375091\pi$
$174$	0	0
$175$	2.17117	0.164125
$176$	0	0
$177$	4.77704	0.359065
$178$	0	0
$179$	12.0961	0.904103	0.452051	−	0.891992i	$-0.350692\pi$
$179$	12.0961	0.904103	0.452051	+	0.891992i	$0.350692\pi$
$180$	0	0
$181$	−3.77666	−0.280717	−0.140358	−	0.990101i	$-0.544825\pi$
$181$	−3.77666	−0.280717	−0.140358	+	0.990101i	$0.544825\pi$
$182$	0	0
$183$	8.67252	0.641091
$184$	0	0
$185$	5.23167	0.384640
$186$	0	0
$187$	−5.88269	−0.430185
$188$	0	0
$189$	−9.25849	−0.673456
$190$	0	0
$191$	0.0106466	0.000770359 0	0.000385179	−	1.00000i	$-0.499877\pi$
$191$	0.0106466	0.000770359 0	0.000385179		1.00000i	$0.499877\pi$
$192$	0	0
$193$	7.75149	0.557964	0.278982	−	0.960296i	$-0.410003\pi$
$193$	7.75149	0.557964	0.278982	+	0.960296i	$0.410003\pi$
$194$	0	0
$195$	0.532328	0.0381208
$196$	0	0
$197$	26.0365	1.85502	0.927512	−	0.373794i	$-0.121943\pi$
$197$	26.0365	1.85502	0.927512	+	0.373794i	$0.121943\pi$
$198$	0	0
$199$	−0.0624479	−0.00442681	−0.00221341	−	0.999998i	$-0.500705\pi$
$199$	−0.0624479	−0.00442681	−0.00221341	+	0.999998i	$0.500705\pi$
$200$	0	0
$201$	−25.3663	−1.78920
$202$	0	0
$203$	−15.0020	−1.05293
$204$	0	0
$205$	−1.37318	−0.0959070
$206$	0	0
$207$	4.06277	0.282382
$208$	0	0
$209$	−8.23794	−0.569830
$210$	0	0
$211$	14.9859	1.03167	0.515834	−	0.856688i	$-0.327482\pi$
$211$	14.9859	1.03167	0.515834	+	0.856688i	$0.327482\pi$
$212$	0	0
$213$	−13.5151	−0.926041
$214$	0	0
$215$	−2.99770	−0.204442
$216$	0	0
$217$	19.5159	1.32483
$218$	0	0
$219$	7.79071	0.526447
$220$	0	0
$221$	1.59854	0.107530
$222$	0	0
$223$	−1.56518	−0.104813	−0.0524063	−	0.998626i	$-0.516689\pi$
$223$	−1.56518	−0.104813	−0.0524063	+	0.998626i	$0.516689\pi$
$224$	0	0
$225$	0.817489	0.0544993
$226$	0	0
$227$	13.7611	0.913357	0.456678	−	0.889632i	$-0.349039\pi$
$227$	13.7611	0.913357	0.456678	+	0.889632i	$0.349039\pi$
$228$	0	0
$229$	23.8675	1.57721	0.788604	−	0.614901i	$-0.210804\pi$
$229$	23.8675	1.57721	0.788604	+	0.614901i	$0.210804\pi$
$230$	0	0
$231$	4.25329	0.279846
$232$	0	0
$233$	−4.76607	−0.312235	−0.156118	−	0.987738i	$-0.549898\pi$
$233$	−4.76607	−0.312235	−0.156118	+	0.987738i	$0.549898\pi$
$234$	0	0
$235$	2.43702	0.158973
$236$	0	0
$237$	−18.7902	−1.22056
$238$	0	0
$239$	7.18157	0.464537	0.232269	−	0.972652i	$-0.425385\pi$
$239$	7.18157	0.464537	0.232269	+	0.972652i	$0.425385\pi$
$240$	0	0
$241$	−7.59767	−0.489409	−0.244705	−	0.969598i	$-0.578691\pi$
$241$	−7.59767	−0.489409	−0.244705	+	0.969598i	$0.578691\pi$
$242$	0	0
$243$	−8.27772	−0.531016
$244$	0	0
$245$	2.28600	0.146047
$246$	0	0
$247$	2.23855	0.142436
$248$	0	0
$249$	−23.0896	−1.46325
$250$	0	0
$251$	27.6754	1.74686	0.873429	−	0.486952i	$-0.161891\pi$
$251$	27.6754	1.74686	0.873429	+	0.486952i	$0.161891\pi$
$252$	0	0
$253$	4.98289	0.313272
$254$	0	0
$255$	11.4637	0.717883
$256$	0	0
$257$	−9.35864	−0.583776	−0.291888	−	0.956452i	$-0.594283\pi$
$257$	−9.35864	−0.583776	−0.291888	+	0.956452i	$0.594283\pi$
$258$	0	0
$259$	−11.3589	−0.705805
$260$	0	0
$261$	−5.64855	−0.349636
$262$	0	0
$263$	−28.4491	−1.75425	−0.877124	−	0.480264i	$-0.840541\pi$
$263$	−28.4491	−1.75425	−0.877124	+	0.480264i	$0.840541\pi$
$264$	0	0
$265$	−6.76735	−0.415715
$266$	0	0
$267$	−23.4344	−1.43416
$268$	0	0
$269$	11.5587	0.704746	0.352373	−	0.935860i	$-0.385375\pi$
$269$	11.5587	0.704746	0.352373	+	0.935860i	$0.385375\pi$
$270$	0	0
$271$	19.1377	1.16253	0.581266	−	0.813714i	$-0.302558\pi$
$271$	19.1377	1.16253	0.581266	+	0.813714i	$0.302558\pi$
$272$	0	0
