LMFDB - Embedded newform 4020.2.a.c.1.3

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.98117.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 4x + 1 $ x^4 - x^3 - 8x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.668475$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +0.668475 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +0.668475 q^{7} +1.00000 q^{9} -4.16442 q^{11} +5.04908 q^{13} -1.00000 q^{15} -6.22162 q^{17} -2.55314 q^{19} +0.668475 q^{21} -3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.66847 q^{29} +8.54503 q^{31} -4.16442 q^{33} -0.668475 q^{35} +4.33695 q^{37} +5.04908 q^{39} -0.446858 q^{41} -0.942801 q^{43} -1.00000 q^{45} -4.73110 q^{47} -6.55314 q^{49} -6.22162 q^{51} -9.54503 q^{53} +4.16442 q^{55} -2.55314 q^{57} -10.3752 q^{59} -10.5531 q^{61} +0.668475 q^{63} -5.04908 q^{65} -1.00000 q^{67} -3.00000 q^{69} +5.93918 q^{71} -10.5505 q^{73} +1.00000 q^{75} -2.78381 q^{77} +2.04908 q^{79} +1.00000 q^{81} +9.15088 q^{83} +6.22162 q^{85} -1.66847 q^{87} -16.1428 q^{89} +3.37519 q^{91} +8.54503 q^{93} +2.55314 q^{95} -8.14545 q^{97} -4.16442 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} - 5 q^{11} - 5 q^{13} - 4 q^{15} - 8 q^{17} + 5 q^{19} + q^{21} - 12 q^{23} + 4 q^{25} + 4 q^{27} - 5 q^{29} - q^{31} - 5 q^{33} - q^{35} + 14 q^{37} - 5 q^{39} - 17 q^{41} - 9 q^{43} - 4 q^{45} - 7 q^{47} - 11 q^{49} - 8 q^{51} - 3 q^{53} + 5 q^{55} + 5 q^{57} - 23 q^{59} - 27 q^{61} + q^{63} + 5 q^{65} - 4 q^{67} - 12 q^{69} - 20 q^{71} - 2 q^{73} + 4 q^{75} - 23 q^{77} - 17 q^{79} + 4 q^{81} + 10 q^{83} + 8 q^{85} - 5 q^{87} - 18 q^{89} - 5 q^{91} - q^{93} - 5 q^{95} - 11 q^{97} - 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 - 5 * q^11 - 5 * q^13 - 4 * q^15 - 8 * q^17 + 5 * q^19 + q^21 - 12 * q^23 + 4 * q^25 + 4 * q^27 - 5 * q^29 - q^31 - 5 * q^33 - q^35 + 14 * q^37 - 5 * q^39 - 17 * q^41 - 9 * q^43 - 4 * q^45 - 7 * q^47 - 11 * q^49 - 8 * q^51 - 3 * q^53 + 5 * q^55 + 5 * q^57 - 23 * q^59 - 27 * q^61 + q^63 + 5 * q^65 - 4 * q^67 - 12 * q^69 - 20 * q^71 - 2 * q^73 + 4 * q^75 - 23 * q^77 - 17 * q^79 + 4 * q^81 + 10 * q^83 + 8 * q^85 - 5 * q^87 - 18 * q^89 - 5 * q^91 - q^93 - 5 * q^95 - 11 * 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
$7$	0.668475	0.252660	0.126330	−	0.991988i	$-0.459680\pi$
$7$	0.668475	0.252660	0.126330	+	0.991988i	$0.459680\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.16442	−1.25562	−0.627810	−	0.778367i	$-0.716048\pi$
$11$	−4.16442	−1.25562	−0.627810	+	0.778367i	$0.716048\pi$
$12$	0	0
$13$	5.04908	1.40036	0.700182	−	0.713964i	$-0.253102\pi$
$13$	5.04908	1.40036	0.700182	+	0.713964i	$0.253102\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−6.22162	−1.50896	−0.754482	−	0.656321i	$-0.772112\pi$
$17$	−6.22162	−1.50896	−0.754482	+	0.656321i	$0.772112\pi$
