LMFDB - Embedded newform 4020.2.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.195727752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 15x^{4} - 25x^{3} - 3x^{2} + 9x + 3 $ x^6 - 15x^4 - 25x^3 - 3x^2 + 9x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.587102$ of defining polynomial
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +1.68378 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +1.68378 q^{7} +1.00000 q^{9} +6.11470 q^{11} +4.10984 q^{13} -1.00000 q^{15} +1.36271 q^{17} +4.20852 q^{19} +1.68378 q^{21} -5.55706 q^{23} +1.00000 q^{25} +1.00000 q^{27} +9.42850 q^{29} -0.853132 q^{31} +6.11470 q^{33} -1.68378 q^{35} -7.11228 q^{37} +4.10984 q^{39} -3.25216 q^{41} -9.08422 q^{43} -1.00000 q^{45} -3.42194 q^{47} -4.16489 q^{49} +1.36271 q^{51} +10.9929 q^{53} -6.11470 q^{55} +4.20852 q^{57} +4.21022 q^{59} -4.10814 q^{61} +1.68378 q^{63} -4.10984 q^{65} +1.00000 q^{67} -5.55706 q^{69} +7.68408 q^{71} -15.3490 q^{73} +1.00000 q^{75} +10.2958 q^{77} +16.1254 q^{79} +1.00000 q^{81} -0.430916 q^{83} -1.36271 q^{85} +9.42850 q^{87} -18.3674 q^{89} +6.92007 q^{91} -0.853132 q^{93} -4.20852 q^{95} +9.36000 q^{97} +6.11470 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9	$ 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 3 q^{13} - 6 q^{15} + 6 q^{17} - 3 q^{19} + 3 q^{21} + 6 q^{23} + 6 q^{25} + 6 q^{27} + 21 q^{29} - 3 q^{31} + 3 q^{33} - 3 q^{35} + 3 q^{39} + 9 q^{41} - 3 q^{43} - 6 q^{45} + 21 q^{47} + 3 q^{49} + 6 q^{51} + 15 q^{53} - 3 q^{55} - 3 q^{57} + 15 q^{59} + 15 q^{61} + 3 q^{63} - 3 q^{65} + 6 q^{67} + 6 q^{69} + 18 q^{71} - 6 q^{73} + 6 q^{75} + 33 q^{77} + 9 q^{79} + 6 q^{81} + 24 q^{83} - 6 q^{85} + 21 q^{87} + 24 q^{89} - 21 q^{91} - 3 q^{93} + 3 q^{95} + 9 q^{97} + 3 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9 + 3 * q^11 + 3 * q^13 - 6 * q^15 + 6 * q^17 - 3 * q^19 + 3 * q^21 + 6 * q^23 + 6 * q^25 + 6 * q^27 + 21 * q^29 - 3 * q^31 + 3 * q^33 - 3 * q^35 + 3 * q^39 + 9 * q^41 - 3 * q^43 - 6 * q^45 + 21 * q^47 + 3 * q^49 + 6 * q^51 + 15 * q^53 - 3 * q^55 - 3 * q^57 + 15 * q^59 + 15 * q^61 + 3 * q^63 - 3 * q^65 + 6 * q^67 + 6 * q^69 + 18 * q^71 - 6 * q^73 + 6 * q^75 + 33 * q^77 + 9 * q^79 + 6 * q^81 + 24 * q^83 - 6 * q^85 + 21 * q^87 + 24 * q^89 - 21 * q^91 - 3 * q^93 + 3 * q^95 + 9 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.68378	0.636409	0.318204	−	0.948022i	$-0.396920\pi$
$7$	1.68378	0.636409	0.318204	+	0.948022i	$0.396920\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.11470	1.84365	0.921825	−	0.387606i	$-0.126698\pi$
$11$	6.11470	1.84365	0.921825	+	0.387606i	$0.126698\pi$
$12$	0	0
$13$	4.10984	1.13987	0.569933	−	0.821691i	$-0.306969\pi$
$13$	4.10984	1.13987	0.569933	+	0.821691i	$0.306969\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.36271	0.330505	0.165252	−	0.986251i	$-0.447156\pi$
$17$	1.36271	0.330505	0.165252	+	0.986251i	$0.447156\pi$
$18$	0	0
$19$	4.20852	0.965502	0.482751	−	0.875758i	$-0.339638\pi$
$19$	4.20852	0.965502	0.482751	+	0.875758i	$0.339638\pi$
$20$	0	0
$21$	1.68378	0.367431
$22$	0	0
$23$	−5.55706	−1.15873	−0.579364	−	0.815069i	$-0.696699\pi$
$23$	−5.55706	−1.15873	−0.579364	+	0.815069i	$0.696699\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	9.42850	1.75083	0.875414	−	0.483374i	$-0.160589\pi$
$29$	9.42850	1.75083	0.875414	+	0.483374i	$0.160589\pi$
$30$	0	0
$31$	−0.853132	−0.153227	−0.0766135	−	0.997061i	$-0.524411\pi$
$31$	−0.853132	−0.153227	−0.0766135	+	0.997061i	$0.524411\pi$
$32$	0	0
$33$	6.11470	1.06443
$34$	0	0
$35$	−1.68378	−0.284611
$36$	0	0
$37$	−7.11228	−1.16925	−0.584626	−	0.811303i	$-0.698759\pi$
$37$	−7.11228	−1.16925	−0.584626	+	0.811303i	$0.698759\pi$
$38$	0	0
$39$	4.10984	0.658102
$40$	0	0
$41$	−3.25216	−0.507902	−0.253951	−	0.967217i	$-0.581730\pi$
$41$	−3.25216	−0.507902	−0.253951	+	0.967217i	$0.581730\pi$
$42$	0	0
$43$	−9.08422	−1.38533	−0.692666	−	0.721259i	$-0.743564\pi$
$43$	−9.08422	−1.38533	−0.692666	+	0.721259i	$0.743564\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−3.42194	−0.499142	−0.249571	−	0.968357i	$-0.580290\pi$
$47$	−3.42194	−0.499142	−0.249571	+	0.968357i	$0.580290\pi$
$48$	0	0
$49$	−4.16489	−0.594984
$50$	0	0
$51$	1.36271	0.190817
$52$	0	0
$53$	10.9929	1.50999	0.754994	−	0.655732i	$-0.227640\pi$
$53$	10.9929	1.50999	0.754994	+	0.655732i	$0.227640\pi$
$54$	0	0
$55$	−6.11470	−0.824505
$56$	0	0
$57$	4.20852	0.557433
$58$	0	0
$59$	4.21022	0.548125	0.274062	−	0.961712i	$-0.411633\pi$
$59$	4.21022	0.548125	0.274062	+	0.961712i	$0.411633\pi$
$60$	0	0
$61$	−4.10814	−0.525994	−0.262997	−	0.964797i	$-0.584711\pi$
$61$	−4.10814	−0.525994	−0.262997	+	0.964797i	$0.584711\pi$
$62$	0	0
$63$	1.68378	0.212136