$273$	−1.15578	−0.0699507
$274$	0	0
$275$	1.00263	0.0604610
$276$	0	0
$277$	31.0194	1.86377	0.931886	−	0.362751i	$-0.118162\pi$
$277$	31.0194	1.86377	0.931886	+	0.362751i	$0.118162\pi$
$278$	0	0
$279$	7.34812	0.439920
$280$	0	0
$281$	−21.1765	−1.26328	−0.631642	−	0.775261i	$-0.717619\pi$
$281$	−21.1765	−1.26328	−0.631642	+	0.775261i	$0.717619\pi$
$282$	0	0
$283$	−29.0778	−1.72850	−0.864249	−	0.503065i	$-0.832206\pi$
$283$	−29.0778	−1.72850	−0.864249	+	0.503065i	$0.832206\pi$
$284$	0	0
$285$	16.0534	0.950919
$286$	0	0
$287$	2.98141	0.175987
$288$	0	0
$289$	17.4246	1.02498
$290$	0	0
$291$	20.2916	1.18951
$292$	0	0
$293$	6.26365	0.365926	0.182963	−	0.983120i	$-0.441431\pi$
$293$	6.26365	0.365926	0.182963	+	0.983120i	$0.441431\pi$
$294$	0	0
$295$	−2.44495	−0.142351
$296$	0	0
$297$	−4.27550	−0.248090
$298$	0	0
$299$	−1.35404	−0.0783059
$300$	0	0
$301$	6.50854	0.375146
$302$	0	0
$303$	−9.01888	−0.518121
$304$	0	0
$305$	−4.43871	−0.254160
$306$	0	0
$307$	−0.321321	−0.0183388	−0.00916938	−	0.999958i	$-0.502919\pi$
$307$	−0.321321	−0.0183388	−0.00916938	+	0.999958i	$0.502919\pi$
$308$	0	0
$309$	−1.70737	−0.0971288
$310$	0	0
$311$	−31.0402	−1.76013	−0.880065	−	0.474853i	$-0.842501\pi$
$311$	−31.0402	−1.76013	−0.880065	+	0.474853i	$0.842501\pi$
$312$	0	0
$313$	9.56373	0.540574	0.270287	−	0.962780i	$-0.412881\pi$
$313$	9.56373	0.540574	0.270287	+	0.962780i	$0.412881\pi$
$314$	0	0
$315$	−1.77491	−0.100005
$316$	0	0
$317$	0.417696	0.0234601	0.0117301	−	0.999931i	$-0.496266\pi$
$317$	0.417696	0.0234601	0.0117301	+	0.999931i	$0.496266\pi$
$318$	0	0
$319$	−6.92782	−0.387883
$320$	0	0
$321$	−17.9262	−1.00054
$322$	0	0
$323$	48.2072	2.68232
$324$	0	0
$325$	−0.272452	−0.0151129
$326$	0	0
$327$	−5.17185	−0.286004
$328$	0	0
$329$	−5.29118	−0.291712
$330$	0	0
$331$	−4.17130	−0.229275	−0.114638	−	0.993407i	$-0.536571\pi$
$331$	−4.17130	−0.229275	−0.114638	+	0.993407i	$0.536571\pi$
$332$	0	0
$333$	−4.27683	−0.234369
$334$	0	0
$335$	12.9828	0.709326
$336$	0	0
$337$	17.0601	0.929325	0.464662	−	0.885488i	$-0.346176\pi$
$337$	17.0601	0.929325	0.464662	+	0.885488i	$0.346176\pi$
$338$	0	0
$339$	−13.3565	−0.725424
$340$	0	0
$341$	9.01231	0.488044
$342$	0	0
$343$	−20.1615	−1.08862
$344$	0	0
$345$	−9.71022	−0.522780
$346$	0	0
$347$	7.24761	0.389072	0.194536	−	0.980895i	$-0.437680\pi$
$347$	7.24761	0.389072	0.194536	+	0.980895i	$0.437680\pi$
$348$	0	0
$349$	−2.40862	−0.128930	−0.0644652	−	0.997920i	$-0.520534\pi$
$349$	−2.40862	−0.128930	−0.0644652	+	0.997920i	$0.520534\pi$
$350$	0	0
$351$	1.16181	0.0620129
$352$	0	0
$353$	−8.53463	−0.454253	−0.227126	−	0.973865i	$-0.572933\pi$
$353$	−8.53463	−0.454253	−0.227126	+	0.973865i	$0.572933\pi$
$354$	0	0
$355$	6.91721	0.367128
$356$	0	0
$357$	−24.8896	−1.31730
$358$	0	0
$359$	−25.6473	−1.35361	−0.676805	−	0.736162i	$-0.736636\pi$
$359$	−25.6473	−1.35361	−0.676805	+	0.736162i	$0.736636\pi$
$360$	0	0
$361$	48.5078	2.55304
$362$	0	0
$363$	−19.5281	−1.02496
$364$	0	0
$365$	−3.98738	−0.208709
$366$	0	0
$367$	−3.98511	−0.208021	−0.104011	−	0.994576i	$-0.533168\pi$
$367$	−3.98511	−0.208021	−0.104011	+	0.994576i	$0.533168\pi$
$368$	0	0
$369$	1.12256	0.0584381
$370$	0	0
$371$	14.6931	0.762828
$372$	0	0
$373$	6.36807	0.329726	0.164863	−	0.986316i	$-0.447282\pi$
$373$	6.36807	0.329726	0.164863	+	0.986316i	$0.447282\pi$
$374$	0	0
$375$	−1.95384	−0.100896
$376$	0	0
$377$	1.88254	0.0969559
$378$	0	0
$379$	−32.6022	−1.67466	−0.837331	−	0.546696i	$-0.815885\pi$
