LMFDB - Embedded newform 4020.2.a.i.1.6

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 21x^{5} - 12x^{4} + 93x^{3} + 18x^{2} - 120x + 48 $ x^7 - 21x^5 - 12x^4 + 93x^3 + 18x^2 - 120*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.67614$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +1.75949 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +1.75949 q^{7} +1.00000 q^{9} -2.83692 q^{11} +4.33587 q^{13} +1.00000 q^{15} +6.47973 q^{17} +2.33943 q^{19} +1.75949 q^{21} -2.17363 q^{23} +1.00000 q^{25} +1.00000 q^{27} +5.93312 q^{29} -10.2846 q^{31} -2.83692 q^{33} +1.75949 q^{35} +7.52591 q^{37} +4.33587 q^{39} +9.89919 q^{41} -10.9145 q^{43} +1.00000 q^{45} -7.41430 q^{47} -3.90421 q^{49} +6.47973 q^{51} -4.31811 q^{53} -2.83692 q^{55} +2.33943 q^{57} +5.22845 q^{59} +8.15142 q^{61} +1.75949 q^{63} +4.33587 q^{65} -1.00000 q^{67} -2.17363 q^{69} +7.49665 q^{71} -11.6035 q^{73} +1.00000 q^{75} -4.99152 q^{77} -0.855299 q^{79} +1.00000 q^{81} +2.27485 q^{83} +6.47973 q^{85} +5.93312 q^{87} +7.22280 q^{89} +7.62890 q^{91} -10.2846 q^{93} +2.33943 q^{95} -15.3746 q^{97} -2.83692 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9	$ 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9} + 5 q^{11} + 9 q^{13} + 7 q^{15} + 11 q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 7 q^{25} + 7 q^{27} + 8 q^{29} + 9 q^{31} + 5 q^{33} + q^{35} + 7 q^{37} + 9 q^{39} + 9 q^{41} + 5 q^{43} + 7 q^{45} + 2 q^{47} + 12 q^{49} + 11 q^{51} + 15 q^{53} + 5 q^{55} + 2 q^{57} + 11 q^{61} + q^{63} + 9 q^{65} - 7 q^{67} + 7 q^{69} + 18 q^{71} + 21 q^{73} + 7 q^{75} + 5 q^{77} + 5 q^{79} + 7 q^{81} + 6 q^{83} + 11 q^{85} + 8 q^{87} + 7 q^{89} - 9 q^{91} + 9 q^{93} + 2 q^{95} - 9 q^{97} + 5 q^{99}+O(q^{100}) $ 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9 + 5 * q^11 + 9 * q^13 + 7 * q^15 + 11 * q^17 + 2 * q^19 + q^21 + 7 * q^23 + 7 * q^25 + 7 * q^27 + 8 * q^29 + 9 * q^31 + 5 * q^33 + q^35 + 7 * q^37 + 9 * q^39 + 9 * q^41 + 5 * q^43 + 7 * q^45 + 2 * q^47 + 12 * q^49 + 11 * q^51 + 15 * q^53 + 5 * q^55 + 2 * q^57 + 11 * q^61 + q^63 + 9 * q^65 - 7 * q^67 + 7 * q^69 + 18 * q^71 + 21 * q^73 + 7 * q^75 + 5 * q^77 + 5 * q^79 + 7 * q^81 + 6 * q^83 + 11 * q^85 + 8 * q^87 + 7 * q^89 - 9 * q^91 + 9 * q^93 + 2 * q^95 - 9 * q^97 + 5 * 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
$7$	1.75949	0.665023	0.332512	−	0.943099i	$-0.392104\pi$
$7$	1.75949	0.665023	0.332512	+	0.943099i	$0.392104\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.83692	−0.855364	−0.427682	−	0.903929i	$-0.640670\pi$
$11$	−2.83692	−0.855364	−0.427682	+	0.903929i	$0.640670\pi$
$12$	0	0
$13$	4.33587	1.20255	0.601277	−	0.799041i	$-0.294659\pi$
$13$	4.33587	1.20255	0.601277	+	0.799041i	$0.294659\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	6.47973	1.57156	0.785782	−	0.618503i	$-0.212261\pi$
$17$	6.47973	1.57156	0.785782	+	0.618503i	$0.212261\pi$
$18$	0	0
$19$	2.33943	0.536703	0.268351	−	0.963321i	$-0.413521\pi$
$19$	2.33943	0.536703	0.268351	+	0.963321i	$0.413521\pi$
$20$	0	0
$21$	1.75949	0.383951
$22$	0	0
$23$	−2.17363	−0.453233	−0.226617	−	0.973984i	$-0.572766\pi$
$23$	−2.17363	−0.453233	−0.226617	+	0.973984i	$0.572766\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.93312	1.10175	0.550876	−	0.834587i	$-0.314294\pi$
$29$	5.93312	1.10175	0.550876	+	0.834587i	$0.314294\pi$
$30$	0	0
$31$	−10.2846	−1.84716	−0.923581	−	0.383404i	$-0.874752\pi$
$31$	−10.2846	−1.84716	−0.923581	+	0.383404i	$0.874752\pi$
$32$	0	0
$33$	−2.83692	−0.493845
$34$	0	0
$35$	1.75949	0.297408
$36$	0	0
$37$	7.52591	1.23725	0.618626	−	0.785685i	$-0.287690\pi$
$37$	7.52591	1.23725	0.618626	+	0.785685i	$0.287690\pi$
$38$	0	0
$39$	4.33587	0.694295
$40$	0	0
$41$	9.89919	1.54599	0.772996	−	0.634410i	$-0.218757\pi$
$41$	9.89919	1.54599	0.772996	+	0.634410i	$0.218757\pi$
$42$	0	0
$43$	−10.9145	−1.66445	−0.832224	−	0.554440i	$-0.812933\pi$
$43$	−10.9145	−1.66445	−0.832224	+	0.554440i	$0.812933\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−7.41430	−1.08149	−0.540744	−	0.841187i	$-0.681857\pi$
$47$	−7.41430	−1.08149	−0.540744	+	0.841187i	$0.681857\pi$
$48$	0	0
$49$	−3.90421	−0.557744
$50$	0	0
$51$	6.47973	0.907343
$52$	0	0
$53$	−4.31811	−0.593138	−0.296569	−	0.955011i	$-0.595842\pi$
$53$	−4.31811	−0.593138	−0.296569	+	0.955011i	$0.595842\pi$
$54$	0	0
$55$	−2.83692	−0.382530
$56$	0	0
$57$	2.33943	0.309865
$58$	0	0
$59$	5.22845	0.680686	0.340343	−	0.940301i	$-0.389457\pi$
$59$	5.22845	0.680686	0.340343	+	0.940301i	$0.389457\pi$
$60$	0	0
$61$	8.15142	1.04368	0.521841	−	0.853043i	$-0.325245\pi$
$61$	8.15142	1.04368	0.521841	+	0.853043i	$0.325245\pi$
$62$	0	0
$63$	1.75949	0.221674
$64$	0	0