$64$	0	0
$65$	−4.10984	−0.509763
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	−5.55706	−0.668991
$70$	0	0
$71$	7.68408	0.911933	0.455966	−	0.889997i	$-0.349294\pi$
$71$	7.68408	0.911933	0.455966	+	0.889997i	$0.349294\pi$
$72$	0	0
$73$	−15.3490	−1.79646	−0.898231	−	0.439523i	$-0.855147\pi$
$73$	−15.3490	−1.79646	−0.898231	+	0.439523i	$0.855147\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	10.2958	1.17332
$78$	0	0
$79$	16.1254	1.81425	0.907125	−	0.420862i	$-0.138272\pi$
$79$	16.1254	1.81425	0.907125	+	0.420862i	$0.138272\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.430916	−0.0472991	−0.0236496	−	0.999720i	$-0.507529\pi$
$83$	−0.430916	−0.0472991	−0.0236496	+	0.999720i	$0.507529\pi$
$84$	0	0
$85$	−1.36271	−0.147806
$86$	0	0
$87$	9.42850	1.01084
$88$	0	0
$89$	−18.3674	−1.94694	−0.973472	−	0.228807i	$-0.926518\pi$
$89$	−18.3674	−1.94694	−0.973472	+	0.228807i	$0.926518\pi$
$90$	0	0
$91$	6.92007	0.725420
$92$	0	0
$93$	−0.853132	−0.0884656
$94$	0	0
$95$	−4.20852	−0.431785
$96$	0	0
$97$	9.36000	0.950364	0.475182	−	0.879887i	$-0.342382\pi$
$97$	9.36000	0.950364	0.475182	+	0.879887i	$0.342382\pi$
$98$	0	0
$99$	6.11470	0.614550
$100$	0	0
$101$	8.85114	0.880722	0.440361	−	0.897821i	$-0.354851\pi$
$101$	8.85114	0.880722	0.440361	+	0.897821i	$0.354851\pi$
$102$	0	0
$103$	−5.18461	−0.510854	−0.255427	−	0.966828i	$-0.582216\pi$
$103$	−5.18461	−0.510854	−0.255427	+	0.966828i	$0.582216\pi$
$104$	0	0
$105$	−1.68378	−0.164320
$106$	0	0
$107$	3.49497	0.337872	0.168936	−	0.985627i	$-0.445967\pi$
$107$	3.49497	0.337872	0.168936	+	0.985627i	$0.445967\pi$
$108$	0	0
$109$	−15.4223	−1.47719	−0.738596	−	0.674149i	$-0.764511\pi$
$109$	−15.4223	−1.47719	−0.738596	+	0.674149i	$0.764511\pi$
$110$	0	0
$111$	−7.11228	−0.675067
$112$	0	0
$113$	−11.0139	−1.03610	−0.518050	−	0.855350i	$-0.673342\pi$
$113$	−11.0139	−1.03610	−0.518050	+	0.855350i	$0.673342\pi$
$114$	0	0
$115$	5.55706	0.518199
$116$	0	0
$117$	4.10984	0.379955
$118$	0	0
$119$	2.29450	0.210336
$120$	0	0
$121$	26.3895	2.39905
$122$	0	0
$123$	−3.25216	−0.293238
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.635592	−0.0563997	−0.0281998	−	0.999602i	$-0.508977\pi$
$127$	−0.635592	−0.0563997	−0.0281998	+	0.999602i	$0.508977\pi$
$128$	0	0
$129$	−9.08422	−0.799821
$130$	0	0
$131$	8.44054	0.737453	0.368727	−	0.929538i	$-0.379794\pi$
$131$	8.44054	0.737453	0.368727	+	0.929538i	$0.379794\pi$
$132$	0	0
$133$	7.08623	0.614454
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−0.877703	−0.0749872	−0.0374936	−	0.999297i	$-0.511937\pi$
$137$	−0.877703	−0.0749872	−0.0374936	+	0.999297i	$0.511937\pi$
$138$	0	0
$139$	1.86595	0.158268	0.0791339	−	0.996864i	$-0.474785\pi$
$139$	1.86595	0.158268	0.0791339	+	0.996864i	$0.474785\pi$
$140$	0	0
$141$	−3.42194	−0.288180
$142$	0	0
$143$	25.1304	2.10151
$144$	0	0
$145$	−9.42850	−0.782994
$146$	0	0
$147$	−4.16489	−0.343514
$148$	0	0
$149$	−7.00258	−0.573673	−0.286837	−	0.957979i	$-0.592604\pi$
$149$	−7.00258	−0.573673	−0.286837	+	0.957979i	$0.592604\pi$
$150$	0	0
$151$	−4.48460	−0.364952	−0.182476	−	0.983210i	$-0.558411\pi$
$151$	−4.48460	−0.364952	−0.182476	+	0.983210i	$0.558411\pi$
$152$	0	0
$153$	1.36271	0.110168
$154$	0	0
$155$	0.853132	0.0685252
$156$	0	0
$157$	11.7486	0.937638	0.468819	−	0.883294i	$-0.344680\pi$
$157$	11.7486	0.937638	0.468819	+	0.883294i	$0.344680\pi$
$158$	0	0
$159$	10.9929	0.871791
$160$	0	0
$161$	−9.35686	−0.737424
$162$	0	0
$163$	−2.88301	−0.225815	−0.112908	−	0.993605i	$-0.536016\pi$
$163$	−2.88301	−0.225815	−0.112908	+	0.993605i	$0.536016\pi$
$164$	0	0
$165$	−6.11470	−0.476028
$166$	0	0
$167$	22.8217	1.76600	0.882998	−	0.469377i	$-0.155522\pi$
$167$	22.8217	1.76600	0.882998	+	0.469377i	$0.155522\pi$
$168$	0	0
$169$	3.89081	0.299293
$170$	0	0
$171$	4.20852	0.321834
$172$	0	0
$173$	11.5810	0.880487	0.440243	−	0.897878i	$-0.354892\pi$
$173$	11.5810	0.880487	0.440243	+	0.897878i	$0.354892\pi$
$174$	0	0
$175$	1.68378	0.127282
$176$	0	0
$177$	4.21022	0.316460
$178$	0	0
$179$	12.7037	0.949518	0.474759	−	0.880116i	$-0.342535\pi$
$179$	12.7037	0.949518	0.474759	+	0.880116i	$0.342535\pi$
$180$	0	0
$181$	22.4511	1.66878	0.834388	−	0.551178i	$-0.185821\pi$
$181$	22.4511	1.66878	0.834388	+	0.551178i	$0.185821\pi$
$182$	0	0
$183$	−4.10814	−0.303683
$184$	0	0
$185$	7.11228	0.522905
$186$	0	0
$187$	8.33254	0.609335
$188$	0	0
$189$	1.68378	0.122477
$190$	0	0
$191$	−24.4540	−1.76943	−0.884715	−	0.466133i	$-0.845647\pi$
$191$	−24.4540	−1.76943	−0.884715	+	0.466133i	$0.845647\pi$
$192$	0	0
$193$	−12.8874	−0.927656	−0.463828	−	0.885925i	$-0.653524\pi$
$193$	−12.8874	−0.927656	−0.463828	+	0.885925i	$0.653524\pi$
$194$	0	0
$195$	−4.10984	−0.294312
$196$	0	0