$379$	−32.6022	−1.67466	−0.837331	+	0.546696i	$0.815885\pi$
$380$	0	0
$381$	1.36056	0.0697038
$382$	0	0
$383$	−33.1331	−1.69302	−0.846512	−	0.532370i	$-0.821302\pi$
$383$	−33.1331	−1.69302	−0.846512	+	0.532370i	$0.821302\pi$
$384$	0	0
$385$	−2.17689	−0.110945
$386$	0	0
$387$	2.45059	0.124571
$388$	0	0
$389$	−18.7940	−0.952896	−0.476448	−	0.879203i	$-0.658076\pi$
$389$	−18.7940	−0.952896	−0.476448	+	0.879203i	$0.658076\pi$
$390$	0	0
$391$	−29.1591	−1.47464
$392$	0	0
$393$	1.40818	0.0710335
$394$	0	0
$395$	9.61707	0.483887
$396$	0	0
$397$	10.5313	0.528549	0.264274	−	0.964448i	$-0.414868\pi$
$397$	10.5313	0.528549	0.264274	+	0.964448i	$0.414868\pi$
$398$	0	0
$399$	−34.8546	−1.74491
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−2.44898	−0.121992
$404$	0	0
$405$	10.7842	0.535870
$406$	0	0
$407$	−5.24544	−0.260007
$408$	0	0
$409$	−14.3231	−0.708230	−0.354115	−	0.935202i	$-0.615218\pi$
$409$	−14.3231	−0.708230	−0.354115	+	0.935202i	$0.615218\pi$
$410$	0	0
$411$	−23.1401	−1.14142
$412$	0	0
$413$	5.30842	0.261210
$414$	0	0
$415$	11.8176	0.580101
$416$	0	0
$417$	10.3939	0.508990
$418$	0	0
$419$	−22.3285	−1.09082	−0.545408	−	0.838171i	$-0.683625\pi$
$419$	−22.3285	−1.09082	−0.545408	+	0.838171i	$0.683625\pi$
$420$	0	0
$421$	31.6709	1.54354	0.771772	−	0.635900i	$-0.219371\pi$
$421$	31.6709	1.54354	0.771772	+	0.635900i	$0.219371\pi$
$422$	0	0
$423$	−1.99223	−0.0968657
$424$	0	0
$425$	−5.86725	−0.284603
$426$	0	0
$427$	9.63720	0.466377
$428$	0	0
$429$	−0.533729	−0.0257687
$430$	0	0
$431$	−14.9334	−0.719318	−0.359659	−	0.933084i	$-0.617107\pi$
$431$	−14.9334	−0.719318	−0.359659	+	0.933084i	$0.617107\pi$
$432$	0	0
$433$	−35.7623	−1.71863	−0.859314	−	0.511449i	$-0.829109\pi$
$433$	−35.7623	−1.71863	−0.859314	+	0.511449i	$0.829109\pi$
$434$	0	0
$435$	13.5003	0.647290
$436$	0	0
$437$	−40.8335	−1.95333
$438$	0	0
$439$	6.34741	0.302945	0.151473	−	0.988461i	$-0.451598\pi$
$439$	6.34741	0.302945	0.151473	+	0.988461i	$0.451598\pi$
$440$	0	0
$441$	−1.86878	−0.0889897
$442$	0	0
$443$	−6.90002	−0.327830	−0.163915	−	0.986474i	$-0.552412\pi$
$443$	−6.90002	−0.327830	−0.163915	+	0.986474i	$0.552412\pi$
$444$	0	0
$445$	11.9940	0.568572
$446$	0	0
$447$	18.1765	0.859718
$448$	0	0
$449$	−1.51352	−0.0714276	−0.0357138	−	0.999362i	$-0.511370\pi$
$449$	−1.51352	−0.0714276	−0.0357138	+	0.999362i	$0.511370\pi$
$450$	0	0
$451$	1.37679	0.0648307
$452$	0	0
$453$	4.53838	0.213232
$454$	0	0
$455$	0.591541	0.0277319
$456$	0	0
$457$	35.2276	1.64788	0.823940	−	0.566677i	$-0.191771\pi$
$457$	35.2276	1.64788	0.823940	+	0.566677i	$0.191771\pi$
$458$	0	0
$459$	25.0196	1.16781
$460$	0	0
$461$	−17.4197	−0.811318	−0.405659	−	0.914025i	$-0.632958\pi$
$461$	−17.4197	−0.811318	−0.405659	+	0.914025i	$0.632958\pi$
$462$	0	0
$463$	−20.8534	−0.969140	−0.484570	−	0.874752i	$-0.661024\pi$
$463$	−20.8534	−0.969140	−0.484570	+	0.874752i	$0.661024\pi$
$464$	0	0
$465$	−17.5624	−0.814436
$466$	0	0
$467$	29.8140	1.37963	0.689813	−	0.723988i	$-0.257693\pi$
$467$	29.8140	1.37963	0.689813	+	0.723988i	$0.257693\pi$
$468$	0	0
$469$	−28.1879	−1.30160
$470$	0	0
$471$	−28.3846	−1.30789
$472$	0	0
$473$	3.00560	0.138197
$474$	0	0
$475$	−8.21631	−0.376990
$476$	0	0
$477$	5.53224	0.253304
$478$	0	0
$479$	25.3757	1.15945	0.579724	−	0.814813i	$-0.303160\pi$
$479$	25.3757	1.15945	0.579724	+	0.814813i	$0.303160\pi$
$480$	0	0
$481$	1.42538	0.0649916
$482$	0	0
$483$	21.0826	0.959290
$484$	0	0
$485$	−10.3855	−0.471581
$486$	0	0