$65$	4.33587	0.537798
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	−2.17363	−0.261674
$70$	0	0
$71$	7.49665	0.889688	0.444844	−	0.895608i	$-0.353259\pi$
$71$	7.49665	0.889688	0.444844	+	0.895608i	$0.353259\pi$
$72$	0	0
$73$	−11.6035	−1.35809	−0.679044	−	0.734098i	$-0.737605\pi$
$73$	−11.6035	−1.35809	−0.679044	+	0.734098i	$0.737605\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−4.99152	−0.568837
$78$	0	0
$79$	−0.855299	−0.0962286	−0.0481143	−	0.998842i	$-0.515321\pi$
$79$	−0.855299	−0.0962286	−0.0481143	+	0.998842i	$0.515321\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.27485	0.249697	0.124849	−	0.992176i	$-0.460156\pi$
$83$	2.27485	0.249697	0.124849	+	0.992176i	$0.460156\pi$
$84$	0	0
$85$	6.47973	0.702825
$86$	0	0
$87$	5.93312	0.636097
$88$	0	0
$89$	7.22280	0.765615	0.382808	−	0.923828i	$-0.374957\pi$
$89$	7.22280	0.765615	0.382808	+	0.923828i	$0.374957\pi$
$90$	0	0
$91$	7.62890	0.799726
$92$	0	0
$93$	−10.2846	−1.06646
$94$	0	0
$95$	2.33943	0.240021
$96$	0	0
$97$	−15.3746	−1.56105	−0.780527	−	0.625122i	$-0.785049\pi$
$97$	−15.3746	−1.56105	−0.780527	+	0.625122i	$0.785049\pi$
$98$	0	0
$99$	−2.83692	−0.285121
$100$	0	0
$101$	6.57379	0.654117	0.327058	−	0.945004i	$-0.393943\pi$
$101$	6.57379	0.654117	0.327058	+	0.945004i	$0.393943\pi$
$102$	0	0
$103$	16.1001	1.58639	0.793195	−	0.608967i	$-0.208416\pi$
$103$	16.1001	1.58639	0.793195	+	0.608967i	$0.208416\pi$
$104$	0	0
$105$	1.75949	0.171708
$106$	0	0
$107$	0.749720	0.0724782	0.0362391	−	0.999343i	$-0.488462\pi$
$107$	0.749720	0.0724782	0.0362391	+	0.999343i	$0.488462\pi$
$108$	0	0
$109$	0.116296	0.0111392	0.00556959	−	0.999984i	$-0.498227\pi$
$109$	0.116296	0.0111392	0.00556959	+	0.999984i	$0.498227\pi$
$110$	0	0
$111$	7.52591	0.714328
$112$	0	0
$113$	15.1661	1.42671	0.713355	−	0.700803i	$-0.247175\pi$
$113$	15.1661	1.42671	0.713355	+	0.700803i	$0.247175\pi$
$114$	0	0
$115$	−2.17363	−0.202692
$116$	0	0
$117$	4.33587	0.400851
$118$	0	0
$119$	11.4010	1.04513
$120$	0	0
$121$	−2.95188	−0.268353
$122$	0	0
$123$	9.89919	0.892579
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.0405	−1.42336	−0.711681	−	0.702503i	$-0.752066\pi$
$127$	−16.0405	−1.42336	−0.711681	+	0.702503i	$0.752066\pi$
$128$	0	0
$129$	−10.9145	−0.960969
$130$	0	0
$131$	3.40070	0.297121	0.148560	−	0.988903i	$-0.452536\pi$
$131$	3.40070	0.297121	0.148560	+	0.988903i	$0.452536\pi$
$132$	0	0
$133$	4.11620	0.356920
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	4.74247	0.405177	0.202588	−	0.979264i	$-0.435065\pi$
$137$	4.74247	0.405177	0.202588	+	0.979264i	$0.435065\pi$
$138$	0	0
$139$	6.17578	0.523823	0.261912	−	0.965092i	$-0.415647\pi$
$139$	6.17578	0.523823	0.261912	+	0.965092i	$0.415647\pi$
$140$	0	0
$141$	−7.41430	−0.624397
$142$	0	0
$143$	−12.3005	−1.02862
$144$	0	0
$145$	5.93312	0.492719
$146$	0	0
$147$	−3.90421	−0.322014
$148$	0	0
$149$	−5.26774	−0.431550	−0.215775	−	0.976443i	$-0.569228\pi$
$149$	−5.26774	−0.431550	−0.215775	+	0.976443i	$0.569228\pi$
$150$	0	0
$151$	2.64901	0.215573	0.107787	−	0.994174i	$-0.465624\pi$
$151$	2.64901	0.215573	0.107787	+	0.994174i	$0.465624\pi$
$152$	0	0
$153$	6.47973	0.523855
$154$	0	0
$155$	−10.2846	−0.826076
$156$	0	0
$157$	17.3094	1.38144	0.690721	−	0.723121i	$-0.257293\pi$
$157$	17.3094	1.38144	0.690721	+	0.723121i	$0.257293\pi$
$158$	0	0
$159$	−4.31811	−0.342448
$160$	0	0
$161$	−3.82447	−0.301411
$162$	0	0
$163$	25.2582	1.97838	0.989189	−	0.146649i	$-0.0468487\pi$
$163$	25.2582	1.97838	0.989189	+	0.146649i	$0.0468487\pi$
$164$	0	0
$165$	−2.83692	−0.220854
$166$	0	0
$167$	0.214443	0.0165941	0.00829705	−	0.999966i	$-0.497359\pi$
$167$	0.214443	0.0165941	0.00829705	+	0.999966i	$0.497359\pi$
$168$	0	0
$169$	5.79975	0.446135
$170$	0	0
$171$	2.33943	0.178901
$172$	0	0
$173$	−3.43038	−0.260807	−0.130404	−	0.991461i	$-0.541627\pi$
$173$	−3.43038	−0.260807	−0.130404	+	0.991461i	$0.541627\pi$
$174$	0	0
$175$	1.75949	0.133005
$176$	0	0
$177$	5.22845	0.392994
$178$	0	0
$179$	5.46601	0.408549	0.204275	−	0.978914i	$-0.434516\pi$
$179$	5.46601	0.408549	0.204275	+	0.978914i	$0.434516\pi$
$180$	0	0
$181$	−7.45209	−0.553909	−0.276955	−	0.960883i	$-0.589325\pi$
$181$	−7.45209	−0.553909	−0.276955	+	0.960883i	$0.589325\pi$
$182$	0	0
$183$	8.15142	0.602570
$184$	0	0
$185$	7.52591	0.553316
$186$	0	0
$187$	−18.3825	−1.34426
$188$	0	0
$189$	1.75949	0.127984
$190$	0	0
$191$	−14.0037	−1.01327	−0.506635	−	0.862161i	$-0.669111\pi$
$191$	−14.0037	−1.01327	−0.506635	+	0.862161i	$0.669111\pi$
$192$	0	0
$193$	−4.33713	−0.312194	−0.156097	−	0.987742i	$-0.549891\pi$
$193$	−4.33713	−0.312194	−0.156097	+	0.987742i	$0.549891\pi$
$194$	0	0
$195$	4.33587	0.310498
$196$	0	0
$197$	−7.54149	−0.537309	−0.268654	−	0.963237i	$-0.586579\pi$