$197$	23.2419	1.65592	0.827959	−	0.560788i	$-0.189502\pi$
$197$	23.2419	1.65592	0.827959	+	0.560788i	$0.189502\pi$
$198$	0	0
$199$	−0.0452949	−0.00321087	−0.00160543	−	0.999999i	$-0.500511\pi$
$199$	−0.0452949	−0.00321087	−0.00160543	+	0.999999i	$0.500511\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	15.8755	1.11424
$204$	0	0
$205$	3.25216	0.227141
$206$	0	0
$207$	−5.55706	−0.386242
$208$	0	0
$209$	25.7338	1.78005
$210$	0	0
$211$	−27.5553	−1.89699	−0.948494	−	0.316795i	$-0.897393\pi$
$211$	−27.5553	−1.89699	−0.948494	+	0.316795i	$0.897393\pi$
$212$	0	0
$213$	7.68408	0.526505
$214$	0	0
$215$	9.08422	0.619539
$216$	0	0
$217$	−1.43649	−0.0975150
$218$	0	0
$219$	−15.3490	−1.03719
$220$	0	0
$221$	5.60051	0.376731
$222$	0	0
$223$	−0.793666	−0.0531478	−0.0265739	−	0.999647i	$-0.508460\pi$
$223$	−0.793666	−0.0531478	−0.0265739	+	0.999647i	$0.508460\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	11.7905	0.782561	0.391280	−	0.920272i	$-0.372032\pi$
$227$	11.7905	0.782561	0.391280	+	0.920272i	$0.372032\pi$
$228$	0	0
$229$	26.1956	1.73105	0.865526	−	0.500864i	$-0.166984\pi$
$229$	26.1956	1.73105	0.865526	+	0.500864i	$0.166984\pi$
$230$	0	0
$231$	10.2958	0.677414
$232$	0	0
$233$	−11.0205	−0.721974	−0.360987	−	0.932571i	$-0.617560\pi$
$233$	−11.0205	−0.721974	−0.360987	+	0.932571i	$0.617560\pi$
$234$	0	0
$235$	3.42194	0.223223
$236$	0	0
$237$	16.1254	1.04746
$238$	0	0
$239$	10.6728	0.690366	0.345183	−	0.938535i	$-0.387817\pi$
$239$	10.6728	0.690366	0.345183	+	0.938535i	$0.387817\pi$
$240$	0	0
$241$	14.1618	0.912244	0.456122	−	0.889917i	$-0.349238\pi$
$241$	14.1618	0.912244	0.456122	+	0.889917i	$0.349238\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	4.16489	0.266085
$246$	0	0
$247$	17.2964	1.10054
$248$	0	0
$249$	−0.430916	−0.0273082
$250$	0	0
$251$	8.06733	0.509205	0.254603	−	0.967046i	$-0.418055\pi$
$251$	8.06733	0.509205	0.254603	+	0.967046i	$0.418055\pi$
$252$	0	0
$253$	−33.9797	−2.13629
$254$	0	0
$255$	−1.36271	−0.0853360
$256$	0	0
$257$	−29.2808	−1.82649	−0.913243	−	0.407415i	$-0.866430\pi$
$257$	−29.2808	−1.82649	−0.913243	+	0.407415i	$0.866430\pi$
$258$	0	0
$259$	−11.9755	−0.744122
$260$	0	0
$261$	9.42850	0.583609
$262$	0	0
$263$	−8.04621	−0.496151	−0.248075	−	0.968741i	$-0.579798\pi$
$263$	−8.04621	−0.496151	−0.248075	+	0.968741i	$0.579798\pi$
$264$	0	0
$265$	−10.9929	−0.675287
$266$	0	0
$267$	−18.3674	−1.12407
$268$	0	0
$269$	−9.85856	−0.601087	−0.300543	−	0.953768i	$-0.597168\pi$
$269$	−9.85856	−0.601087	−0.300543	+	0.953768i	$0.597168\pi$
$270$	0	0
$271$	−31.5828	−1.91852	−0.959258	−	0.282530i	$-0.908826\pi$
$271$	−31.5828	−1.91852	−0.959258	+	0.282530i	$0.908826\pi$
$272$	0	0
$273$	6.92007	0.418822
$274$	0	0
$275$	6.11470	0.368730
$276$	0	0
$277$	12.8986	0.775000	0.387500	−	0.921870i	$-0.373339\pi$
$277$	12.8986	0.775000	0.387500	+	0.921870i	$0.373339\pi$
$278$	0	0
$279$	−0.853132	−0.0510757
$280$	0	0
$281$	−14.0115	−0.835855	−0.417927	−	0.908480i	$-0.637243\pi$
$281$	−14.0115	−0.835855	−0.417927	+	0.908480i	$0.637243\pi$
$282$	0	0
$283$	−24.2273	−1.44016	−0.720080	−	0.693891i	$-0.755895\pi$
$283$	−24.2273	−1.44016	−0.720080	+	0.693891i	$0.755895\pi$
$284$	0	0
$285$	−4.20852	−0.249291
$286$	0	0
$287$	−5.47592	−0.323234
$288$	0	0
$289$	−15.1430	−0.890766
$290$	0	0
$291$	9.36000	0.548693
$292$	0	0
$293$	21.0313	1.22866	0.614330	−	0.789049i	$-0.289426\pi$
$293$	21.0313	1.22866	0.614330	+	0.789049i	$0.289426\pi$
$294$	0	0
$295$	−4.21022	−0.245129
$296$	0	0
$297$	6.11470	0.354811
$298$	0	0
$299$	−22.8386	−1.32079
$300$	0	0
$301$	−15.2958	−0.881637
$302$	0	0
$303$	8.85114	0.508485
$304$	0	0
$305$	4.10814	0.235232
$306$	0	0
$307$	−25.0428	−1.42927	−0.714633	−	0.699499i	$-0.753406\pi$
$307$	−25.0428	−1.42927	−0.714633	+	0.699499i	$0.753406\pi$
$308$	0	0
$309$	−5.18461	−0.294942
$310$	0	0
$311$	18.4378	1.04551	0.522757	−	0.852482i	$-0.324903\pi$
$311$	18.4378	1.04551	0.522757	+	0.852482i	$0.324903\pi$
$312$	0	0
$313$	12.3318	0.697034	0.348517	−	0.937302i	$-0.386685\pi$
$313$	12.3318	0.697034	0.348517	+	0.937302i	$0.386685\pi$
$314$	0	0
$315$	−1.68378	−0.0948702
$316$	0	0
$317$	−23.9455	−1.34491	−0.672457	−	0.740136i	$-0.734761\pi$
$317$	−23.9455	−1.34491	−0.672457	+	0.740136i	$0.734761\pi$
$318$	0	0
$319$	57.6524	3.22791
$320$	0	0
$321$	3.49497	0.195070
$322$	0	0
$323$	5.73498	0.319103
$324$	0	0
$325$	4.10984	0.227973
$326$	0	0
$327$	−15.4223	−0.852857
$328$	0	0
$329$	−5.76180	−0.317658
$330$	0	0
$331$	−2.71526	−0.149244	−0.0746220	−	0.997212i	$-0.523775\pi$
$331$	−2.71526	−0.149244	−0.0746220	+	0.997212i	$0.523775\pi$
$332$	0	0
$333$	−7.11228	−0.389750
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−29.7986	−1.62323	−0.811617	−	0.584190i	$-0.801412\pi$