$18$	0	0
$19$	−2.55314	−0.585731	−0.292865	−	0.956154i	$-0.594609\pi$
$19$	−2.55314	−0.585731	−0.292865	+	0.956154i	$0.594609\pi$
$20$	0	0
$21$	0.668475	0.145873
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.66847	−0.309828	−0.154914	−	0.987928i	$-0.549510\pi$
$29$	−1.66847	−0.309828	−0.154914	+	0.987928i	$0.549510\pi$
$30$	0	0
$31$	8.54503	1.53473	0.767366	−	0.641209i	$-0.221567\pi$
$31$	8.54503	1.53473	0.767366	+	0.641209i	$0.221567\pi$
$32$	0	0
$33$	−4.16442	−0.724932
$34$	0	0
$35$	−0.668475	−0.112993
$36$	0	0
$37$	4.33695	0.712990	0.356495	−	0.934297i	$-0.383972\pi$
$37$	4.33695	0.712990	0.356495	+	0.934297i	$0.383972\pi$
$38$	0	0
$39$	5.04908	0.808501
$40$	0	0
$41$	−0.446858	−0.0697876	−0.0348938	−	0.999391i	$-0.511109\pi$
$41$	−0.446858	−0.0697876	−0.0348938	+	0.999391i	$0.511109\pi$
$42$	0	0
$43$	−0.942801	−0.143776	−0.0718879	−	0.997413i	$-0.522902\pi$
$43$	−0.942801	−0.143776	−0.0718879	+	0.997413i	$0.522902\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−4.73110	−0.690102	−0.345051	−	0.938584i	$-0.612138\pi$
$47$	−4.73110	−0.690102	−0.345051	+	0.938584i	$0.612138\pi$
$48$	0	0
$49$	−6.55314	−0.936163
$50$	0	0
$51$	−6.22162	−0.871201
$52$	0	0
$53$	−9.54503	−1.31111	−0.655555	−	0.755147i	$-0.727565\pi$
$53$	−9.54503	−1.31111	−0.655555	+	0.755147i	$0.727565\pi$
$54$	0	0
$55$	4.16442	0.561530
$56$	0	0
$57$	−2.55314	−0.338172
$58$	0	0
$59$	−10.3752	−1.35073	−0.675367	−	0.737482i	$-0.736015\pi$
$59$	−10.3752	−1.35073	−0.675367	+	0.737482i	$0.736015\pi$
$60$	0	0
$61$	−10.5531	−1.35119	−0.675596	−	0.737272i	$-0.736113\pi$
$61$	−10.5531	−1.35119	−0.675596	+	0.737272i	$0.736113\pi$
$62$	0	0
$63$	0.668475	0.0842199
$64$	0	0
$65$	−5.04908	−0.626262
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	5.93918	0.704850	0.352425	−	0.935840i	$-0.385357\pi$
$71$	5.93918	0.704850	0.352425	+	0.935840i	$0.385357\pi$
$72$	0	0
$73$	−10.5505	−1.23484	−0.617418	−	0.786635i	$-0.711821\pi$
$73$	−10.5505	−1.23484	−0.617418	+	0.786635i	$0.711821\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−2.78381	−0.317244
$78$	0	0
$79$	2.04908	0.230540	0.115270	−	0.993334i	$-0.463227\pi$
$79$	2.04908	0.230540	0.115270	+	0.993334i	$0.463227\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.15088	1.00444	0.502220	−	0.864740i	$-0.332517\pi$
$83$	9.15088	1.00444	0.502220	+	0.864740i	$0.332517\pi$
$84$	0	0
$85$	6.22162	0.674829
$86$	0	0
$87$	−1.66847	−0.178879
$88$	0	0
$89$	−16.1428	−1.71113	−0.855565	−	0.517696i	$-0.826790\pi$
$89$	−16.1428	−1.71113	−0.855565	+	0.517696i	$0.826790\pi$
$90$	0	0
$91$	3.37519	0.353816
$92$	0	0
$93$	8.54503	0.886078
$94$	0	0
$95$	2.55314	0.261947
$96$	0	0
$97$	−8.14545	−0.827046	−0.413523	−	0.910494i	$-0.635702\pi$
$97$	−8.14545	−0.827046	−0.413523	+	0.910494i	$0.635702\pi$