$487$	−33.8037	−1.53179	−0.765895	−	0.642966i	$-0.777704\pi$
$487$	−33.8037	−1.53179	−0.765895	+	0.642966i	$0.777704\pi$
$488$	0	0
$489$	−0.935288	−0.0422952
$490$	0	0
$491$	−18.9206	−0.853876	−0.426938	−	0.904281i	$-0.640408\pi$
$491$	−18.9206	−0.853876	−0.426938	+	0.904281i	$0.640408\pi$
$492$	0	0
$493$	40.5405	1.82585
$494$	0	0
$495$	−0.819641	−0.0368401
$496$	0	0
$497$	−15.0185	−0.673671
$498$	0	0
$499$	26.5558	1.18880	0.594401	−	0.804169i	$-0.297389\pi$
$499$	26.5558	1.18880	0.594401	+	0.804169i	$0.297389\pi$
$500$	0	0
$501$	−40.9925	−1.83141
$502$	0	0
$503$	−10.1522	−0.452663	−0.226331	−	0.974050i	$-0.572673\pi$
$503$	−10.1522	−0.452663	−0.226331	+	0.974050i	$0.572673\pi$
$504$	0	0
$505$	4.61598	0.205408
$506$	0	0
$507$	−25.2549	−1.12161
$508$	0	0
$509$	19.7875	0.877067	0.438534	−	0.898715i	$-0.355498\pi$
$509$	19.7875	0.877067	0.438534	+	0.898715i	$0.355498\pi$
$510$	0	0
$511$	8.65730	0.382977
$512$	0	0
$513$	35.0366	1.54690
$514$	0	0
$515$	0.873853	0.0385066
$516$	0	0
$517$	−2.44343	−0.107462
$518$	0	0
$519$	19.6554	0.862775
$520$	0	0
$521$	−6.91212	−0.302825	−0.151413	−	0.988471i	$-0.548382\pi$
$521$	−6.91212	−0.302825	−0.151413	+	0.988471i	$0.548382\pi$
$522$	0	0
$523$	10.0776	0.440665	0.220332	−	0.975425i	$-0.429286\pi$
$523$	10.0776	0.440665	0.220332	+	0.975425i	$0.429286\pi$
$524$	0	0
$525$	4.24213	0.185142
$526$	0	0
$527$	−52.7386	−2.29733
$528$	0	0
$529$	1.69903	0.0738711
$530$	0	0
$531$	1.99872	0.0867371
$532$	0	0
$533$	−0.374126	−0.0162052
$534$	0	0
$535$	9.17487	0.396664
$536$	0	0
$537$	23.6338	1.01987
$538$	0	0
$539$	−2.29202	−0.0987244
$540$	0	0
$541$	3.29253	0.141557	0.0707785	−	0.997492i	$-0.477452\pi$
$541$	3.29253	0.141557	0.0707785	+	0.997492i	$0.477452\pi$
$542$	0	0
$543$	−7.37898	−0.316662
$544$	0	0
$545$	2.64702	0.113386
$546$	0	0
$547$	−12.2102	−0.522070	−0.261035	−	0.965329i	$-0.584064\pi$
$547$	−12.2102	−0.522070	−0.261035	+	0.965329i	$0.584064\pi$
$548$	0	0
$549$	3.62859	0.154865
$550$	0	0
$551$	56.7717	2.41855
$552$	0	0
$553$	−20.8803	−0.887922
$554$	0	0
$555$	10.2218	0.433893
$556$	0	0
$557$	−9.81173	−0.415736	−0.207868	−	0.978157i	$-0.566652\pi$
$557$	−9.81173	−0.415736	−0.207868	+	0.978157i	$0.566652\pi$
$558$	0	0
$559$	−0.816731	−0.0345440
$560$	0	0
$561$	−11.4938	−0.485270
$562$	0	0
$563$	−28.1600	−1.18680	−0.593402	−	0.804906i	$-0.702215\pi$
$563$	−28.1600	−1.18680	−0.593402	+	0.804906i	$0.702215\pi$
$564$	0	0
$565$	6.83601	0.287593
$566$	0	0
$567$	−23.4143	−0.983309
$568$	0	0
$569$	27.0409	1.13361	0.566806	−	0.823851i	$-0.308179\pi$
$569$	27.0409	1.13361	0.566806	+	0.823851i	$0.308179\pi$
$570$	0	0
$571$	−25.6087	−1.07169	−0.535846	−	0.844316i	$-0.680007\pi$
$571$	−25.6087	−1.07169	−0.535846	+	0.844316i	$0.680007\pi$
$572$	0	0
$573$	0.0208017	0.000869003 0
$574$	0	0
$575$	4.96981	0.207255
$576$	0	0
$577$	−4.56709	−0.190130	−0.0950652	−	0.995471i	$-0.530306\pi$
$577$	−4.56709	−0.190130	−0.0950652	+	0.995471i	$0.530306\pi$
$578$	0	0
$579$	15.1452	0.629412
$580$	0	0
$581$	−25.6580	−1.06447
$582$	0	0
$583$	6.78517	0.281013
$584$	0	0
$585$	0.222727	0.00920861
$586$	0	0
$587$	30.7643	1.26978	0.634890	−	0.772603i	$-0.281045\pi$
$587$	30.7643	1.26978	0.634890	+	0.772603i	$0.281045\pi$
$588$	0	0
$589$	−73.8535	−3.04308
$590$	0	0
$591$	50.8711	2.09256
$592$	0	0
$593$	−34.9191	−1.43395	−0.716977	−	0.697097i	$-0.754475\pi$
$593$	−34.9191	−1.43395	−0.716977	+	0.697097i	$0.754475\pi$
$594$	0	0
$595$	12.7388	0.522241
$596$	0	0
$597$	−0.122013	−0.00499367