$337$	−29.7986	−1.62323	−0.811617	+	0.584190i	$0.801412\pi$
$338$	0	0
$339$	−11.0139	−0.598193
$340$	0	0
$341$	−5.21664	−0.282497
$342$	0	0
$343$	−18.7992	−1.01506
$344$	0	0
$345$	5.55706	0.299182
$346$	0	0
$347$	34.8173	1.86909	0.934545	−	0.355844i	$-0.115807\pi$
$347$	34.8173	1.86909	0.934545	+	0.355844i	$0.115807\pi$
$348$	0	0
$349$	11.1377	0.596187	0.298094	−	0.954537i	$-0.403649\pi$
$349$	11.1377	0.596187	0.298094	+	0.954537i	$0.403649\pi$
$350$	0	0
$351$	4.10984	0.219367
$352$	0	0
$353$	3.44529	0.183374	0.0916871	−	0.995788i	$-0.470774\pi$
$353$	3.44529	0.183374	0.0916871	+	0.995788i	$0.470774\pi$
$354$	0	0
$355$	−7.68408	−0.407829
$356$	0	0
$357$	2.29450	0.121438
$358$	0	0
$359$	−16.3305	−0.861888	−0.430944	−	0.902379i	$-0.641819\pi$
$359$	−16.3305	−0.861888	−0.430944	+	0.902379i	$0.641819\pi$
$360$	0	0
$361$	−1.28833	−0.0678067
$362$	0	0
$363$	26.3895	1.38509
$364$	0	0
$365$	15.3490	0.803402
$366$	0	0
$367$	2.28479	0.119265	0.0596326	−	0.998220i	$-0.481007\pi$
$367$	2.28479	0.119265	0.0596326	+	0.998220i	$0.481007\pi$
$368$	0	0
$369$	−3.25216	−0.169301
$370$	0	0
$371$	18.5096	0.960969
$372$	0	0
$373$	6.59108	0.341273	0.170637	−	0.985334i	$-0.445418\pi$
$373$	6.59108	0.341273	0.170637	+	0.985334i	$0.445418\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	38.7496	1.99571
$378$	0	0
$379$	−27.0678	−1.39038	−0.695190	−	0.718826i	$-0.744680\pi$
$379$	−27.0678	−1.39038	−0.695190	+	0.718826i	$0.744680\pi$
$380$	0	0
$381$	−0.635592	−0.0325624
$382$	0	0
$383$	−27.3917	−1.39965	−0.699826	−	0.714313i	$-0.746739\pi$
$383$	−27.3917	−1.39965	−0.699826	+	0.714313i	$0.746739\pi$
$384$	0	0
$385$	−10.2958	−0.524723
$386$	0	0
$387$	−9.08422	−0.461777
$388$	0	0
$389$	24.7548	1.25512	0.627560	−	0.778568i	$-0.284054\pi$
$389$	24.7548	1.25512	0.627560	+	0.778568i	$0.284054\pi$
$390$	0	0
$391$	−7.57264	−0.382965
$392$	0	0
$393$	8.44054	0.425769
$394$	0	0
$395$	−16.1254	−0.811357
$396$	0	0
$397$	−6.09854	−0.306077	−0.153038	−	0.988220i	$-0.548906\pi$
$397$	−6.09854	−0.306077	−0.153038	+	0.988220i	$0.548906\pi$
$398$	0	0
$399$	7.08623	0.354755
$400$	0	0
$401$	26.5654	1.32661	0.663307	−	0.748348i	$-0.269152\pi$
$401$	26.5654	1.32661	0.663307	+	0.748348i	$0.269152\pi$
$402$	0	0
$403$	−3.50624	−0.174658
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−43.4894	−2.15569
$408$	0	0
$409$	11.9795	0.592349	0.296175	−	0.955134i	$-0.404289\pi$
$409$	11.9795	0.592349	0.296175	+	0.955134i	$0.404289\pi$
$410$	0	0
$411$	−0.877703	−0.0432939
$412$	0	0
$413$	7.08909	0.348831
$414$	0	0
$415$	0.430916	0.0211528
$416$	0	0
$417$	1.86595	0.0913760
$418$	0	0
$419$	−16.8849	−0.824880	−0.412440	−	0.910985i	$-0.635323\pi$
$419$	−16.8849	−0.824880	−0.412440	+	0.910985i	$0.635323\pi$
$420$	0	0
$421$	18.4114	0.897315	0.448657	−	0.893704i	$-0.351902\pi$
$421$	18.4114	0.897315	0.448657	+	0.893704i	$0.351902\pi$
$422$	0	0
$423$	−3.42194	−0.166381
$424$	0	0
$425$	1.36271	0.0661010
$426$	0	0
$427$	−6.91721	−0.334747
$428$	0	0
$429$	25.1304	1.21331
$430$	0	0
$431$	8.15777	0.392946	0.196473	−	0.980509i	$-0.437051\pi$
$431$	8.15777	0.392946	0.196473	+	0.980509i	$0.437051\pi$
$432$	0	0
$433$	14.9239	0.717198	0.358599	−	0.933492i	$-0.383254\pi$
$433$	14.9239	0.717198	0.358599	+	0.933492i	$0.383254\pi$
$434$	0	0
$435$	−9.42850	−0.452062
$436$	0	0
$437$	−23.3870	−1.11875
$438$	0	0
$439$	−26.3883	−1.25944	−0.629722	−	0.776820i	$-0.716831\pi$
$439$	−26.3883	−1.25944	−0.629722	+	0.776820i	$0.716831\pi$
$440$	0	0
$441$	−4.16489	−0.198328
$442$	0	0
$443$	7.38673	0.350954	0.175477	−	0.984484i	$-0.443853\pi$
$443$	7.38673	0.350954	0.175477	+	0.984484i	$0.443853\pi$
$444$	0	0
$445$	18.3674	0.870700
$446$	0	0
$447$	−7.00258	−0.331211
$448$	0	0
$449$	1.25576	0.0592628	0.0296314	−	0.999561i	$-0.490567\pi$
$449$	1.25576	0.0592628	0.0296314	+	0.999561i	$0.490567\pi$
$450$	0	0
$451$	−19.8860	−0.936394
$452$	0	0
$453$	−4.48460	−0.210705
$454$	0	0
$455$	−6.92007	−0.324418
$456$	0	0
$457$	8.57465	0.401105	0.200553	−	0.979683i	$-0.435726\pi$
$457$	8.57465	0.401105	0.200553	+	0.979683i	$0.435726\pi$
$458$	0	0
$459$	1.36271	0.0636057
$460$	0	0
$461$	−37.1178	−1.72875	−0.864374	−	0.502849i	$-0.832285\pi$
$461$	−37.1178	−1.72875	−0.864374	+	0.502849i	$0.832285\pi$
$462$	0	0
$463$	−21.5094	−0.999629	−0.499814	−	0.866133i	$-0.666598\pi$
$463$	−21.5094	−0.999629	−0.499814	+	0.866133i	$0.666598\pi$
$464$	0	0
$465$	0.853132	0.0395630
$466$	0	0
$467$	11.9981	0.555208	0.277604	−	0.960696i	$-0.410460\pi$
$467$	11.9981	0.555208	0.277604	+	0.960696i	$0.410460\pi$
$468$	0	0
$469$	1.68378	0.0777497
$470$	0	0
$471$	11.7486	0.541345
$472$	0	0
$473$	−55.5473	−2.55407
$474$	0	0
$475$	4.20852	0.193100
$476$	0	0