$197$	−7.54149	−0.537309	−0.268654	+	0.963237i	$0.586579\pi$
$198$	0	0
$199$	10.0259	0.710716	0.355358	−	0.934730i	$-0.384359\pi$
$199$	10.0259	0.710716	0.355358	+	0.934730i	$0.384359\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	10.4392	0.732691
$204$	0	0
$205$	9.89919	0.691389
$206$	0	0
$207$	−2.17363	−0.151078
$208$	0	0
$209$	−6.63678	−0.459076
$210$	0	0
$211$	−11.8497	−0.815766	−0.407883	−	0.913034i	$-0.633733\pi$
$211$	−11.8497	−0.815766	−0.407883	+	0.913034i	$0.633733\pi$
$212$	0	0
$213$	7.49665	0.513662
$214$	0	0
$215$	−10.9145	−0.744364
$216$	0	0
$217$	−18.0955	−1.22841
$218$	0	0
$219$	−11.6035	−0.784092
$220$	0	0
$221$	28.0952	1.88989
$222$	0	0
$223$	−7.59727	−0.508751	−0.254375	−	0.967106i	$-0.581870\pi$
$223$	−7.59727	−0.508751	−0.254375	+	0.967106i	$0.581870\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−6.83851	−0.453888	−0.226944	−	0.973908i	$-0.572873\pi$
$227$	−6.83851	−0.453888	−0.226944	+	0.973908i	$0.572873\pi$
$228$	0	0
$229$	−1.97023	−0.130196	−0.0650982	−	0.997879i	$-0.520736\pi$
$229$	−1.97023	−0.130196	−0.0650982	+	0.997879i	$0.520736\pi$
$230$	0	0
$231$	−4.99152	−0.328418
$232$	0	0
$233$	−5.48742	−0.359493	−0.179747	−	0.983713i	$-0.557528\pi$
$233$	−5.48742	−0.359493	−0.179747	+	0.983713i	$0.557528\pi$
$234$	0	0
$235$	−7.41430	−0.483656
$236$	0	0
$237$	−0.855299	−0.0555576
$238$	0	0
$239$	−9.82995	−0.635847	−0.317923	−	0.948116i	$-0.602985\pi$
$239$	−9.82995	−0.635847	−0.317923	+	0.948116i	$0.602985\pi$
$240$	0	0
$241$	−26.2878	−1.69335	−0.846674	−	0.532111i	$-0.821399\pi$
$241$	−26.2878	−1.69335	−0.846674	+	0.532111i	$0.821399\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.90421	−0.249431
$246$	0	0
$247$	10.1435	0.645413
$248$	0	0
$249$	2.27485	0.144163
$250$	0	0
$251$	−20.8357	−1.31514	−0.657569	−	0.753394i	$-0.728415\pi$
$251$	−20.8357	−1.31514	−0.657569	+	0.753394i	$0.728415\pi$
$252$	0	0
$253$	6.16642	0.387679
$254$	0	0
$255$	6.47973	0.405776
$256$	0	0
$257$	23.5239	1.46738	0.733690	−	0.679484i	$-0.237796\pi$
$257$	23.5239	1.46738	0.733690	+	0.679484i	$0.237796\pi$
$258$	0	0
$259$	13.2417	0.822802
$260$	0	0
$261$	5.93312	0.367251
$262$	0	0
$263$	16.7341	1.03187	0.515935	−	0.856628i	$-0.327445\pi$
$263$	16.7341	1.03187	0.515935	+	0.856628i	$0.327445\pi$
$264$	0	0
$265$	−4.31811	−0.265259
$266$	0	0
$267$	7.22280	0.442028
$268$	0	0
$269$	−26.3670	−1.60763	−0.803813	−	0.594882i	$-0.797199\pi$
$269$	−26.3670	−1.60763	−0.803813	+	0.594882i	$0.797199\pi$
$270$	0	0
$271$	26.5928	1.61540	0.807700	−	0.589594i	$-0.200712\pi$
$271$	26.5928	1.61540	0.807700	+	0.589594i	$0.200712\pi$
$272$	0	0
$273$	7.62890	0.461722
$274$	0	0
$275$	−2.83692	−0.171073
$276$	0	0
$277$	−32.2764	−1.93930	−0.969651	−	0.244495i	$-0.921378\pi$
$277$	−32.2764	−1.93930	−0.969651	+	0.244495i	$0.921378\pi$
$278$	0	0
$279$	−10.2846	−0.615720
$280$	0	0
$281$	8.92253	0.532274	0.266137	−	0.963935i	$-0.414253\pi$
$281$	8.92253	0.532274	0.266137	+	0.963935i	$0.414253\pi$
$282$	0	0
$283$	7.65190	0.454858	0.227429	−	0.973795i	$-0.426968\pi$
$283$	7.65190	0.454858	0.227429	+	0.973795i	$0.426968\pi$
$284$	0	0
$285$	2.33943	0.138576
$286$	0	0
$287$	17.4175	1.02812
$288$	0	0
$289$	24.9869	1.46982
$290$	0	0
$291$	−15.3746	−0.901275
$292$	0	0
$293$	−14.6909	−0.858250	−0.429125	−	0.903245i	$-0.641178\pi$
$293$	−14.6909	−0.858250	−0.429125	+	0.903245i	$0.641178\pi$
$294$	0	0
$295$	5.22845	0.304412
$296$	0	0
$297$	−2.83692	−0.164615
$298$	0	0
$299$	−9.42457	−0.545037
$300$	0	0
$301$	−19.2039	−1.10690
$302$	0	0
$303$	6.57379	0.377654
$304$	0	0
$305$	8.15142	0.466749
$306$	0	0
$307$	2.92871	0.167150	0.0835752	−	0.996501i	$-0.473366\pi$
$307$	2.92871	0.167150	0.0835752	+	0.996501i	$0.473366\pi$
$308$	0	0
$309$	16.1001	0.915903
$310$	0	0
$311$	11.3044	0.641015	0.320508	−	0.947246i	$-0.396147\pi$
$311$	11.3044	0.641015	0.320508	+	0.947246i	$0.396147\pi$
$312$	0	0
$313$	25.8647	1.46196	0.730980	−	0.682399i	$-0.239063\pi$
$313$	25.8647	1.46196	0.730980	+	0.682399i	$0.239063\pi$
$314$	0	0
$315$	1.75949	0.0991358
$316$	0	0
$317$	10.8793	0.611043	0.305522	−	0.952185i	$-0.401169\pi$
$317$	10.8793	0.611043	0.305522	+	0.952185i	$0.401169\pi$
$318$	0	0
$319$	−16.8318	−0.942399
$320$	0	0
$321$	0.749720	0.0418453
$322$	0	0
$323$	15.1589	0.843463
$324$	0	0
$325$	4.33587	0.240511
$326$	0	0
$327$	0.116296	0.00643121
$328$	0	0
$329$	−13.0454	−0.719214
$330$	0	0
$331$	12.0540	0.662545	0.331273	−	0.943535i	$-0.392522\pi$
$331$	12.0540	0.662545	0.331273	+	0.943535i	$0.392522\pi$
$332$	0	0
$333$	7.52591	0.412417
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	15.4304	0.840550	0.420275	−	0.907397i	$-0.361934\pi$