$598$	0	0
$599$	39.9110	1.63072	0.815359	−	0.578956i	$-0.196540\pi$
$599$	39.9110	1.63072	0.815359	+	0.578956i	$0.196540\pi$
$600$	0	0
$601$	12.2767	0.500778	0.250389	−	0.968145i	$-0.419442\pi$
$601$	12.2767	0.500778	0.250389	+	0.968145i	$0.419442\pi$
$602$	0	0
$603$	−10.6133	−0.432207
$604$	0	0
$605$	9.99473	0.406343
$606$	0	0
$607$	−42.9536	−1.74343	−0.871716	−	0.490012i	$-0.836992\pi$
$607$	−42.9536	−1.74343	−0.871716	+	0.490012i	$0.836992\pi$
$608$	0	0
$609$	−29.3115	−1.18776
$610$	0	0
$611$	0.663970	0.0268614
$612$	0	0
$613$	−29.1808	−1.17860	−0.589300	−	0.807914i	$-0.700597\pi$
$613$	−29.1808	−1.17860	−0.589300	+	0.807914i	$0.700597\pi$
$614$	0	0
$615$	−2.68297	−0.108188
$616$	0	0
$617$	−30.1451	−1.21360	−0.606799	−	0.794855i	$-0.707547\pi$
$617$	−30.1451	−1.21360	−0.606799	+	0.794855i	$0.707547\pi$
$618$	0	0
$619$	−1.51630	−0.0609452	−0.0304726	−	0.999536i	$-0.509701\pi$
$619$	−1.51630	−0.0609452	−0.0304726	+	0.999536i	$0.509701\pi$
$620$	0	0
$621$	−21.1927	−0.850432
$622$	0	0
$623$	−26.0412	−1.04332
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−16.0956	−0.642797
$628$	0	0
$629$	30.6955	1.22391
$630$	0	0
$631$	−8.17323	−0.325371	−0.162686	−	0.986678i	$-0.552016\pi$
$631$	−8.17323	−0.325371	−0.162686	+	0.986678i	$0.552016\pi$
$632$	0	0
$633$	29.2800	1.16377
$634$	0	0
$635$	−0.696354	−0.0276340
$636$	0	0
$637$	0.622827	0.0246773
$638$	0	0
$639$	−5.65475	−0.223698
$640$	0	0
$641$	26.4926	1.04640	0.523198	−	0.852211i	$-0.324739\pi$
$641$	26.4926	1.04640	0.523198	+	0.852211i	$0.324739\pi$
$642$	0	0
$643$	11.2349	0.443063	0.221531	−	0.975153i	$-0.428894\pi$
$643$	11.2349	0.443063	0.221531	+	0.975153i	$0.428894\pi$
$644$	0	0
$645$	−5.85703	−0.230620
$646$	0	0
$647$	−36.4248	−1.43201	−0.716004	−	0.698096i	$-0.754031\pi$
$647$	−36.4248	−1.43201	−0.716004	+	0.698096i	$0.754031\pi$
$648$	0	0
$649$	2.45139	0.0962254
$650$	0	0
$651$	38.1310	1.49447
$652$	0	0
$653$	11.3383	0.443700	0.221850	−	0.975081i	$-0.428790\pi$
$653$	11.3383	0.443700	0.221850	+	0.975081i	$0.428790\pi$
$654$	0	0
$655$	−0.720727	−0.0281611
$656$	0	0
$657$	3.25964	0.127171
$658$	0	0
$659$	24.0738	0.937784	0.468892	−	0.883256i	$-0.344653\pi$
$659$	24.0738	0.937784	0.468892	+	0.883256i	$0.344653\pi$
$660$	0	0
$661$	−12.6937	−0.493727	−0.246864	−	0.969050i	$-0.579400\pi$
$661$	−12.6937	−0.493727	−0.246864	+	0.969050i	$0.579400\pi$
$662$	0	0
$663$	3.12330	0.121299
$664$	0	0
$665$	17.8390	0.691768
$666$	0	0
$667$	−34.3396	−1.32963
$668$	0	0
$669$	−3.05812	−0.118234
$670$	0	0
$671$	4.45039	0.171805
$672$	0	0
$673$	27.7973	1.07151	0.535754	−	0.844374i	$-0.320027\pi$
$673$	27.7973	1.07151	0.535754	+	0.844374i	$0.320027\pi$
$674$	0	0
$675$	−4.26428	−0.164132
$676$	0	0
$677$	9.33163	0.358644	0.179322	−	0.983790i	$-0.442610\pi$
$677$	9.33163	0.358644	0.179322	+	0.983790i	$0.442610\pi$
$678$	0	0
$679$	22.5487	0.865339
$680$	0	0
$681$	26.8870	1.03031
$682$	0	0
$683$	27.8946	1.06736	0.533678	−	0.845688i	$-0.320810\pi$
$683$	27.8946	1.06736	0.533678	+	0.845688i	$0.320810\pi$
$684$	0	0
$685$	11.8434	0.452512
$686$	0	0
$687$	46.6333	1.77917
$688$	0	0
$689$	−1.84378	−0.0702424
$690$	0	0
$691$	33.4529	1.27261	0.636304	−	0.771438i	$-0.280462\pi$
$691$	33.4529	1.27261	0.636304	+	0.771438i	$0.280462\pi$
$692$	0	0
$693$	1.77958	0.0676008
$694$	0	0
$695$	−5.31971	−0.201788
$696$	0	0
$697$	−8.05679	−0.305173
$698$	0	0
$699$	−9.31213	−0.352217
$700$	0	0
$701$	−23.6802	−0.894389	−0.447194	−	0.894437i	$-0.647577\pi$