$477$	10.9929	0.503329
$478$	0	0
$479$	22.7998	1.04175	0.520874	−	0.853634i	$-0.325606\pi$
$479$	22.7998	1.04175	0.520874	+	0.853634i	$0.325606\pi$
$480$	0	0
$481$	−29.2303	−1.33279
$482$	0	0
$483$	−9.35686	−0.425752
$484$	0	0
$485$	−9.36000	−0.425016
$486$	0	0
$487$	1.65582	0.0750324	0.0375162	−	0.999296i	$-0.488055\pi$
$487$	1.65582	0.0750324	0.0375162	+	0.999296i	$0.488055\pi$
$488$	0	0
$489$	−2.88301	−0.130374
$490$	0	0
$491$	17.0234	0.768253	0.384127	−	0.923280i	$-0.374503\pi$
$491$	17.0234	0.768253	0.384127	+	0.923280i	$0.374503\pi$
$492$	0	0
$493$	12.8483	0.578657
$494$	0	0
$495$	−6.11470	−0.274835
$496$	0	0
$497$	12.9383	0.580362
$498$	0	0
$499$	−2.00688	−0.0898405	−0.0449202	−	0.998991i	$-0.514303\pi$
$499$	−2.00688	−0.0898405	−0.0449202	+	0.998991i	$0.514303\pi$
$500$	0	0
$501$	22.8217	1.01960
$502$	0	0
$503$	6.16523	0.274894	0.137447	−	0.990509i	$-0.456110\pi$
$503$	6.16523	0.274894	0.137447	+	0.990509i	$0.456110\pi$
$504$	0	0
$505$	−8.85114	−0.393871
$506$	0	0
$507$	3.89081	0.172797
$508$	0	0
$509$	−10.5360	−0.467002	−0.233501	−	0.972357i	$-0.575018\pi$
$509$	−10.5360	−0.467002	−0.233501	+	0.972357i	$0.575018\pi$
$510$	0	0
$511$	−25.8443	−1.14328
$512$	0	0
$513$	4.20852	0.185811
$514$	0	0
$515$	5.18461	0.228461
$516$	0	0
$517$	−20.9241	−0.920243
$518$	0	0
$519$	11.5810	0.508349
$520$	0	0
$521$	−16.4522	−0.720782	−0.360391	−	0.932801i	$-0.617357\pi$
$521$	−16.4522	−0.720782	−0.360391	+	0.932801i	$0.617357\pi$
$522$	0	0
$523$	−26.7513	−1.16975	−0.584877	−	0.811122i	$-0.698857\pi$
$523$	−26.7513	−1.16975	−0.584877	+	0.811122i	$0.698857\pi$
$524$	0	0
$525$	1.68378	0.0734862
$526$	0	0
$527$	−1.16257	−0.0506423
$528$	0	0
$529$	7.88092	0.342649
$530$	0	0
$531$	4.21022	0.182708
$532$	0	0
$533$	−13.3659	−0.578940
$534$	0	0
$535$	−3.49497	−0.151101
$536$	0	0
$537$	12.7037	0.548204
$538$	0	0
$539$	−25.4670	−1.09694
$540$	0	0
$541$	43.7572	1.88127	0.940635	−	0.339420i	$-0.110231\pi$
$541$	43.7572	1.88127	0.940635	+	0.339420i	$0.110231\pi$
$542$	0	0
$543$	22.4511	0.963468
$544$	0	0
$545$	15.4223	0.660620
$546$	0	0
$547$	40.6700	1.73892	0.869462	−	0.494000i	$-0.164466\pi$
$547$	40.6700	1.73892	0.869462	+	0.494000i	$0.164466\pi$
$548$	0	0
$549$	−4.10814	−0.175331
$550$	0	0
$551$	39.6801	1.69043
$552$	0	0
$553$	27.1516	1.15460
$554$	0	0
$555$	7.11228	0.301899
$556$	0	0
$557$	−16.4695	−0.697836	−0.348918	−	0.937153i	$-0.613451\pi$
$557$	−16.4695	−0.697836	−0.348918	+	0.937153i	$0.613451\pi$
$558$	0	0
$559$	−37.3347	−1.57909
$560$	0	0
$561$	8.33254	0.351800
$562$	0	0
$563$	−37.7901	−1.59266	−0.796331	−	0.604861i	$-0.793229\pi$
$563$	−37.7901	−1.59266	−0.796331	+	0.604861i	$0.793229\pi$
$564$	0	0
$565$	11.0139	0.463358
$566$	0	0
$567$	1.68378	0.0707121
$568$	0	0
$569$	26.0225	1.09092	0.545460	−	0.838137i	$-0.316355\pi$
$569$	26.0225	1.09092	0.545460	+	0.838137i	$0.316355\pi$
$570$	0	0
$571$	−10.1738	−0.425759	−0.212879	−	0.977078i	$-0.568284\pi$
$571$	−10.1738	−0.425759	−0.212879	+	0.977078i	$0.568284\pi$
$572$	0	0
$573$	−24.4540	−1.02158
$574$	0	0
$575$	−5.55706	−0.231745
$576$	0	0
$577$	28.9914	1.20693	0.603464	−	0.797390i	$-0.293787\pi$
$577$	28.9914	1.20693	0.603464	+	0.797390i	$0.293787\pi$
$578$	0	0
$579$	−12.8874	−0.535582
$580$	0	0
$581$	−0.725567	−0.0301016
$582$	0	0
$583$	67.2181	2.78389
$584$	0	0
$585$	−4.10984	−0.169921
$586$	0	0
$587$	−9.84516	−0.406353	−0.203177	−	0.979142i	$-0.565127\pi$
$587$	−9.84516	−0.406353	−0.203177	+	0.979142i	$0.565127\pi$
$588$	0	0
$589$	−3.59042	−0.147941
$590$	0	0
$591$	23.2419	0.956045
$592$	0	0
$593$	−24.9890	−1.02618	−0.513088	−	0.858336i	$-0.671498\pi$
$593$	−24.9890	−1.02618	−0.513088	+	0.858336i	$0.671498\pi$
$594$	0	0
$595$	−2.29450	−0.0940652
$596$	0	0
$597$	−0.0452949	−0.00185380
$598$	0	0
$599$	−29.3564	−1.19947	−0.599734	−	0.800199i	$-0.704727\pi$
$599$	−29.3564	−1.19947	−0.599734	+	0.800199i	$0.704727\pi$
$600$	0	0
$601$	−6.68161	−0.272548	−0.136274	−	0.990671i	$-0.543513\pi$
$601$	−6.68161	−0.272548	−0.136274	+	0.990671i	$0.543513\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	−26.3895	−1.07289
$606$	0	0
$607$	−33.2561	−1.34982	−0.674911	−	0.737899i	$-0.735818\pi$
$607$	−33.2561	−1.34982	−0.674911	+	0.737899i	$0.735818\pi$
$608$	0	0
$609$	15.8755	0.643308
$610$	0	0
$611$	−14.0637	−0.568954
$612$	0	0
$613$	−7.71775	−0.311717	−0.155859	−	0.987779i	$-0.549814\pi$
$613$	−7.71775	−0.311717	−0.155859	+	0.987779i	$0.549814\pi$
$614$	0	0
$615$	3.25216	0.131140
$616$	0	0
$617$	41.7140	1.67934	0.839671	−	0.543096i	$-0.182748\pi$
$617$	41.7140	1.67934	0.839671	+	0.543096i	$0.182748\pi$
$618$	0	0
$619$	1.62916	0.0654815	0.0327407	−	0.999464i	$-0.489576\pi$
$619$	1.62916	0.0654815	0.0327407	+	0.999464i	$0.489576\pi$