$337$	15.4304	0.840550	0.420275	+	0.907397i	$0.361934\pi$
$338$	0	0
$339$	15.1661	0.823711
$340$	0	0
$341$	29.1765	1.58000
$342$	0	0
$343$	−19.1858	−1.03594
$344$	0	0
$345$	−2.17363	−0.117024
$346$	0	0
$347$	12.3072	0.660684	0.330342	−	0.943861i	$-0.392836\pi$
$347$	12.3072	0.660684	0.330342	+	0.943861i	$0.392836\pi$
$348$	0	0
$349$	−0.371187	−0.0198692	−0.00993458	−	0.999951i	$-0.503162\pi$
$349$	−0.371187	−0.0198692	−0.00993458	+	0.999951i	$0.503162\pi$
$350$	0	0
$351$	4.33587	0.231432
$352$	0	0
$353$	19.7859	1.05310	0.526548	−	0.850145i	$-0.323486\pi$
$353$	19.7859	1.05310	0.526548	+	0.850145i	$0.323486\pi$
$354$	0	0
$355$	7.49665	0.397881
$356$	0	0
$357$	11.4010	0.603405
$358$	0	0
$359$	19.0947	1.00778	0.503889	−	0.863768i	$-0.331902\pi$
$359$	19.0947	1.00778	0.503889	+	0.863768i	$0.331902\pi$
$360$	0	0
$361$	−13.5271	−0.711950
$362$	0	0
$363$	−2.95188	−0.154933
$364$	0	0
$365$	−11.6035	−0.607355
$366$	0	0
$367$	−4.73017	−0.246913	−0.123457	−	0.992350i	$-0.539398\pi$
$367$	−4.73017	−0.246913	−0.123457	+	0.992350i	$0.539398\pi$
$368$	0	0
$369$	9.89919	0.515331
$370$	0	0
$371$	−7.59765	−0.394450
$372$	0	0
$373$	−20.4780	−1.06031	−0.530156	−	0.847900i	$-0.677867\pi$
$373$	−20.4780	−1.06031	−0.530156	+	0.847900i	$0.677867\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	25.7252	1.32492
$378$	0	0
$379$	−5.76637	−0.296198	−0.148099	−	0.988973i	$-0.547316\pi$
$379$	−5.76637	−0.296198	−0.148099	+	0.988973i	$0.547316\pi$
$380$	0	0
$381$	−16.0405	−0.821779
$382$	0	0
$383$	13.1544	0.672159	0.336080	−	0.941834i	$-0.390899\pi$
$383$	13.1544	0.672159	0.336080	+	0.941834i	$0.390899\pi$
$384$	0	0
$385$	−4.99152	−0.254392
$386$	0	0
$387$	−10.9145	−0.554816
$388$	0	0
$389$	−0.456309	−0.0231358	−0.0115679	−	0.999933i	$-0.503682\pi$
$389$	−0.456309	−0.0231358	−0.0115679	+	0.999933i	$0.503682\pi$
$390$	0	0
$391$	−14.0845	−0.712285
$392$	0	0
$393$	3.40070	0.171543
$394$	0	0
$395$	−0.855299	−0.0430347
$396$	0	0
$397$	−23.9311	−1.20107	−0.600535	−	0.799599i	$-0.705046\pi$
$397$	−23.9311	−1.20107	−0.600535	+	0.799599i	$0.705046\pi$
$398$	0	0
$399$	4.11620	0.206068
$400$	0	0
$401$	−36.1828	−1.80688	−0.903440	−	0.428714i	$-0.858967\pi$
$401$	−36.1828	−1.80688	−0.903440	+	0.428714i	$0.858967\pi$
$402$	0	0
$403$	−44.5925	−2.22131
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−21.3504	−1.05830
$408$	0	0
$409$	26.4624	1.30848	0.654240	−	0.756287i	$-0.272989\pi$
$409$	26.4624	1.30848	0.654240	+	0.756287i	$0.272989\pi$
$410$	0	0
$411$	4.74247	0.233929
$412$	0	0
$413$	9.19939	0.452672
$414$	0	0
$415$	2.27485	0.111668
$416$	0	0
$417$	6.17578	0.302429
$418$	0	0
$419$	−7.67533	−0.374964	−0.187482	−	0.982268i	$-0.560033\pi$
$419$	−7.67533	−0.374964	−0.187482	+	0.982268i	$0.560033\pi$
$420$	0	0
$421$	−3.81761	−0.186059	−0.0930296	−	0.995663i	$-0.529655\pi$
$421$	−3.81761	−0.186059	−0.0930296	+	0.995663i	$0.529655\pi$
$422$	0	0
$423$	−7.41430	−0.360496
$424$	0	0
$425$	6.47973	0.314313
$426$	0	0
$427$	14.3423	0.694073
$428$	0	0
$429$	−12.3005	−0.593874
$430$	0	0
$431$	21.9546	1.05751	0.528757	−	0.848773i	$-0.322658\pi$
$431$	21.9546	1.05751	0.528757	+	0.848773i	$0.322658\pi$
$432$	0	0
$433$	0.475087	0.0228312	0.0114156	−	0.999935i	$-0.496366\pi$
$433$	0.475087	0.0228312	0.0114156	+	0.999935i	$0.496366\pi$
$434$	0	0
$435$	5.93312	0.284471
$436$	0	0
$437$	−5.08506	−0.243251
$438$	0	0
$439$	−18.9532	−0.904586	−0.452293	−	0.891869i	$-0.649394\pi$
$439$	−18.9532	−0.904586	−0.452293	+	0.891869i	$0.649394\pi$
$440$	0	0
$441$	−3.90421	−0.185915
$442$	0	0
$443$	38.2460	1.81712	0.908562	−	0.417750i	$-0.137181\pi$
$443$	38.2460	1.81712	0.908562	+	0.417750i	$0.137181\pi$
$444$	0	0
$445$	7.22280	0.342394
$446$	0	0
$447$	−5.26774	−0.249156
$448$	0	0
$449$	−19.2095	−0.906552	−0.453276	−	0.891370i	$-0.649745\pi$
$449$	−19.2095	−0.906552	−0.453276	+	0.891370i	$0.649745\pi$
$450$	0	0
$451$	−28.0832	−1.32239
$452$	0	0
$453$	2.64901	0.124461
$454$	0	0
$455$	7.62890	0.357648
$456$	0	0
$457$	−15.7713	−0.737749	−0.368875	−	0.929479i	$-0.620257\pi$
$457$	−15.7713	−0.737749	−0.368875	+	0.929479i	$0.620257\pi$
$458$	0	0
$459$	6.47973	0.302448
$460$	0	0
$461$	−34.1088	−1.58861	−0.794303	−	0.607522i	$-0.792164\pi$
$461$	−34.1088	−1.58861	−0.794303	+	0.607522i	$0.792164\pi$
$462$	0	0
$463$	7.34941	0.341556	0.170778	−	0.985310i	$-0.445372\pi$
$463$	7.34941	0.341556	0.170778	+	0.985310i	$0.445372\pi$
$464$	0	0
$465$	−10.2846	−0.476935
$466$	0	0
$467$	−34.3294	−1.58857	−0.794287	−	0.607543i	$-0.792155\pi$
$467$	−34.3294	−1.58857	−0.794287	+	0.607543i	$0.792155\pi$
$468$	0	0
$469$	−1.75949	−0.0812455
$470$	0	0
$471$	17.3094	0.797576
$472$	0	0
$473$	30.9636	1.42371
$474$	0	0
$475$	2.33943	0.107341
$476$	0	0
$477$	−4.31811	−0.197713