$701$	−23.6802	−0.894389	−0.447194	+	0.894437i	$0.647577\pi$
$702$	0	0
$703$	42.9850	1.62121
$704$	0	0
$705$	4.76154	0.179330
$706$	0	0
$707$	−10.0221	−0.376919
$708$	0	0
$709$	39.8489	1.49656	0.748278	−	0.663385i	$-0.230881\pi$
$709$	39.8489	1.49656	0.748278	+	0.663385i	$0.230881\pi$
$710$	0	0
$711$	−7.86185	−0.294842
$712$	0	0
$713$	44.6719	1.67298
$714$	0	0
$715$	0.273169	0.0102160
$716$	0	0
$717$	14.0316	0.524021
$718$	0	0
$719$	25.2914	0.943209	0.471605	−	0.881810i	$-0.343675\pi$
$719$	25.2914	0.943209	0.471605	+	0.881810i	$0.343675\pi$
$720$	0	0
$721$	−1.89729	−0.0706586
$722$	0	0
$723$	−14.8446	−0.552078
$724$	0	0
$725$	−6.90963	−0.256617
$726$	0	0
$727$	48.2882	1.79091	0.895454	−	0.445154i	$-0.146851\pi$
$727$	48.2882	1.79091	0.895454	+	0.445154i	$0.146851\pi$
$728$	0	0
$729$	16.1792	0.599229
$730$	0	0
$731$	−17.5883	−0.650526
$732$	0	0
$733$	42.3662	1.56483	0.782416	−	0.622757i	$-0.213987\pi$
$733$	42.3662	1.56483	0.782416	+	0.622757i	$0.213987\pi$
$734$	0	0
$735$	4.46649	0.164749
$736$	0	0
$737$	−13.0170	−0.479487
$738$	0	0
$739$	−22.6921	−0.834743	−0.417371	−	0.908736i	$-0.637049\pi$
$739$	−22.6921	−0.834743	−0.417371	+	0.908736i	$0.637049\pi$
$740$	0	0
$741$	4.37377	0.160674
$742$	0	0
$743$	31.9848	1.17341	0.586703	−	0.809802i	$-0.300425\pi$
$743$	31.9848	1.17341	0.586703	+	0.809802i	$0.300425\pi$
$744$	0	0
$745$	−9.30295	−0.340834
$746$	0	0
$747$	−9.66073	−0.353467
$748$	0	0
$749$	−19.9202	−0.727870
$750$	0	0
$751$	−6.31847	−0.230564	−0.115282	−	0.993333i	$-0.536777\pi$
$751$	−6.31847	−0.230564	−0.115282	+	0.993333i	$0.536777\pi$
$752$	0	0
$753$	54.0733	1.97054
$754$	0	0
$755$	−2.32280	−0.0845354
$756$	0	0
$757$	14.0729	0.511488	0.255744	−	0.966745i	$-0.417680\pi$
$757$	14.0729	0.511488	0.255744	+	0.966745i	$0.417680\pi$
$758$	0	0
$759$	9.73578	0.353386
$760$	0	0
$761$	10.1389	0.367535	0.183768	−	0.982970i	$-0.441171\pi$
$761$	10.1389	0.367535	0.183768	+	0.982970i	$0.441171\pi$
$762$	0	0
$763$	−5.74714	−0.208060
$764$	0	0
$765$	4.79641	0.173415
$766$	0	0
$767$	−0.666132	−0.0240526
$768$	0	0
$769$	6.73480	0.242863	0.121432	−	0.992600i	$-0.461251\pi$
$769$	6.73480	0.242863	0.121432	+	0.992600i	$0.461251\pi$
$770$	0	0
$771$	−18.2853	−0.658528
$772$	0	0
$773$	28.6933	1.03203	0.516013	−	0.856581i	$-0.327416\pi$
$773$	28.6933	1.03203	0.516013	+	0.856581i	$0.327416\pi$
$774$	0	0
$775$	8.98865	0.322882
$776$	0	0
$777$	−22.1934	−0.796183
$778$	0	0
$779$	−11.2825	−0.404237
$780$	0	0
$781$	−6.93542	−0.248169
$782$	0	0
$783$	29.4646	1.05298
$784$	0	0
$785$	14.5276	0.518513
$786$	0	0
$787$	0.345425	0.0123131	0.00615654	−	0.999981i	$-0.498040\pi$
$787$	0.345425	0.0123131	0.00615654	+	0.999981i	$0.498040\pi$
$788$	0	0
$789$	−55.5850	−1.97888
$790$	0	0
$791$	−14.8422	−0.527727
$792$	0	0
$793$	−1.20933	−0.0429447
$794$	0	0
$795$	−13.2223	−0.468947
$796$	0	0
$797$	−40.4292	−1.43208	−0.716038	−	0.698061i	$-0.754046\pi$
$797$	−40.4292	−1.43208	−0.716038	+	0.698061i	$0.754046\pi$
$798$	0	0
$799$	14.2986	0.505847
$800$	0	0
$801$	−9.80500	−0.346443
$802$	0	0
$803$	3.99788	0.141082
$804$	0	0
$805$	−10.7903	−0.380309
$806$	0	0
$807$	22.5838	0.794988
$808$	0	0
$809$	−16.7938	−0.590438	−0.295219	−	0.955430i	$-0.595393\pi$
$809$	−16.7938	−0.590438	−0.295219	+	0.955430i	$0.595393\pi$
$810$	0	0
$811$	28.9934	1.01809	0.509047	−	0.860739i	$-0.329998\pi$
$811$	28.9934	1.01809	0.509047	+	0.860739i	$0.329998\pi$
$812$	0	0
$813$	37.3920	1.31139
$814$	0	0
$815$	0.478693	0.0167679