$620$	0	0
$621$	−5.55706	−0.222997
$622$	0	0
$623$	−30.9267	−1.23905
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	25.7338	1.02771
$628$	0	0
$629$	−9.69195	−0.386443
$630$	0	0
$631$	13.8009	0.549405	0.274703	−	0.961529i	$-0.411421\pi$
$631$	13.8009	0.549405	0.274703	+	0.961529i	$0.411421\pi$
$632$	0	0
$633$	−27.5553	−1.09523
$634$	0	0
$635$	0.635592	0.0252227
$636$	0	0
$637$	−17.1170	−0.678201
$638$	0	0
$639$	7.68408	0.303978
$640$	0	0
$641$	14.7711	0.583422	0.291711	−	0.956507i	$-0.405775\pi$
$641$	14.7711	0.583422	0.291711	+	0.956507i	$0.405775\pi$
$642$	0	0
$643$	21.3977	0.843842	0.421921	−	0.906633i	$-0.361356\pi$
$643$	21.3977	0.843842	0.421921	+	0.906633i	$0.361356\pi$
$644$	0	0
$645$	9.08422	0.357691
$646$	0	0
$647$	34.0591	1.33900	0.669501	−	0.742811i	$-0.266508\pi$
$647$	34.0591	1.33900	0.669501	+	0.742811i	$0.266508\pi$
$648$	0	0
$649$	25.7442	1.01055
$650$	0	0
$651$	−1.43649	−0.0563003
$652$	0	0
$653$	−12.3883	−0.484792	−0.242396	−	0.970177i	$-0.577933\pi$
$653$	−12.3883	−0.484792	−0.242396	+	0.970177i	$0.577933\pi$
$654$	0	0
$655$	−8.44054	−0.329799
$656$	0	0
$657$	−15.3490	−0.598821
$658$	0	0
$659$	30.6626	1.19445	0.597223	−	0.802075i	$-0.296271\pi$
$659$	30.6626	1.19445	0.597223	+	0.802075i	$0.296271\pi$
$660$	0	0
$661$	−15.0667	−0.586028	−0.293014	−	0.956108i	$-0.594658\pi$
$661$	−15.0667	−0.586028	−0.293014	+	0.956108i	$0.594658\pi$
$662$	0	0
$663$	5.60051	0.217506
$664$	0	0
$665$	−7.08623	−0.274792
$666$	0	0
$667$	−52.3947	−2.02873
$668$	0	0
$669$	−0.793666	−0.0306849
$670$	0	0
$671$	−25.1200	−0.969748
$672$	0	0
$673$	−36.6315	−1.41204	−0.706021	−	0.708191i	$-0.749512\pi$
$673$	−36.6315	−1.41204	−0.706021	+	0.708191i	$0.749512\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	45.7094	1.75676	0.878378	−	0.477967i	$-0.158626\pi$
$677$	45.7094	1.75676	0.878378	+	0.477967i	$0.158626\pi$
$678$	0	0
$679$	15.7602	0.604820
$680$	0	0
$681$	11.7905	0.451812
$682$	0	0
$683$	−19.2871	−0.738001	−0.369001	−	0.929429i	$-0.620300\pi$
$683$	−19.2871	−0.738001	−0.369001	+	0.929429i	$0.620300\pi$
$684$	0	0
$685$	0.877703	0.0335353
$686$	0	0
$687$	26.1956	0.999424
$688$	0	0
$689$	45.1790	1.72118
$690$	0	0
$691$	−11.9656	−0.455192	−0.227596	−	0.973756i	$-0.573087\pi$
$691$	−11.9656	−0.455192	−0.227596	+	0.973756i	$0.573087\pi$
$692$	0	0
$693$	10.2958	0.391105
$694$	0	0
$695$	−1.86595	−0.0707796
$696$	0	0
$697$	−4.43174	−0.167864
$698$	0	0
$699$	−11.0205	−0.416832
$700$	0	0
$701$	−51.5056	−1.94534	−0.972670	−	0.232193i	$-0.925410\pi$
$701$	−51.5056	−1.94534	−0.972670	+	0.232193i	$0.925410\pi$
$702$	0	0
$703$	−29.9322	−1.12891
$704$	0	0
$705$	3.42194	0.128878
$706$	0	0
$707$	14.9034	0.560499
$708$	0	0
$709$	−47.7096	−1.79177	−0.895887	−	0.444283i	$-0.853459\pi$
$709$	−47.7096	−1.79177	−0.895887	+	0.444283i	$0.853459\pi$
$710$	0	0
$711$	16.1254	0.604750
$712$	0	0
$713$	4.74090	0.177548
$714$	0	0
$715$	−25.1304	−0.939825
$716$	0	0
$717$	10.6728	0.398583
$718$	0	0
$719$	28.8600	1.07630	0.538148	−	0.842850i	$-0.319124\pi$
$719$	28.8600	1.07630	0.538148	+	0.842850i	$0.319124\pi$
$720$	0	0
$721$	−8.72973	−0.325112
$722$	0	0
$723$	14.1618	0.526684
$724$	0	0
$725$	9.42850	0.350166
$726$	0	0
$727$	11.2483	0.417176	0.208588	−	0.978004i	$-0.433113\pi$
$727$	11.2483	0.417176	0.208588	+	0.978004i	$0.433113\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.3791	−0.457859
$732$	0	0
$733$	3.76604	0.139102	0.0695509	−	0.997578i	$-0.477843\pi$
$733$	3.76604	0.139102	0.0695509	+	0.997578i	$0.477843\pi$
$734$	0	0
$735$	4.16489	0.153624
$736$	0	0
$737$	6.11470	0.225238
$738$	0	0
$739$	20.2231	0.743920	0.371960	−	0.928249i	$-0.378686\pi$
$739$	20.2231	0.743920	0.371960	+	0.928249i	$0.378686\pi$
$740$	0	0
$741$	17.2964	0.635398
$742$	0	0
$743$	−28.7791	−1.05580	−0.527901	−	0.849306i	$-0.677021\pi$
$743$	−28.7791	−1.05580	−0.527901	+	0.849306i	$0.677021\pi$
$744$	0	0
$745$	7.00258	0.256555
$746$	0	0
$747$	−0.430916	−0.0157664
$748$	0	0
$749$	5.88477	0.215025
$750$	0	0
$751$	41.7062	1.52188	0.760941	−	0.648821i	$-0.224738\pi$
$751$	41.7062	1.52188	0.760941	+	0.648821i	$0.224738\pi$
$752$	0	0
$753$	8.06733	0.293990
$754$	0	0
$755$	4.48460	0.163211
$756$	0	0
$757$	19.2242	0.698715	0.349357	−	0.936990i	$-0.386400\pi$
$757$	19.2242	0.698715	0.349357	+	0.936990i	$0.386400\pi$
$758$	0	0
$759$	−33.9797	−1.23339
$760$	0	0
$761$	29.2522	1.06039	0.530196	−	0.847875i	$-0.322118\pi$
$761$	29.2522	1.06039	0.530196	+	0.847875i	$0.322118\pi$
$762$	0	0
$763$	−25.9678	−0.940098
$764$	0	0
$765$	−1.36271	−0.0492688
$766$	0	0
$767$	17.3034	0.624788
$768$	0	0
$769$	−32.9051	−1.18659	−0.593294	−	0.804986i	$-0.702173\pi$
$769$	−32.9051	−1.18659	−0.593294	+	0.804986i	$0.702173\pi$
$770$	0	0