$98$	0	0
$99$	−4.16442	−0.418540
$100$	0	0
$101$	17.4270	1.73405	0.867026	−	0.498263i	$-0.166029\pi$
$101$	17.4270	1.73405	0.867026	+	0.498263i	$0.166029\pi$
$102$	0	0
$103$	−0.279751	−0.0275647	−0.0137823	−	0.999905i	$-0.504387\pi$
$103$	−0.279751	−0.0275647	−0.0137823	+	0.999905i	$0.504387\pi$
$104$	0	0
$105$	−0.668475	−0.0652365
$106$	0	0
$107$	−17.3725	−1.67946	−0.839731	−	0.543002i	$-0.817288\pi$
$107$	−17.3725	−1.67946	−0.839731	+	0.543002i	$0.817288\pi$
$108$	0	0
$109$	3.15088	0.301799	0.150900	−	0.988549i	$-0.451783\pi$
$109$	3.15088	0.301799	0.150900	+	0.988549i	$0.451783\pi$
$110$	0	0
$111$	4.33695	0.411645
$112$	0	0
$113$	9.10086	0.856137	0.428068	−	0.903746i	$-0.359194\pi$
$113$	9.10086	0.856137	0.428068	+	0.903746i	$0.359194\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	5.04908	0.466788
$118$	0	0
$119$	−4.15899	−0.381254
$120$	0	0
$121$	6.34237	0.576579
$122$	0	0
$123$	−0.446858	−0.0402919
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.61128	−0.142978	−0.0714888	−	0.997441i	$-0.522775\pi$
$127$	−1.61128	−0.142978	−0.0714888	+	0.997441i	$0.522775\pi$
$128$	0	0
$129$	−0.942801	−0.0830090
$130$	0	0
$131$	15.1864	1.32684	0.663422	−	0.748245i	$-0.269103\pi$
$131$	15.1864	1.32684	0.663422	+	0.748245i	$0.269103\pi$
$132$	0	0
$133$	−1.70671	−0.147991
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−5.59680	−0.478167	−0.239084	−	0.970999i	$-0.576847\pi$
$137$	−5.59680	−0.478167	−0.239084	+	0.970999i	$0.576847\pi$
$138$	0	0
$139$	4.33426	0.367627	0.183814	−	0.982961i	$-0.441156\pi$
$139$	4.33426	0.367627	0.183814	+	0.982961i	$0.441156\pi$
$140$	0	0
$141$	−4.73110	−0.398430
$142$	0	0
$143$	−21.0265	−1.75832
$144$	0	0
$145$	1.66847	0.138559
$146$	0	0
$147$	−6.55314	−0.540494
$148$	0	0
$149$	2.88198	0.236101	0.118050	−	0.993008i	$-0.462336\pi$
$149$	2.88198	0.236101	0.118050	+	0.993008i	$0.462336\pi$
$150$	0	0
$151$	−18.9121	−1.53904	−0.769522	−	0.638620i	$-0.779506\pi$
$151$	−18.9121	−1.53904	−0.769522	+	0.638620i	$0.779506\pi$
$152$	0	0
$153$	−6.22162	−0.502988
$154$	0	0
$155$	−8.54503	−0.686353
$156$	0	0
$157$	−5.66847	−0.452394	−0.226197	−	0.974082i	$-0.572629\pi$
$157$	−5.66847	−0.452394	−0.226197	+	0.974082i	$0.572629\pi$
$158$	0	0
$159$	−9.54503	−0.756970
$160$	0	0
$161$	−2.00542	−0.158050
$162$	0	0
$163$	−1.39146	−0.108987	−0.0544937	−	0.998514i	$-0.517354\pi$
$163$	−1.39146	−0.108987	−0.0544937	+	0.998514i	$0.517354\pi$
$164$	0	0
$165$	4.16442	0.324199
$166$	0	0
$167$	−2.28244	−0.176621	−0.0883103	−	0.996093i	$-0.528147\pi$
$167$	−2.28244	−0.176621	−0.0883103	+	0.996093i	$0.528147\pi$
$168$	0	0
$169$	12.4933	0.961019
$170$	0	0
$171$	−2.55314	−0.195244
$172$	0	0
$173$	−15.8630	−1.20604	−0.603021	−	0.797725i	$-0.706037\pi$
$173$	−15.8630	−1.20604	−0.603021	+	0.797725i	$0.706037\pi$
$174$	0	0