$478$	0	0
$479$	−5.73187	−0.261896	−0.130948	−	0.991389i	$-0.541802\pi$
$479$	−5.73187	−0.261896	−0.130948	+	0.991389i	$0.541802\pi$
$480$	0	0
$481$	32.6314	1.48786
$482$	0	0
$483$	−3.82447	−0.174020
$484$	0	0
$485$	−15.3746	−0.698125
$486$	0	0
$487$	17.8107	0.807079	0.403539	−	0.914962i	$-0.367780\pi$
$487$	17.8107	0.807079	0.403539	+	0.914962i	$0.367780\pi$
$488$	0	0
$489$	25.2582	1.14222
$490$	0	0
$491$	29.3867	1.32620	0.663101	−	0.748530i	$-0.269240\pi$
$491$	29.3867	1.32620	0.663101	+	0.748530i	$0.269240\pi$
$492$	0	0
$493$	38.4450	1.73147
$494$	0	0
$495$	−2.83692	−0.127510
$496$	0	0
$497$	13.1903	0.591664
$498$	0	0
$499$	−7.41429	−0.331909	−0.165955	−	0.986133i	$-0.553071\pi$
$499$	−7.41429	−0.331909	−0.165955	+	0.986133i	$0.553071\pi$
$500$	0	0
$501$	0.214443	0.00958061
$502$	0	0
$503$	1.28305	0.0572084	0.0286042	−	0.999591i	$-0.490894\pi$
$503$	1.28305	0.0572084	0.0286042	+	0.999591i	$0.490894\pi$
$504$	0	0
$505$	6.57379	0.292530
$506$	0	0
$507$	5.79975	0.257576
$508$	0	0
$509$	−15.4978	−0.686929	−0.343464	−	0.939166i	$-0.611600\pi$
$509$	−15.4978	−0.686929	−0.343464	+	0.939166i	$0.611600\pi$
$510$	0	0
$511$	−20.4162	−0.903160
$512$	0	0
$513$	2.33943	0.103288
$514$	0	0
$515$	16.1001	0.709455
$516$	0	0
$517$	21.0338	0.925065
$518$	0	0
$519$	−3.43038	−0.150577
$520$	0	0
$521$	−4.90416	−0.214855	−0.107428	−	0.994213i	$-0.534261\pi$
$521$	−4.90416	−0.214855	−0.107428	+	0.994213i	$0.534261\pi$
$522$	0	0
$523$	−25.5558	−1.11748	−0.558739	−	0.829344i	$-0.688714\pi$
$523$	−25.5558	−1.11748	−0.558739	+	0.829344i	$0.688714\pi$
$524$	0	0
$525$	1.75949	0.0767903
$526$	0	0
$527$	−66.6411	−2.90293
$528$	0	0
$529$	−18.2753	−0.794580
$530$	0	0
$531$	5.22845	0.226895
$532$	0	0
$533$	42.9216	1.85914
$534$	0	0
$535$	0.749720	0.0324132
$536$	0	0
$537$	5.46601	0.235876
$538$	0	0
$539$	11.0759	0.477074
$540$	0	0
$541$	−13.2144	−0.568131	−0.284066	−	0.958805i	$-0.591683\pi$
$541$	−13.2144	−0.568131	−0.284066	+	0.958805i	$0.591683\pi$
$542$	0	0
$543$	−7.45209	−0.319800
$544$	0	0
$545$	0.116296	0.00498159
$546$	0	0
$547$	−34.9534	−1.49450	−0.747250	−	0.664543i	$-0.768626\pi$
$547$	−34.9534	−1.49450	−0.747250	+	0.664543i	$0.768626\pi$
$548$	0	0
$549$	8.15142	0.347894
$550$	0	0
$551$	13.8801	0.591313
$552$	0	0
$553$	−1.50489	−0.0639943
$554$	0	0
$555$	7.52591	0.319457
$556$	0	0
$557$	19.8898	0.842757	0.421378	−	0.906885i	$-0.361546\pi$
$557$	19.8898	0.842757	0.421378	+	0.906885i	$0.361546\pi$
$558$	0	0
$559$	−47.3239	−2.00159
$560$	0	0
$561$	−18.3825	−0.776109
$562$	0	0
$563$	−7.13716	−0.300796	−0.150398	−	0.988626i	$-0.548055\pi$
$563$	−7.13716	−0.300796	−0.150398	+	0.988626i	$0.548055\pi$
$564$	0	0
$565$	15.1661	0.638044
$566$	0	0
$567$	1.75949	0.0738915
$568$	0	0
$569$	−29.2985	−1.22826	−0.614129	−	0.789205i	$-0.710493\pi$
$569$	−29.2985	−1.22826	−0.614129	+	0.789205i	$0.710493\pi$
$570$	0	0
$571$	−41.4368	−1.73408	−0.867038	−	0.498243i	$-0.833979\pi$
$571$	−41.4368	−1.73408	−0.867038	+	0.498243i	$0.833979\pi$
$572$	0	0
$573$	−14.0037	−0.585012
$574$	0	0
$575$	−2.17363	−0.0906467
$576$	0	0
$577$	−26.8464	−1.11763	−0.558814	−	0.829293i	$-0.688744\pi$
$577$	−26.8464	−1.11763	−0.558814	+	0.829293i	$0.688744\pi$
$578$	0	0
$579$	−4.33713	−0.180245
$580$	0	0
$581$	4.00256	0.166054
$582$	0	0
$583$	12.2501	0.507348
$584$	0	0
$585$	4.33587	0.179266
$586$	0	0
$587$	−41.7777	−1.72435	−0.862176	−	0.506609i	$-0.830899\pi$
$587$	−41.7777	−1.72435	−0.862176	+	0.506609i	$0.830899\pi$
$588$	0	0
$589$	−24.0600	−0.991376
$590$	0	0
$591$	−7.54149	−0.310215
$592$	0	0
$593$	12.1718	0.499835	0.249917	−	0.968267i	$-0.419597\pi$
$593$	12.1718	0.499835	0.249917	+	0.968267i	$0.419597\pi$
$594$	0	0
$595$	11.4010	0.467395
$596$	0	0
$597$	10.0259	0.410332
$598$	0	0
$599$	11.7878	0.481638	0.240819	−	0.970570i	$-0.422584\pi$
$599$	11.7878	0.481638	0.240819	+	0.970570i	$0.422584\pi$
$600$	0	0
$601$	−30.8507	−1.25843	−0.629213	−	0.777233i	$-0.716623\pi$
$601$	−30.8507	−1.25843	−0.629213	+	0.777233i	$0.716623\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−2.95188	−0.120011
$606$	0	0
$607$	−29.4254	−1.19434	−0.597170	−	0.802115i	$-0.703708\pi$
$607$	−29.4254	−1.19434	−0.597170	+	0.802115i	$0.703708\pi$
$608$	0	0
$609$	10.4392	0.423019
$610$	0	0
$611$	−32.1474	−1.30055
$612$	0	0
$613$	2.52218	0.101870	0.0509350	−	0.998702i	$-0.483780\pi$
$613$	2.52218	0.101870	0.0509350	+	0.998702i	$0.483780\pi$
$614$	0	0
$615$	9.89919	0.399174
$616$	0	0
$617$	−28.2605	−1.13773	−0.568863	−	0.822433i	$-0.692616\pi$
$617$	−28.2605	−1.13773	−0.568863	+	0.822433i	$0.692616\pi$
$618$	0	0
$619$	20.9926	0.843765	0.421882	−	0.906651i	$-0.361370\pi$
$619$	20.9926	0.843765	0.421882	+	0.906651i	$0.361370\pi$