$816$	0	0
$817$	−24.6301	−0.861697
$818$	0	0
$819$	−0.483578	−0.0168976
$820$	0	0
$821$	−15.2016	−0.530539	−0.265269	−	0.964174i	$-0.585461\pi$
$821$	−15.2016	−0.530539	−0.265269	+	0.964174i	$0.585461\pi$
$822$	0	0
$823$	8.28104	0.288659	0.144329	−	0.989530i	$-0.453898\pi$
$823$	8.28104	0.288659	0.144329	+	0.989530i	$0.453898\pi$
$824$	0	0
$825$	1.95898	0.0682030
$826$	0	0
$827$	34.3396	1.19410	0.597052	−	0.802202i	$-0.296338\pi$
$827$	34.3396	1.19410	0.597052	+	0.802202i	$0.296338\pi$
$828$	0	0
$829$	10.5395	0.366050	0.183025	−	0.983108i	$-0.441411\pi$
$829$	10.5395	0.366050	0.183025	+	0.983108i	$0.441411\pi$
$830$	0	0
$831$	60.6068	2.10243
$832$	0	0
$833$	13.4126	0.464718
$834$	0	0
$835$	20.9805	0.726059
$836$	0	0
$837$	−38.3301	−1.32488
$838$	0	0
$839$	−32.6721	−1.12797	−0.563983	−	0.825786i	$-0.690732\pi$
$839$	−32.6721	−1.12797	−0.563983	+	0.825786i	$0.690732\pi$
$840$	0	0
$841$	18.7430	0.646310
$842$	0	0
$843$	−41.3755	−1.42505
$844$	0	0
$845$	12.9258	0.444660
$846$	0	0
$847$	−21.7003	−0.745631
$848$	0	0
$849$	−56.8134	−1.94983
$850$	0	0
$851$	−26.0004	−0.891282
$852$	0	0
$853$	−24.8395	−0.850487	−0.425243	−	0.905079i	$-0.639812\pi$
$853$	−24.8395	−0.850487	−0.425243	+	0.905079i	$0.639812\pi$
$854$	0	0
$855$	6.71674	0.229708
$856$	0	0
$857$	−31.1677	−1.06467	−0.532335	−	0.846534i	$-0.678685\pi$
$857$	−31.1677	−1.06467	−0.532335	+	0.846534i	$0.678685\pi$
$858$	0	0
$859$	29.8739	1.01928	0.509642	−	0.860386i	$-0.329778\pi$
$859$	29.8739	1.01928	0.509642	+	0.860386i	$0.329778\pi$
$860$	0	0
$861$	5.82520	0.198522
$862$	0	0
$863$	42.7868	1.45648	0.728239	−	0.685323i	$-0.240339\pi$
$863$	42.7868	1.45648	0.728239	+	0.685323i	$0.240339\pi$
$864$	0	0
$865$	−10.0599	−0.342046
$866$	0	0
$867$	34.0449	1.15623
$868$	0	0
$869$	−9.64238	−0.327096
$870$	0	0
$871$	3.53719	0.119853
$872$	0	0
$873$	8.49002	0.287344
$874$	0	0
$875$	−2.17117	−0.0733991
$876$	0	0
$877$	−22.0240	−0.743698	−0.371849	−	0.928293i	$-0.621276\pi$
$877$	−22.0240	−0.743698	−0.371849	+	0.928293i	$0.621276\pi$
$878$	0	0
$879$	12.2382	0.412783
$880$	0	0
$881$	−9.14501	−0.308103	−0.154052	−	0.988063i	$-0.549232\pi$
$881$	−9.14501	−0.308103	−0.154052	+	0.988063i	$0.549232\pi$
$882$	0	0
$883$	41.1298	1.38413	0.692064	−	0.721836i	$-0.256701\pi$
$883$	41.1298	1.38413	0.692064	+	0.721836i	$0.256701\pi$
$884$	0	0
$885$	−4.77704	−0.160579
$886$	0	0
$887$	44.1121	1.48114	0.740569	−	0.671980i	$-0.234556\pi$
$887$	44.1121	1.48114	0.740569	+	0.671980i	$0.234556\pi$
$888$	0	0
$889$	1.51190	0.0507077
$890$	0	0
$891$	−10.8126	−0.362235
$892$	0	0
$893$	20.0233	0.670054
$894$	0	0
$895$	−12.0961	−0.404327
$896$	0	0
$897$	−2.64557	−0.0883330
$898$	0	0
$899$	−62.1082	−2.07143
$900$	0	0
$901$	−39.7058	−1.32279
$902$	0	0
$903$	12.7166	0.423183
$904$	0	0
$905$	3.77666	0.125540
$906$	0	0
$907$	31.0518	1.03106	0.515528	−	0.856872i	$-0.327596\pi$
$907$	31.0518	1.03106	0.515528	+	0.856872i	$0.327596\pi$
$908$	0	0
$909$	−3.77351	−0.125159
$910$	0	0
$911$	−52.9618	−1.75470	−0.877351	−	0.479849i	$-0.840691\pi$
$911$	−52.9618	−1.75470	−0.877351	+	0.479849i	$0.840691\pi$
$912$	0	0
$913$	−11.8487	−0.392134
$914$	0	0
$915$	−8.67252	−0.286705
$916$	0	0
$917$	1.56482	0.0516750
$918$	0	0
$919$	−5.80731	−0.191565	−0.0957827	−	0.995402i	$-0.530535\pi$
$919$	−5.80731	−0.191565	−0.0957827	+	0.995402i	$0.530535\pi$
$920$	0	0
$921$	−0.627810	−0.0206870
$922$	0	0
$923$	1.88461	0.0620327
$924$	0	0
$925$	−5.23167	−0.172016
$926$	0	0