$175$	0.668475	0.0505319
$176$	0	0
$177$	−10.3752	−0.779847
$178$	0	0
$179$	−3.62930	−0.271267	−0.135633	−	0.990759i	$-0.543307\pi$
$179$	−3.62930	−0.271267	−0.135633	+	0.990759i	$0.543307\pi$
$180$	0	0
$181$	−24.5279	−1.82314	−0.911571	−	0.411143i	$-0.865130\pi$
$181$	−24.5279	−1.82314	−0.911571	+	0.411143i	$0.865130\pi$
$182$	0	0
$183$	−10.5531	−0.780111
$184$	0	0
$185$	−4.33695	−0.318859
$186$	0	0
$187$	25.9094	1.89468
$188$	0	0
$189$	0.668475	0.0486244
$190$	0	0
$191$	25.3225	1.83227	0.916137	−	0.400866i	$-0.131291\pi$
$191$	25.3225	1.83227	0.916137	+	0.400866i	$0.131291\pi$
$192$	0	0
$193$	−21.6185	−1.55613	−0.778067	−	0.628182i	$-0.783800\pi$
$193$	−21.6185	−1.55613	−0.778067	+	0.628182i	$0.783800\pi$
$194$	0	0
$195$	−5.04908	−0.361572
$196$	0	0
$197$	24.3608	1.73563	0.867816	−	0.496886i	$-0.165523\pi$
$197$	24.3608	1.73563	0.867816	+	0.496886i	$0.165523\pi$
$198$	0	0
$199$	3.00269	0.212855	0.106428	−	0.994320i	$-0.466059\pi$
$199$	3.00269	0.212855	0.106428	+	0.994320i	$0.466059\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	−1.11533	−0.0782810
$204$	0	0
$205$	0.446858	0.0312100
$206$	0	0
$207$	−3.00000	−0.208514
$208$	0	0
$209$	10.6323	0.735455
$210$	0	0
$211$	−7.46887	−0.514178	−0.257089	−	0.966388i	$-0.582763\pi$
$211$	−7.46887	−0.514178	−0.257089	+	0.966388i	$0.582763\pi$
$212$	0	0
$213$	5.93918	0.406946
$214$	0	0
$215$	0.942801	0.0642985
$216$	0	0
$217$	5.71213	0.387765
$218$	0	0
$219$	−10.5505	−0.712933
$220$	0	0
$221$	−31.4135	−2.11310
$222$	0	0
$223$	14.8630	0.995301	0.497651	−	0.867378i	$-0.334196\pi$
$223$	14.8630	0.995301	0.497651	+	0.867378i	$0.334196\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−29.1990	−1.93801	−0.969004	−	0.247046i	$-0.920540\pi$
$227$	−29.1990	−1.93801	−0.969004	+	0.247046i	$0.920540\pi$
$228$	0	0
$229$	−26.5725	−1.75596	−0.877979	−	0.478700i	$-0.841108\pi$
$229$	−26.5725	−1.75596	−0.877979	+	0.478700i	$0.841108\pi$
$230$	0	0
$231$	−2.78381	−0.183161
$232$	0	0
$233$	10.9428	0.716887	0.358443	−	0.933551i	$-0.383308\pi$
$233$	10.9428	0.716887	0.358443	+	0.933551i	$0.383308\pi$
$234$	0	0
$235$	4.73110	0.308623
$236$	0	0
$237$	2.04908	0.133102
$238$	0	0
$239$	−2.77296	−0.179368	−0.0896839	−	0.995970i	$-0.528586\pi$
$239$	−2.77296	−0.179368	−0.0896839	+	0.995970i	$0.528586\pi$
$240$	0	0
$241$	20.1729	1.29945	0.649725	−	0.760169i	$-0.274884\pi$
$241$	20.1729	1.29945	0.649725	+	0.760169i	$0.274884\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.55314	0.418665
$246$	0	0
$247$	−12.8910	−0.820236
$248$	0	0
$249$	9.15088	0.579913
$250$	0	0
$251$	3.74464	0.236359	0.118180	−	0.992992i	$-0.462294\pi$
$251$	3.74464	0.236359	0.118180	+	0.992992i	$0.462294\pi$
$252$	0	0
$253$	12.4933	0.785444
$254$	0	0
$255$	6.22162	0.389613
$256$	0	0
$257$	15.7450	0.982146	0.491073	−	0.871119i	$-0.336605\pi$