$620$	0	0
$621$	−2.17363	−0.0872248
$622$	0	0
$623$	12.7084	0.509152
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.63678	−0.265048
$628$	0	0
$629$	48.7659	1.94442
$630$	0	0
$631$	9.69159	0.385816	0.192908	−	0.981217i	$-0.438208\pi$
$631$	9.69159	0.385816	0.192908	+	0.981217i	$0.438208\pi$
$632$	0	0
$633$	−11.8497	−0.470982
$634$	0	0
$635$	−16.0405	−0.636547
$636$	0	0
$637$	−16.9281	−0.670717
$638$	0	0
$639$	7.49665	0.296563
$640$	0	0
$641$	31.0349	1.22581	0.612903	−	0.790158i	$-0.290002\pi$
$641$	31.0349	1.22581	0.612903	+	0.790158i	$0.290002\pi$
$642$	0	0
$643$	−47.1871	−1.86088	−0.930438	−	0.366450i	$-0.880573\pi$
$643$	−47.1871	−1.86088	−0.930438	+	0.366450i	$0.880573\pi$
$644$	0	0
$645$	−10.9145	−0.429759
$646$	0	0
$647$	36.1189	1.41998	0.709990	−	0.704211i	$-0.248699\pi$
$647$	36.1189	1.41998	0.709990	+	0.704211i	$0.248699\pi$
$648$	0	0
$649$	−14.8327	−0.582234
$650$	0	0
$651$	−18.0955	−0.709220
$652$	0	0
$653$	43.7380	1.71160	0.855801	−	0.517306i	$-0.173065\pi$
$653$	43.7380	1.71160	0.855801	+	0.517306i	$0.173065\pi$
$654$	0	0
$655$	3.40070	0.132876
$656$	0	0
$657$	−11.6035	−0.452696
$658$	0	0
$659$	9.33050	0.363465	0.181732	−	0.983348i	$-0.441830\pi$
$659$	9.33050	0.363465	0.181732	+	0.983348i	$0.441830\pi$
$660$	0	0
$661$	−15.0834	−0.586675	−0.293338	−	0.956009i	$-0.594766\pi$
$661$	−15.0834	−0.586675	−0.293338	+	0.956009i	$0.594766\pi$
$662$	0	0
$663$	28.0952	1.09113
$664$	0	0
$665$	4.11620	0.159619
$666$	0	0
$667$	−12.8964	−0.499351
$668$	0	0
$669$	−7.59727	−0.293727
$670$	0	0
$671$	−23.1249	−0.892728
$672$	0	0
$673$	31.8419	1.22741	0.613707	−	0.789534i	$-0.289678\pi$
$673$	31.8419	1.22741	0.613707	+	0.789534i	$0.289678\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−21.3517	−0.820612	−0.410306	−	0.911948i	$-0.634578\pi$
$677$	−21.3517	−0.820612	−0.410306	+	0.911948i	$0.634578\pi$
$678$	0	0
$679$	−27.0514	−1.03814
$680$	0	0
$681$	−6.83851	−0.262052
$682$	0	0
$683$	32.8451	1.25678	0.628392	−	0.777897i	$-0.283714\pi$
$683$	32.8451	1.25678	0.628392	+	0.777897i	$0.283714\pi$
$684$	0	0
$685$	4.74247	0.181201
$686$	0	0
$687$	−1.97023	−0.0751689
$688$	0	0
$689$	−18.7227	−0.713280
$690$	0	0
$691$	−19.8427	−0.754852	−0.377426	−	0.926040i	$-0.623191\pi$
$691$	−19.8427	−0.754852	−0.377426	+	0.926040i	$0.623191\pi$
$692$	0	0
$693$	−4.99152	−0.189612
$694$	0	0
$695$	6.17578	0.234261
$696$	0	0
$697$	64.1440	2.42963
$698$	0	0
$699$	−5.48742	−0.207553
$700$	0	0
$701$	8.76976	0.331229	0.165615	−	0.986191i	$-0.447039\pi$
$701$	8.76976	0.331229	0.165615	+	0.986191i	$0.447039\pi$
$702$	0	0
$703$	17.6064	0.664037
$704$	0	0
$705$	−7.41430	−0.279239
$706$	0	0
$707$	11.5665	0.435003
$708$	0	0
$709$	32.4479	1.21861	0.609304	−	0.792937i	$-0.291449\pi$
$709$	32.4479	1.21861	0.609304	+	0.792937i	$0.291449\pi$
$710$	0	0
$711$	−0.855299	−0.0320762
$712$	0	0
$713$	22.3548	0.837195
$714$	0	0
$715$	−12.3005	−0.460013
$716$	0	0
$717$	−9.82995	−0.367106
$718$	0	0
$719$	0.849632	0.0316859	0.0158430	−	0.999874i	$-0.494957\pi$
$719$	0.849632	0.0316859	0.0158430	+	0.999874i	$0.494957\pi$
$720$	0	0
$721$	28.3279	1.05499
$722$	0	0
$723$	−26.2878	−0.977655
$724$	0	0
$725$	5.93312	0.220350
$726$	0	0
$727$	−25.7586	−0.955333	−0.477667	−	0.878541i	$-0.658517\pi$
$727$	−25.7586	−0.955333	−0.477667	+	0.878541i	$0.658517\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−70.7231	−2.61579
$732$	0	0
$733$	−20.4130	−0.753971	−0.376986	−	0.926219i	$-0.623039\pi$
$733$	−20.4130	−0.753971	−0.376986	+	0.926219i	$0.623039\pi$
$734$	0	0
$735$	−3.90421	−0.144009
$736$	0	0
$737$	2.83692	0.104499
$738$	0	0
$739$	−0.815981	−0.0300164	−0.0150082	−	0.999887i	$-0.504777\pi$
$739$	−0.815981	−0.0300164	−0.0150082	+	0.999887i	$0.504777\pi$
$740$	0	0
$741$	10.1435	0.372630
$742$	0	0
$743$	0.792383	0.0290697	0.0145349	−	0.999894i	$-0.495373\pi$
$743$	0.792383	0.0290697	0.0145349	+	0.999894i	$0.495373\pi$
$744$	0	0
$745$	−5.26774	−0.192995
$746$	0	0
$747$	2.27485	0.0832323
$748$	0	0
$749$	1.31912	0.0481997
$750$	0	0
$751$	−1.68657	−0.0615437	−0.0307719	−	0.999526i	$-0.509797\pi$
$751$	−1.68657	−0.0615437	−0.0307719	+	0.999526i	$0.509797\pi$
$752$	0	0
$753$	−20.8357	−0.759295
$754$	0	0
$755$	2.64901	0.0964073
$756$	0	0
$757$	26.0275	0.945987	0.472994	−	0.881066i	$-0.343173\pi$
$757$	26.0275	0.945987	0.472994	+	0.881066i	$0.343173\pi$
$758$	0	0
$759$	6.16642	0.223827
$760$	0	0
$761$	13.9189	0.504559	0.252279	−	0.967654i	$-0.418820\pi$
$761$	13.9189	0.504559	0.252279	+	0.967654i	$0.418820\pi$
$762$	0	0
$763$	0.204622	0.00740781
$764$	0	0
$765$	6.47973	0.234275
$766$	0	0
$767$	22.6699	0.818561
$768$	0	0
$769$	−28.7081	−1.03524	−0.517619	−	0.855611i	$-0.673182\pi$