$927$	−0.714365	−0.0234628
$928$	0	0
$929$	23.2136	0.761615	0.380807	−	0.924654i	$-0.375646\pi$
$929$	23.2136	0.761615	0.380807	+	0.924654i	$0.375646\pi$
$930$	0	0
$931$	18.7825	0.615572
$932$	0	0
$933$	−60.6476	−1.98551
$934$	0	0
$935$	5.88269	0.192385
$936$	0	0
$937$	−22.9324	−0.749169	−0.374584	−	0.927193i	$-0.622215\pi$
$937$	−22.9324	−0.749169	−0.374584	+	0.927193i	$0.622215\pi$
$938$	0	0
$939$	18.6860	0.609794
$940$	0	0
$941$	−41.3910	−1.34931	−0.674654	−	0.738134i	$-0.735707\pi$
$941$	−41.3910	−1.34931	−0.674654	+	0.738134i	$0.735707\pi$
$942$	0	0
$943$	6.82445	0.222235
$944$	0	0
$945$	9.25849	0.301179
$946$	0	0
$947$	1.94821	0.0633083	0.0316542	−	0.999499i	$-0.489922\pi$
$947$	1.94821	0.0633083	0.0316542	+	0.999499i	$0.489922\pi$
$948$	0	0
$949$	−1.08637	−0.0352651
$950$	0	0
$951$	0.816110	0.0264642
$952$	0	0
$953$	−19.2668	−0.624114	−0.312057	−	0.950063i	$-0.601018\pi$
$953$	−19.2668	−0.624114	−0.312057	+	0.950063i	$0.601018\pi$
$954$	0	0
$955$	−0.0106466	−0.000344515 0
$956$	0	0
$957$	−13.5358	−0.437552
$958$	0	0
$959$	−25.7140	−0.830350
$960$	0	0
$961$	49.7958	1.60632
$962$	0	0
$963$	−7.50036	−0.241696
$964$	0	0
$965$	−7.75149	−0.249529
$966$	0	0
$967$	−52.3285	−1.68277	−0.841385	−	0.540436i	$-0.818259\pi$
$967$	−52.3285	−1.68277	−0.841385	+	0.540436i	$0.818259\pi$
$968$	0	0
$969$	94.1890	3.02579
$970$	0	0
$971$	16.4905	0.529206	0.264603	−	0.964357i	$-0.414759\pi$
$971$	16.4905	0.529206	0.264603	+	0.964357i	$0.414759\pi$
$972$	0	0
$973$	11.5500	0.370277
$974$	0	0
$975$	−0.532328	−0.0170481
$976$	0	0
$977$	19.5252	0.624666	0.312333	−	0.949973i	$-0.398890\pi$
$977$	19.5252	0.624666	0.312333	+	0.949973i	$0.398890\pi$
$978$	0	0
$979$	−12.0256	−0.384340
$980$	0	0
$981$	−2.16391	−0.0690882
$982$	0	0
$983$	−16.9262	−0.539863	−0.269932	−	0.962879i	$-0.587001\pi$
$983$	−16.9262	−0.539863	−0.269932	+	0.962879i	$0.587001\pi$
$984$	0	0
$985$	−26.0365	−0.829592
$986$	0	0
$987$	−10.3381	−0.329066
$988$	0	0
$989$	14.8980	0.473730
$990$	0	0
$991$	39.6379	1.25914	0.629569	−	0.776945i	$-0.283232\pi$
$991$	39.6379	1.25914	0.629569	+	0.776945i	$0.283232\pi$
$992$	0	0
$993$	−8.15004	−0.258634
$994$	0	0
$995$	0.0624479	0.00197973
$996$	0	0
$997$	55.0315	1.74287	0.871433	−	0.490515i	$-0.163191\pi$
$997$	55.0315	1.74287	0.871433	+	0.490515i	$0.163191\pi$
$998$	0	0
$999$	22.3093	0.705834

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.22	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.22	✓	28	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)

\(f(q)\)	\(=\)	\(q+1.95384 q^{3} -1.00000 q^{5} +2.17117 q^{7} +0.817489 q^{9} +O(q^{10})\)	\(q+1.95384 q^{3} -1.00000 q^{5} +2.17117 q^{7} +0.817489 q^{9} +1.00263 q^{11} -0.272452 q^{13} -1.95384 q^{15} -5.86725 q^{17} -8.21631 q^{19} +4.24213 q^{21} +4.96981 q^{23} +1.00000 q^{25} -4.26428 q^{27} -6.90963 q^{29} +8.98865 q^{31} +1.95898 q^{33} -2.17117 q^{35} -5.23167 q^{37} -0.532328 q^{39} +1.37318 q^{41} +2.99770 q^{43} -0.817489 q^{45} -2.43702 q^{47} -2.28600 q^{49} -11.4637 q^{51} +6.76735 q^{53} -1.00263 q^{55} -16.0534 q^{57} +2.44495 q^{59} +4.43871 q^{61} +1.77491 q^{63} +0.272452 q^{65} -12.9828 q^{67} +9.71022 q^{69} -6.91721 q^{71} +3.98738 q^{73} +1.95384 q^{75} +2.17689 q^{77} -9.61707 q^{79} -10.7842 q^{81} -11.8176 q^{83} +5.86725 q^{85} -13.5003 q^{87} -11.9940 q^{89} -0.591541 q^{91} +17.5624 q^{93} +8.21631 q^{95} +10.3855 q^{97} +0.819641 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more