$257$	15.7450	0.982146	0.491073	+	0.871119i	$0.336605\pi$
$258$	0	0
$259$	2.89914	0.180144
$260$	0	0
$261$	−1.66847	−0.103276
$262$	0	0
$263$	−2.56126	−0.157934	−0.0789669	−	0.996877i	$-0.525162\pi$
$263$	−2.56126	−0.157934	−0.0789669	+	0.996877i	$0.525162\pi$
$264$	0	0
$265$	9.54503	0.586346
$266$	0	0
$267$	−16.1428	−0.987921
$268$	0	0
$269$	0.123492	0.00752941	0.00376471	−	0.999993i	$-0.498802\pi$
$269$	0.123492	0.00752941	0.00376471	+	0.999993i	$0.498802\pi$
$270$	0	0
$271$	−22.4460	−1.36350	−0.681748	−	0.731587i	$-0.738780\pi$
$271$	−22.4460	−1.36350	−0.681748	+	0.731587i	$0.738780\pi$
$272$	0	0
$273$	3.37519	0.204276
$274$	0	0
$275$	−4.16442	−0.251124
$276$	0	0
$277$	−20.9220	−1.25708	−0.628541	−	0.777777i	$-0.716348\pi$
$277$	−20.9220	−1.25708	−0.628541	+	0.777777i	$0.716348\pi$
$278$	0	0
$279$	8.54503	0.511577
$280$	0	0
$281$	−25.5968	−1.52698	−0.763490	−	0.645820i	$-0.776516\pi$
$281$	−25.5968	−1.52698	−0.763490	+	0.645820i	$0.776516\pi$
$282$	0	0
$283$	1.96445	0.116775	0.0583873	−	0.998294i	$-0.481404\pi$
$283$	1.96445	0.116775	0.0583873	+	0.998294i	$0.481404\pi$
$284$	0	0
$285$	2.55314	0.151235
$286$	0	0
$287$	−0.298714	−0.0176325
$288$	0	0
$289$	21.7085	1.27697
$290$	0	0
$291$	−8.14545	−0.477495
$292$	0	0
$293$	26.2981	1.53635	0.768177	−	0.640238i	$-0.221164\pi$
$293$	26.2981	1.53635	0.768177	+	0.640238i	$0.221164\pi$
$294$	0	0
$295$	10.3752	0.604067
$296$	0	0
$297$	−4.16442	−0.241644
$298$	0	0
$299$	−15.1473	−0.875988
$300$	0	0
$301$	−0.630239	−0.0363264
$302$	0	0
$303$	17.4270	1.00116
$304$	0	0
$305$	10.5531	0.604271
$306$	0	0
$307$	−0.290599	−0.0165854	−0.00829269	−	0.999966i	$-0.502640\pi$
$307$	−0.290599	−0.0165854	−0.00829269	+	0.999966i	$0.502640\pi$
$308$	0	0
$309$	−0.279751	−0.0159145
$310$	0	0
$311$	1.03012	0.0584128	0.0292064	−	0.999573i	$-0.490702\pi$
$311$	1.03012	0.0584128	0.0292064	+	0.999573i	$0.490702\pi$
$312$	0	0
$313$	−18.6513	−1.05423	−0.527117	−	0.849793i	$-0.676727\pi$
$313$	−18.6513	−1.05423	−0.527117	+	0.849793i	$0.676727\pi$
$314$	0	0
$315$	−0.668475	−0.0376643
$316$	0	0
$317$	14.1473	0.794589	0.397294	−	0.917691i	$-0.369949\pi$
$317$	14.1473	0.794589	0.397294	+	0.917691i	$0.369949\pi$
$318$	0	0
$319$	6.94823	0.389026
$320$	0	0
$321$	−17.3725	−0.969638
$322$	0	0
$323$	15.8847	0.883847
$324$	0	0
$325$	5.04908	0.280073
$326$	0	0
$327$	3.15088	0.174244
$328$	0	0
$329$	−3.16262	−0.174361
$330$	0	0
$331$	2.31525	0.127258	0.0636289	−	0.997974i	$-0.479733\pi$
$331$	2.31525	0.127258	0.0636289	+	0.997974i	$0.479733\pi$
$332$	0	0
$333$	4.33695	0.237663
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	20.5549	1.11970	0.559849	−	0.828594i	$-0.310859\pi$
$337$	20.5549	1.11970	0.559849	+	0.828594i	$0.310859\pi$
$338$	0	0
$339$	9.10086	0.494291
$340$	0	0
$341$	−35.5851	−1.92704