$771$	−29.2808	−1.05452
$772$	0	0
$773$	23.6443	0.850427	0.425214	−	0.905093i	$-0.360199\pi$
$773$	23.6443	0.850427	0.425214	+	0.905093i	$0.360199\pi$
$774$	0	0
$775$	−0.853132	−0.0306454
$776$	0	0
$777$	−11.9755	−0.429619
$778$	0	0
$779$	−13.6868	−0.490380
$780$	0	0
$781$	46.9858	1.68128
$782$	0	0
$783$	9.42850	0.336947
$784$	0	0
$785$	−11.7486	−0.419324
$786$	0	0
$787$	−44.7904	−1.59661	−0.798303	−	0.602256i	$-0.794269\pi$
$787$	−44.7904	−1.59661	−0.798303	+	0.602256i	$0.794269\pi$
$788$	0	0
$789$	−8.04621	−0.286453
$790$	0	0
$791$	−18.5450	−0.659383
$792$	0	0
$793$	−16.8838	−0.599562
$794$	0	0
$795$	−10.9929	−0.389877
$796$	0	0
$797$	−9.85097	−0.348939	−0.174470	−	0.984663i	$-0.555821\pi$
$797$	−9.85097	−0.348939	−0.174470	+	0.984663i	$0.555821\pi$
$798$	0	0
$799$	−4.66311	−0.164969
$800$	0	0
$801$	−18.3674	−0.648981
$802$	0	0
$803$	−93.8543	−3.31205
$804$	0	0
$805$	9.35686	0.329786
$806$	0	0
$807$	−9.85856	−0.347038
$808$	0	0
$809$	29.4972	1.03707	0.518533	−	0.855058i	$-0.326479\pi$
$809$	29.4972	1.03707	0.518533	+	0.855058i	$0.326479\pi$
$810$	0	0
$811$	−27.8819	−0.979065	−0.489533	−	0.871985i	$-0.662833\pi$
$811$	−27.8819	−0.979065	−0.489533	+	0.871985i	$0.662833\pi$
$812$	0	0
$813$	−31.5828	−1.10766
$814$	0	0
$815$	2.88301	0.100988
$816$	0	0
$817$	−38.2312	−1.33754
$818$	0	0
$819$	6.92007	0.241807
$820$	0	0
$821$	−33.8707	−1.18210	−0.591048	−	0.806636i	$-0.701286\pi$
$821$	−33.8707	−1.18210	−0.591048	+	0.806636i	$0.701286\pi$
$822$	0	0
$823$	−40.7410	−1.42014	−0.710071	−	0.704130i	$-0.751337\pi$
$823$	−40.7410	−1.42014	−0.710071	+	0.704130i	$0.751337\pi$
$824$	0	0
$825$	6.11470	0.212886
$826$	0	0
$827$	−43.4203	−1.50987	−0.754936	−	0.655798i	$-0.772332\pi$
$827$	−43.4203	−1.50987	−0.754936	+	0.655798i	$0.772332\pi$
$828$	0	0
$829$	−26.1871	−0.909514	−0.454757	−	0.890616i	$-0.650274\pi$
$829$	−26.1871	−0.909514	−0.454757	+	0.890616i	$0.650274\pi$
$830$	0	0
$831$	12.8986	0.447446
$832$	0	0
$833$	−5.67552	−0.196645
$834$	0	0
$835$	−22.8217	−0.789777
$836$	0	0
$837$	−0.853132	−0.0294885
$838$	0	0
$839$	3.93389	0.135813	0.0679066	−	0.997692i	$-0.478368\pi$
$839$	3.93389	0.135813	0.0679066	+	0.997692i	$0.478368\pi$
$840$	0	0
$841$	59.8966	2.06540
$842$	0	0
$843$	−14.0115	−0.482581
$844$	0	0
$845$	−3.89081	−0.133848
$846$	0	0
$847$	44.4341	1.52677
$848$	0	0
$849$	−24.2273	−0.831477
$850$	0	0
$851$	39.5234	1.35484
$852$	0	0
$853$	−39.1703	−1.34116	−0.670582	−	0.741835i	$-0.733956\pi$
$853$	−39.1703	−1.34116	−0.670582	+	0.741835i	$0.733956\pi$
$854$	0	0
$855$	−4.20852	−0.143928
$856$	0	0
$857$	22.3914	0.764875	0.382437	−	0.923981i	$-0.375085\pi$
$857$	22.3914	0.764875	0.382437	+	0.923981i	$0.375085\pi$
$858$	0	0
$859$	−7.62901	−0.260299	−0.130149	−	0.991494i	$-0.541546\pi$
$859$	−7.62901	−0.260299	−0.130149	+	0.991494i	$0.541546\pi$
$860$	0	0
$861$	−5.47592	−0.186619
$862$	0	0
$863$	−40.8949	−1.39208	−0.696040	−	0.718003i	$-0.745056\pi$
$863$	−40.8949	−1.39208	−0.696040	+	0.718003i	$0.745056\pi$
$864$	0	0
$865$	−11.5810	−0.393766
$866$	0	0
$867$	−15.1430	−0.514284
$868$	0	0
$869$	98.6019	3.34484
$870$	0	0
$871$	4.10984	0.139257
$872$	0	0
$873$	9.36000	0.316788
$874$	0	0
$875$	−1.68378	−0.0569221
$876$	0	0
$877$	−12.8913	−0.435310	−0.217655	−	0.976026i	$-0.569841\pi$
$877$	−12.8913	−0.435310	−0.217655	+	0.976026i	$0.569841\pi$
$878$	0	0
$879$	21.0313	0.709368
$880$	0	0
$881$	31.1496	1.04946	0.524729	−	0.851270i	$-0.324167\pi$
$881$	31.1496	1.04946	0.524729	+	0.851270i	$0.324167\pi$
$882$	0	0
$883$	19.7880	0.665918	0.332959	−	0.942941i	$-0.391953\pi$
$883$	19.7880	0.665918	0.332959	+	0.942941i	$0.391953\pi$
$884$	0	0
$885$	−4.21022	−0.141525
$886$	0	0
$887$	2.01518	0.0676630	0.0338315	−	0.999428i	$-0.489229\pi$
$887$	2.01518	0.0676630	0.0338315	+	0.999428i	$0.489229\pi$
$888$	0	0
$889$	−1.07020	−0.0358933
$890$	0	0
$891$	6.11470	0.204850
$892$	0	0
$893$	−14.4013	−0.481922
$894$	0	0
$895$	−12.7037	−0.424637
$896$	0	0
$897$	−22.8386	−0.762560
$898$	0	0
$899$	−8.04375	−0.268274
$900$	0	0
$901$	14.9801	0.499058
$902$	0	0
$903$	−15.2958	−0.509013
$904$	0	0
$905$	−22.4511	−0.746299
$906$	0	0
$907$	−10.9777	−0.364509	−0.182254	−	0.983251i	$-0.558339\pi$
$907$	−10.9777	−0.364509	−0.182254	+	0.983251i	$0.558339\pi$
$908$	0	0
$909$	8.85114	0.293574
$910$	0	0
$911$	14.8279	0.491269	0.245634	−	0.969363i	$-0.421004\pi$
$911$	14.8279	0.491269	0.245634	+	0.969363i	$0.421004\pi$
$912$	0	0
$913$	−2.63492	−0.0872031
$914$	0	0
$915$	4.10814	0.135811
$916$	0	0
$917$	14.2120	0.469322
$918$	0	0
$919$	−14.8482	−0.489796	−0.244898	−	0.969549i	$-0.578755\pi$
$919$	−14.8482	−0.489796	−0.244898	+	0.969549i	$0.578755\pi$
$920$	0	0
$921$	−25.0428	−0.825188
$922$	0	0
$923$	31.5804	1.03948