$769$	−28.7081	−1.03524	−0.517619	+	0.855611i	$0.673182\pi$
$770$	0	0
$771$	23.5239	0.847193
$772$	0	0
$773$	−4.36374	−0.156953	−0.0784764	−	0.996916i	$-0.525006\pi$
$773$	−4.36374	−0.156953	−0.0784764	+	0.996916i	$0.525006\pi$
$774$	0	0
$775$	−10.2846	−0.369432
$776$	0	0
$777$	13.2417	0.475045
$778$	0	0
$779$	23.1585	0.829738
$780$	0	0
$781$	−21.2674	−0.761007
$782$	0	0
$783$	5.93312	0.212032
$784$	0	0
$785$	17.3094	0.617800
$786$	0	0
$787$	40.7137	1.45129	0.725643	−	0.688072i	$-0.241543\pi$
$787$	40.7137	1.45129	0.725643	+	0.688072i	$0.241543\pi$
$788$	0	0
$789$	16.7341	0.595750
$790$	0	0
$791$	26.6846	0.948795
$792$	0	0
$793$	35.3435	1.25508
$794$	0	0
$795$	−4.31811	−0.153147
$796$	0	0
$797$	−43.5554	−1.54281	−0.771407	−	0.636343i	$-0.780446\pi$
$797$	−43.5554	−1.54281	−0.771407	+	0.636343i	$0.780446\pi$
$798$	0	0
$799$	−48.0427	−1.69963
$800$	0	0
$801$	7.22280	0.255205
$802$	0	0
$803$	32.9182	1.16166
$804$	0	0
$805$	−3.82447	−0.134795
$806$	0	0
$807$	−26.3670	−0.928163
$808$	0	0
$809$	16.3824	0.575974	0.287987	−	0.957634i	$-0.407014\pi$
$809$	16.3824	0.575974	0.287987	+	0.957634i	$0.407014\pi$
$810$	0	0
$811$	33.2545	1.16772	0.583862	−	0.811853i	$-0.301541\pi$
$811$	33.2545	1.16772	0.583862	+	0.811853i	$0.301541\pi$
$812$	0	0
$813$	26.5928	0.932652
$814$	0	0
$815$	25.2582	0.884757
$816$	0	0
$817$	−25.5338	−0.893313
$818$	0	0
$819$	7.62890	0.266575
$820$	0	0
$821$	35.5710	1.24143	0.620717	−	0.784034i	$-0.286841\pi$
$821$	35.5710	1.24143	0.620717	+	0.784034i	$0.286841\pi$
$822$	0	0
$823$	−29.8351	−1.03998	−0.519992	−	0.854171i	$-0.674065\pi$
$823$	−29.8351	−1.03998	−0.519992	+	0.854171i	$0.674065\pi$
$824$	0	0
$825$	−2.83692	−0.0987689
$826$	0	0
$827$	−29.9287	−1.04072	−0.520362	−	0.853946i	$-0.674203\pi$
$827$	−29.9287	−1.04072	−0.520362	+	0.853946i	$0.674203\pi$
$828$	0	0
$829$	3.64631	0.126642	0.0633208	−	0.997993i	$-0.479831\pi$
$829$	3.64631	0.126642	0.0633208	+	0.997993i	$0.479831\pi$
$830$	0	0
$831$	−32.2764	−1.11966
$832$	0	0
$833$	−25.2982	−0.876531
$834$	0	0
$835$	0.214443	0.00742111
$836$	0	0
$837$	−10.2846	−0.355486
$838$	0	0
$839$	23.6212	0.815493	0.407746	−	0.913095i	$-0.366315\pi$
$839$	23.6212	0.815493	0.407746	+	0.913095i	$0.366315\pi$
$840$	0	0
$841$	6.20188	0.213858
$842$	0	0
$843$	8.92253	0.307308
$844$	0	0
$845$	5.79975	0.199517
$846$	0	0
$847$	−5.19379	−0.178461
$848$	0	0
$849$	7.65190	0.262613
$850$	0	0
$851$	−16.3586	−0.560764
$852$	0	0
$853$	−50.1620	−1.71751	−0.858757	−	0.512383i	$-0.828763\pi$
$853$	−50.1620	−1.71751	−0.858757	+	0.512383i	$0.828763\pi$
$854$	0	0
$855$	2.33943	0.0800069
$856$	0	0
$857$	3.67116	0.125404	0.0627022	−	0.998032i	$-0.480028\pi$
$857$	3.67116	0.125404	0.0627022	+	0.998032i	$0.480028\pi$
$858$	0	0
$859$	14.5603	0.496790	0.248395	−	0.968659i	$-0.420097\pi$
$859$	14.5603	0.496790	0.248395	+	0.968659i	$0.420097\pi$
$860$	0	0
$861$	17.4175	0.593586
$862$	0	0
$863$	−35.2730	−1.20071	−0.600353	−	0.799735i	$-0.704973\pi$
$863$	−35.2730	−1.20071	−0.600353	+	0.799735i	$0.704973\pi$
$864$	0	0
$865$	−3.43038	−0.116636
$866$	0	0
$867$	24.9869	0.848599
$868$	0	0
$869$	2.42642	0.0823105
$870$	0	0
$871$	−4.33587	−0.146915
$872$	0	0
$873$	−15.3746	−0.520351
$874$	0	0
$875$	1.75949	0.0594815
$876$	0	0
$877$	51.3184	1.73290	0.866449	−	0.499266i	$-0.166397\pi$
$877$	51.3184	1.73290	0.866449	+	0.499266i	$0.166397\pi$
$878$	0	0
$879$	−14.6909	−0.495511
$880$	0	0
$881$	−24.6535	−0.830597	−0.415299	−	0.909685i	$-0.636323\pi$
$881$	−24.6535	−0.830597	−0.415299	+	0.909685i	$0.636323\pi$
$882$	0	0
$883$	−9.40549	−0.316520	−0.158260	−	0.987397i	$-0.550588\pi$
$883$	−9.40549	−0.316520	−0.158260	+	0.987397i	$0.550588\pi$
$884$	0	0
$885$	5.22845	0.175752
$886$	0	0
$887$	−45.8749	−1.54033	−0.770164	−	0.637846i	$-0.779826\pi$
$887$	−45.8749	−1.54033	−0.770164	+	0.637846i	$0.779826\pi$
$888$	0	0
$889$	−28.2230	−0.946569
$890$	0	0
$891$	−2.83692	−0.0950404
$892$	0	0
$893$	−17.3453	−0.580437
$894$	0	0
$895$	5.46601	0.182709
$896$	0	0
$897$	−9.42457	−0.314677
$898$	0	0
$899$	−61.0195	−2.03511
$900$	0	0
$901$	−27.9802	−0.932154
$902$	0	0
$903$	−19.2039	−0.639067
$904$	0	0
$905$	−7.45209	−0.247716
$906$	0	0
$907$	−20.0034	−0.664201	−0.332100	−	0.943244i	$-0.607757\pi$
$907$	−20.0034	−0.664201	−0.332100	+	0.943244i	$0.607757\pi$
$908$	0	0
$909$	6.57379	0.218039
$910$	0	0
$911$	30.1264	0.998131	0.499066	−	0.866564i	$-0.333677\pi$
$911$	30.1264	0.998131	0.499066	+	0.866564i	$0.333677\pi$
$912$	0	0
$913$	−6.45356	−0.213582
$914$	0	0
$915$	8.15142	0.269478
$916$	0	0
$917$	5.98349	0.197592
$918$	0	0
$919$	−9.04812	−0.298470	−0.149235	−	0.988802i	$-0.547681\pi$
$919$	−9.04812	−0.298470	−0.149235	+	0.988802i	$0.547681\pi$
$920$	0	0
$921$	2.92871	0.0965043