$342$	0	0
$343$	−9.05993	−0.489190
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	−15.6577	−0.840548	−0.420274	−	0.907397i	$-0.638066\pi$
$347$	−15.6577	−0.840548	−0.420274	+	0.907397i	$0.638066\pi$
$348$	0	0
$349$	16.9040	0.904850	0.452425	−	0.891803i	$-0.350559\pi$
$349$	16.9040	0.904850	0.452425	+	0.891803i	$0.350559\pi$
$350$	0	0
$351$	5.04908	0.269500
$352$	0	0
$353$	2.28244	0.121482	0.0607410	−	0.998154i	$-0.480654\pi$
$353$	2.28244	0.121482	0.0607410	+	0.998154i	$0.480654\pi$
$354$	0	0
$355$	−5.93918	−0.315219
$356$	0	0
$357$	−4.15899	−0.220117
$358$	0	0
$359$	−19.4608	−1.02710	−0.513550	−	0.858060i	$-0.671670\pi$
$359$	−19.4608	−1.02710	−0.513550	+	0.858060i	$0.671670\pi$
$360$	0	0
$361$	−12.4815	−0.656919
$362$	0	0
$363$	6.34237	0.332888
$364$	0	0
$365$	10.5505	0.552236
$366$	0	0
$367$	−6.50406	−0.339509	−0.169755	−	0.985486i	$-0.554297\pi$
$367$	−6.50406	−0.339509	−0.169755	+	0.985486i	$0.554297\pi$
$368$	0	0
$369$	−0.446858	−0.0232625
$370$	0	0
$371$	−6.38061	−0.331265
$372$	0	0
$373$	26.2752	1.36048	0.680239	−	0.732991i	$-0.261876\pi$
$373$	26.2752	1.36048	0.680239	+	0.732991i	$0.261876\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−8.42427	−0.433872
$378$	0	0
$379$	−9.42794	−0.484281	−0.242140	−	0.970241i	$-0.577849\pi$
$379$	−9.42794	−0.484281	−0.242140	+	0.970241i	$0.577849\pi$
$380$	0	0
$381$	−1.61128	−0.0825482
$382$	0	0
$383$	10.2270	0.522577	0.261289	−	0.965261i	$-0.415853\pi$
$383$	10.2270	0.522577	0.261289	+	0.965261i	$0.415853\pi$
$384$	0	0
$385$	2.78381	0.141876
$386$	0	0
$387$	−0.942801	−0.0479253
$388$	0	0
$389$	1.60492	0.0813725	0.0406862	−	0.999172i	$-0.487046\pi$
$389$	1.60492	0.0813725	0.0406862	+	0.999172i	$0.487046\pi$
$390$	0	0
$391$	18.6648	0.943922
$392$	0	0
$393$	15.1864	0.766054
$394$	0	0
$395$	−2.04908	−0.103101
$396$	0	0
$397$	18.1852	0.912688	0.456344	−	0.889803i	$-0.349159\pi$
$397$	18.1852	0.912688	0.456344	+	0.889803i	$0.349159\pi$
$398$	0	0
$399$	−1.70671	−0.0854424
$400$	0	0
$401$	9.93469	0.496115	0.248057	−	0.968745i	$-0.420208\pi$
$401$	9.93469	0.496115	0.248057	+	0.968745i	$0.420208\pi$
$402$	0	0
$403$	43.1446	2.14918
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−18.0609	−0.895244
$408$	0	0
$409$	4.44050	0.219569	0.109784	−	0.993955i	$-0.464984\pi$
$409$	4.44050	0.219569	0.109784	+	0.993955i	$0.464984\pi$
$410$	0	0
$411$	−5.59680	−0.276070
$412$	0	0
$413$	−6.93555	−0.341276
$414$	0	0
$415$	−9.15088	−0.449199
$416$	0	0
$417$	4.33426	0.212250
$418$	0	0
$419$	27.6833	1.35242	0.676209	−	0.736710i	$-0.263622\pi$
$419$	27.6833	1.35242	0.676209	+	0.736710i	$0.263622\pi$
$420$	0	0
$421$	−25.4625	−1.24097	−0.620484	−	0.784219i	$-0.713064\pi$
$421$	−25.4625	−1.24097	−0.620484	+	0.784219i	$0.713064\pi$
$422$	0	0
$423$	−4.73110	−0.230034
$424$	0	0