$924$	0	0
$925$	−7.11228	−0.233850
$926$	0	0
$927$	−5.18461	−0.170285
$928$	0	0
$929$	42.1896	1.38420	0.692099	−	0.721803i	$-0.256686\pi$
$929$	42.1896	1.38420	0.692099	+	0.721803i	$0.256686\pi$
$930$	0	0
$931$	−17.5280	−0.574458
$932$	0	0
$933$	18.4378	0.603628
$934$	0	0
$935$	−8.33254	−0.272503
$936$	0	0
$937$	57.9334	1.89260	0.946301	−	0.323287i	$-0.104788\pi$
$937$	57.9334	1.89260	0.946301	+	0.323287i	$0.104788\pi$
$938$	0	0
$939$	12.3318	0.402433
$940$	0	0
$941$	17.9369	0.584726	0.292363	−	0.956307i	$-0.405558\pi$
$941$	17.9369	0.584726	0.292363	+	0.956307i	$0.405558\pi$
$942$	0	0
$943$	18.0725	0.588520
$944$	0	0
$945$	−1.68378	−0.0547734
$946$	0	0
$947$	−2.65908	−0.0864085	−0.0432043	−	0.999066i	$-0.513757\pi$
$947$	−2.65908	−0.0864085	−0.0432043	+	0.999066i	$0.513757\pi$
$948$	0	0
$949$	−63.0819	−2.04773
$950$	0	0
$951$	−23.9455	−0.776487
$952$	0	0
$953$	−18.0509	−0.584725	−0.292363	−	0.956308i	$-0.594441\pi$
$953$	−18.0509	−0.584725	−0.292363	+	0.956308i	$0.594441\pi$
$954$	0	0
$955$	24.4540	0.791313
$956$	0	0
$957$	57.6524	1.86364
$958$	0	0
$959$	−1.47786	−0.0477225
$960$	0	0
$961$	−30.2722	−0.976521
$962$	0	0
$963$	3.49497	0.112624
$964$	0	0
$965$	12.8874	0.414860
$966$	0	0
$967$	10.1659	0.326915	0.163457	−	0.986550i	$-0.447735\pi$
$967$	10.1659	0.326915	0.163457	+	0.986550i	$0.447735\pi$
$968$	0	0
$969$	5.73498	0.184234
$970$	0	0
$971$	−13.1310	−0.421395	−0.210697	−	0.977551i	$-0.567573\pi$
$971$	−13.1310	−0.421395	−0.210697	+	0.977551i	$0.567573\pi$
$972$	0	0
$973$	3.14185	0.100723
$974$	0	0
$975$	4.10984	0.131620
$976$	0	0
$977$	−34.7252	−1.11096	−0.555478	−	0.831531i	$-0.687465\pi$
$977$	−34.7252	−1.11096	−0.555478	+	0.831531i	$0.687465\pi$
$978$	0	0
$979$	−112.311	−3.58948
$980$	0	0
$981$	−15.4223	−0.492397
$982$	0	0
$983$	45.3175	1.44540	0.722702	−	0.691160i	$-0.242900\pi$
$983$	45.3175	1.44540	0.722702	+	0.691160i	$0.242900\pi$
$984$	0	0
$985$	−23.2419	−0.740549
$986$	0	0
$987$	−5.76180	−0.183400
$988$	0	0
$989$	50.4816	1.60522
$990$	0	0
$991$	−29.5548	−0.938839	−0.469419	−	0.882975i	$-0.655537\pi$
$991$	−29.5548	−0.938839	−0.469419	+	0.882975i	$0.655537\pi$
$992$	0	0
$993$	−2.71526	−0.0861661
$994$	0	0
$995$	0.0452949	0.00143594
$996$	0	0
$997$	−18.9837	−0.601220	−0.300610	−	0.953747i	$-0.597190\pi$
$997$	−18.9837	−0.601220	−0.300610	+	0.953747i	$0.597190\pi$
$998$	0	0
$999$	−7.11228	−0.225022

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.g.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.g.1.4	✓	6	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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.68378 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.68378 q^{7} +1.00000 q^{9} +6.11470 q^{11} +4.10984 q^{13} -1.00000 q^{15} +1.36271 q^{17} +4.20852 q^{19} +1.68378 q^{21} -5.55706 q^{23} +1.00000 q^{25} +1.00000 q^{27} +9.42850 q^{29} -0.853132 q^{31} +6.11470 q^{33} -1.68378 q^{35} -7.11228 q^{37} +4.10984 q^{39} -3.25216 q^{41} -9.08422 q^{43} -1.00000 q^{45} -3.42194 q^{47} -4.16489 q^{49} +1.36271 q^{51} +10.9929 q^{53} -6.11470 q^{55} +4.20852 q^{57} +4.21022 q^{59} -4.10814 q^{61} +1.68378 q^{63} -4.10984 q^{65} +1.00000 q^{67} -5.55706 q^{69} +7.68408 q^{71} -15.3490 q^{73} +1.00000 q^{75} +10.2958 q^{77} +16.1254 q^{79} +1.00000 q^{81} -0.430916 q^{83} -1.36271 q^{85} +9.42850 q^{87} -18.3674 q^{89} +6.92007 q^{91} -0.853132 q^{93} -4.20852 q^{95} +9.36000 q^{97} +6.11470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9	\( 6 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9} + 3 q^{11} + 3 q^{13} - 6 q^{15} + 6 q^{17} - 3 q^{19} + 3 q^{21} + 6 q^{23} + 6 q^{25} + 6 q^{27} + 21 q^{29} - 3 q^{31} + 3 q^{33} - 3 q^{35} + 3 q^{39} + 9 q^{41} - 3 q^{43} - 6 q^{45} + 21 q^{47} + 3 q^{49} + 6 q^{51} + 15 q^{53} - 3 q^{55} - 3 q^{57} + 15 q^{59} + 15 q^{61} + 3 q^{63} - 3 q^{65} + 6 q^{67} + 6 q^{69} + 18 q^{71} - 6 q^{73} + 6 q^{75} + 33 q^{77} + 9 q^{79} + 6 q^{81} + 24 q^{83} - 6 q^{85} + 21 q^{87} + 24 q^{89} - 21 q^{91} - 3 q^{93} + 3 q^{95} + 9 q^{97} + 3 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9 + 3 * q^11 + 3 * q^13 - 6 * q^15 + 6 * q^17 - 3 * q^19 + 3 * q^21 + 6 * q^23 + 6 * q^25 + 6 * q^27 + 21 * q^29 - 3 * q^31 + 3 * q^33 - 3 * q^35 + 3 * q^39 + 9 * q^41 - 3 * q^43 - 6 * q^45 + 21 * q^47 + 3 * q^49 + 6 * q^51 + 15 * q^53 - 3 * q^55 - 3 * q^57 + 15 * q^59 + 15 * q^61 + 3 * q^63 - 3 * q^65 + 6 * q^67 + 6 * q^69 + 18 * q^71 - 6 * q^73 + 6 * q^75 + 33 * q^77 + 9 * q^79 + 6 * q^81 + 24 * q^83 - 6 * q^85 + 21 * q^87 + 24 * q^89 - 21 * q^91 - 3 * q^93 + 3 * q^95 + 9 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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