$922$	0	0
$923$	32.5045	1.06990
$924$	0	0
$925$	7.52591	0.247450
$926$	0	0
$927$	16.1001	0.528797
$928$	0	0
$929$	−38.8841	−1.27575	−0.637873	−	0.770141i	$-0.720186\pi$
$929$	−38.8841	−1.27575	−0.637873	+	0.770141i	$0.720186\pi$
$930$	0	0
$931$	−9.13363	−0.299343
$932$	0	0
$933$	11.3044	0.370090
$934$	0	0
$935$	−18.3825	−0.601171
$936$	0	0
$937$	−13.8620	−0.452851	−0.226425	−	0.974029i	$-0.572704\pi$
$937$	−13.8620	−0.452851	−0.226425	+	0.974029i	$0.572704\pi$
$938$	0	0
$939$	25.8647	0.844063
$940$	0	0
$941$	−22.0408	−0.718508	−0.359254	−	0.933240i	$-0.616969\pi$
$941$	−22.0408	−0.718508	−0.359254	+	0.933240i	$0.616969\pi$
$942$	0	0
$943$	−21.5172	−0.700695
$944$	0	0
$945$	1.75949	0.0572361
$946$	0	0
$947$	−14.6133	−0.474867	−0.237434	−	0.971404i	$-0.576306\pi$
$947$	−14.6133	−0.474867	−0.237434	+	0.971404i	$0.576306\pi$
$948$	0	0
$949$	−50.3113	−1.63317
$950$	0	0
$951$	10.8793	0.352786
$952$	0	0
$953$	−46.8698	−1.51826	−0.759131	−	0.650938i	$-0.774376\pi$
$953$	−46.8698	−1.51826	−0.759131	+	0.650938i	$0.774376\pi$
$954$	0	0
$955$	−14.0037	−0.453148
$956$	0	0
$957$	−16.8318	−0.544094
$958$	0	0
$959$	8.34432	0.269452
$960$	0	0
$961$	74.7721	2.41200
$962$	0	0
$963$	0.749720	0.0241594
$964$	0	0
$965$	−4.33713	−0.139617
$966$	0	0
$967$	−34.1989	−1.09976	−0.549882	−	0.835243i	$-0.685327\pi$
$967$	−34.1989	−1.09976	−0.549882	+	0.835243i	$0.685327\pi$
$968$	0	0
$969$	15.1589	0.486974
$970$	0	0
$971$	−26.7368	−0.858026	−0.429013	−	0.903298i	$-0.641139\pi$
$971$	−26.7368	−0.858026	−0.429013	+	0.903298i	$0.641139\pi$
$972$	0	0
$973$	10.8662	0.348355
$974$	0	0
$975$	4.33587	0.138859
$976$	0	0
$977$	−45.8940	−1.46828	−0.734140	−	0.678998i	$-0.762414\pi$
$977$	−45.8940	−1.46828	−0.734140	+	0.678998i	$0.762414\pi$
$978$	0	0
$979$	−20.4905	−0.654880
$980$	0	0
$981$	0.116296	0.00371306
$982$	0	0
$983$	−23.4297	−0.747293	−0.373646	−	0.927571i	$-0.621893\pi$
$983$	−23.4297	−0.747293	−0.373646	+	0.927571i	$0.621893\pi$
$984$	0	0
$985$	−7.54149	−0.240292
$986$	0	0
$987$	−13.0454	−0.415239
$988$	0	0
$989$	23.7241	0.754383
$990$	0	0
$991$	−12.1199	−0.385000	−0.192500	−	0.981297i	$-0.561660\pi$
$991$	−12.1199	−0.385000	−0.192500	+	0.981297i	$0.561660\pi$
$992$	0	0
$993$	12.0540	0.382521
$994$	0	0
$995$	10.0259	0.317842
$996$	0	0
$997$	23.8600	0.755653	0.377826	−	0.925876i	$-0.376672\pi$
$997$	23.8600	0.755653	0.377826	+	0.925876i	$0.376672\pi$
$998$	0	0
$999$	7.52591	0.238109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.i.1.6	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.i.1.6	✓	7	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.75949 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.75949 q^{7} +1.00000 q^{9} -2.83692 q^{11} +4.33587 q^{13} +1.00000 q^{15} +6.47973 q^{17} +2.33943 q^{19} +1.75949 q^{21} -2.17363 q^{23} +1.00000 q^{25} +1.00000 q^{27} +5.93312 q^{29} -10.2846 q^{31} -2.83692 q^{33} +1.75949 q^{35} +7.52591 q^{37} +4.33587 q^{39} +9.89919 q^{41} -10.9145 q^{43} +1.00000 q^{45} -7.41430 q^{47} -3.90421 q^{49} +6.47973 q^{51} -4.31811 q^{53} -2.83692 q^{55} +2.33943 q^{57} +5.22845 q^{59} +8.15142 q^{61} +1.75949 q^{63} +4.33587 q^{65} -1.00000 q^{67} -2.17363 q^{69} +7.49665 q^{71} -11.6035 q^{73} +1.00000 q^{75} -4.99152 q^{77} -0.855299 q^{79} +1.00000 q^{81} +2.27485 q^{83} +6.47973 q^{85} +5.93312 q^{87} +7.22280 q^{89} +7.62890 q^{91} -10.2846 q^{93} +2.33943 q^{95} -15.3746 q^{97} -2.83692 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9	\( 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9} + 5 q^{11} + 9 q^{13} + 7 q^{15} + 11 q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 7 q^{25} + 7 q^{27} + 8 q^{29} + 9 q^{31} + 5 q^{33} + q^{35} + 7 q^{37} + 9 q^{39} + 9 q^{41} + 5 q^{43} + 7 q^{45} + 2 q^{47} + 12 q^{49} + 11 q^{51} + 15 q^{53} + 5 q^{55} + 2 q^{57} + 11 q^{61} + q^{63} + 9 q^{65} - 7 q^{67} + 7 q^{69} + 18 q^{71} + 21 q^{73} + 7 q^{75} + 5 q^{77} + 5 q^{79} + 7 q^{81} + 6 q^{83} + 11 q^{85} + 8 q^{87} + 7 q^{89} - 9 q^{91} + 9 q^{93} + 2 q^{95} - 9 q^{97} + 5 q^{99}+O(q^{100}) \) 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9 + 5 * q^11 + 9 * q^13 + 7 * q^15 + 11 * q^17 + 2 * q^19 + q^21 + 7 * q^23 + 7 * q^25 + 7 * q^27 + 8 * q^29 + 9 * q^31 + 5 * q^33 + q^35 + 7 * q^37 + 9 * q^39 + 9 * q^41 + 5 * q^43 + 7 * q^45 + 2 * q^47 + 12 * q^49 + 11 * q^51 + 15 * q^53 + 5 * q^55 + 2 * q^57 + 11 * q^61 + q^63 + 9 * q^65 - 7 * q^67 + 7 * q^69 + 18 * q^71 + 21 * q^73 + 7 * q^75 + 5 * q^77 + 5 * q^79 + 7 * q^81 + 6 * q^83 + 11 * q^85 + 8 * q^87 + 7 * q^89 - 9 * q^91 + 9 * q^93 + 2 * q^95 - 9 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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