$425$	−6.22162	−0.301793
$426$	0	0
$427$	−7.05451	−0.341392
$428$	0	0
$429$	−21.0265	−1.01517
$430$	0	0
$431$	−21.3499	−1.02839	−0.514194	−	0.857674i	$-0.671909\pi$
$431$	−21.3499	−1.02839	−0.514194	+	0.857674i	$0.671909\pi$
$432$	0	0
$433$	−13.4499	−0.646361	−0.323181	−	0.946337i	$-0.604752\pi$
$433$	−13.4499	−0.646361	−0.323181	+	0.946337i	$0.604752\pi$
$434$	0	0
$435$	1.66847	0.0799972
$436$	0	0
$437$	7.65942	0.366400
$438$	0	0
$439$	18.8721	0.900714	0.450357	−	0.892848i	$-0.351297\pi$
$439$	18.8721	0.900714	0.450357	+	0.892848i	$0.351297\pi$
$440$	0	0
$441$	−6.55314	−0.312054
$442$	0	0
$443$	14.0473	0.667409	0.333704	−	0.942678i	$-0.391701\pi$
$443$	14.0473	0.667409	0.333704	+	0.942678i	$0.391701\pi$
$444$	0	0
$445$	16.1428	0.765240
$446$	0	0
$447$	2.88198	0.136313
$448$	0	0
$449$	−25.0464	−1.18201	−0.591006	−	0.806667i	$-0.701269\pi$
$449$	−25.0464	−1.18201	−0.591006	+	0.806667i	$0.701269\pi$
$450$	0	0
$451$	1.86091	0.0876266
$452$	0	0
$453$	−18.9121	−0.888568
$454$	0	0
$455$	−3.37519	−0.158231
$456$	0	0
$457$	−33.7022	−1.57652	−0.788261	−	0.615341i	$-0.789018\pi$
$457$	−33.7022	−1.57652	−0.788261	+	0.615341i	$0.789018\pi$
$458$	0	0
$459$	−6.22162	−0.290400
$460$	0	0
$461$	−13.9717	−0.650726	−0.325363	−	0.945589i	$-0.605486\pi$
$461$	−13.9717	−0.650726	−0.325363	+	0.945589i	$0.605486\pi$
$462$	0	0
$463$	30.0883	1.39832	0.699161	−	0.714964i	$-0.253557\pi$
$463$	30.0883	1.39832	0.699161	+	0.714964i	$0.253557\pi$
$464$	0	0
$465$	−8.54503	−0.396266
$466$	0	0
$467$	6.55408	0.303287	0.151643	−	0.988435i	$-0.451544\pi$
$467$	6.55408	0.303287	0.151643	+	0.988435i	$0.451544\pi$
$468$	0	0
$469$	−0.668475	−0.0308673
$470$	0	0
$471$	−5.66847	−0.261190
$472$	0	0
$473$	3.92622	0.180528
$474$	0	0
$475$	−2.55314	−0.117146
$476$	0	0
$477$	−9.54503	−0.437037
$478$	0	0
$479$	31.4562	1.43727	0.718635	−	0.695387i	$-0.244767\pi$
$479$	31.4562	1.43727	0.718635	+	0.695387i	$0.244767\pi$
$480$	0	0
$481$	21.8976	0.998446
$482$	0	0
$483$	−2.00542	−0.0912500
$484$	0	0
$485$	8.14545	0.369866
$486$	0	0
$487$	−7.60223	−0.344490	−0.172245	−	0.985054i	$-0.555102\pi$
$487$	−7.60223	−0.344490	−0.172245	+	0.985054i	$0.555102\pi$
$488$	0	0
$489$	−1.39146	−0.0629239
$490$	0	0
$491$	13.8428	0.624717	0.312359	−	0.949964i	$-0.398881\pi$
$491$	13.8428	0.624717	0.312359	+	0.949964i	$0.398881\pi$
$492$	0	0
$493$	10.3806	0.467519
$494$	0	0
$495$	4.16442	0.187177
$496$	0	0
$497$	3.97019	0.178087
$498$	0	0
$499$	27.4107	1.22707	0.613536	−	0.789666i	$-0.289746\pi$
$499$	27.4107	1.22707	0.613536	+	0.789666i	$0.289746\pi$
$500$	0	0
$501$	−2.28244	−0.101972
$502$	0	0
$503$	13.7901	0.614870	0.307435	−	0.951569i	$-0.400529\pi$
$503$	13.7901	0.614870	0.307435	+	0.951569i	$0.400529\pi$
$504$	0	0
$505$	−17.4270	−0.775491
$506$	0	0
$507$	12.4933	0.554845
$508$	0	0
$509$	−25.5860	−1.13408	−0.567040	−	0.823690i	$-0.691911\pi$
$509$	−25.5860	−1.13408	−0.567040	+	0.823690i	$0.691911\pi$
$510$	0	0
$511$	−7.05271	−0.311994
$512$	0	0
$513$	−2.55314	−0.112724
$514$	0	0
$515$	0.279751	0.0123273
$516$	0	0
$517$	19.7023	0.866505
$518$	0	0
$519$	−15.8630	−0.696309
$520$	0	0
$521$	−32.9964	−1.44560	−0.722800	−	0.691058i	$-0.757145\pi$
$521$	−32.9964	−1.44560	−0.722800	+	0.691058i	$0.757145\pi$
$522$	0	0
$523$	22.8766	1.00032	0.500161	−	0.865932i	$-0.333274\pi$
$523$	22.8766	1.00032	0.500161	+	0.865932i	$0.333274\pi$
$524$	0	0
$525$	0.668475	0.0291746
$526$	0	0
$527$	−53.1639	−2.31585
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−10.3752	−0.450245
$532$	0	0
$533$	−2.25623	−0.0977280
$534$	0	0
$535$	17.3725	0.751078
$536$	0	0
$537$	−3.62930	−0.156616
$538$	0	0
$539$	27.2900	1.17546
$540$	0	0
$541$	2.45041	0.105351	0.0526757	−	0.998612i	$-0.483225\pi$
$541$	2.45041	0.105351	0.0526757	+	0.998612i	$0.483225\pi$
$542$	0	0
$543$	−24.5279	−1.05259
$544$	0	0
$545$	−3.15088	−0.134969
$546$	0	0
$547$	15.9284	0.681048	0.340524	−	0.940236i	$-0.389396\pi$
$547$	15.9284	0.681048	0.340524	+	0.940236i	$0.389396\pi$
$548$	0	0
$549$	−10.5531	−0.450397
$550$	0	0
$551$	4.25985	0.181476
$552$	0	0
$553$	1.36976	0.0582482
$554$	0	0
$555$	−4.33695	−0.184093
$556$	0	0
$557$	−26.3080	−1.11471	−0.557353	−	0.830276i	$-0.688183\pi$
$557$	−26.3080	−1.11471	−0.557353	+	0.830276i	$0.688183\pi$
$558$	0	0
$559$	−4.76028	−0.201339
$560$	0	0
$561$	25.9094	1.09390
$562$	0	0
$563$	17.3222	0.730042	0.365021	−	0.930999i	$-0.381062\pi$
$563$	17.3222	0.730042	0.365021	+	0.930999i	$0.381062\pi$
$564$	0	0
$565$	−9.10086	−0.382876
$566$	0	0
$567$	0.668475	0.0280733
$568$	0	0
$569$	−7.61939	−0.319421	−0.159711	−	0.987164i	$-0.551056\pi$
$569$	−7.61939	−0.319421	−0.159711	+	0.987164i	$0.551056\pi$
$570$	0	0
$571$	30.0991	1.25961	0.629804	−	0.776754i	$-0.283135\pi$
$571$	30.0991	1.25961	0.629804	+	0.776754i	$0.283135\pi$
$572$	0	0
$573$	25.3225	1.05786
$574$	0	0
$575$	−3.00000	−0.125109
$576$	0	0
$577$	18.5914	0.773971	0.386986	−	0.922086i	$-0.373516\pi$
$577$	18.5914	0.773971	0.386986	+	0.922086i	$0.373516\pi$
$578$	0	0
$579$	−21.6185	−0.898434
$580$	0	0
$581$	6.11713	0.253781
$582$	0	0
$583$	39.7495	1.64626
$584$	0	0
$585$	−5.04908	−0.208754
$586$	0	0
$587$	37.4833	1.54710	0.773550	−	0.633735i	$-0.218479\pi$
$587$	37.4833	1.54710	0.773550	+	0.633735i	$0.218479\pi$
$588$	0	0
$589$	−21.8167	−0.898940
$590$	0	0
$591$	24.3608	1.00207
$592$	0	0
$593$	22.1554	0.909812	0.454906	−	0.890539i	$-0.349673\pi$
$593$	22.1554	0.909812	0.454906	+	0.890539i	$0.349673\pi$
$594$	0	0
$595$	4.15899	0.170502
$596$	0	0
$597$	3.00269	0.122892
$598$	0	0
$599$	3.03824	0.124139	0.0620695	−	0.998072i	$-0.480230\pi$
$599$	3.03824	0.124139	0.0620695	+	0.998072i	$0.480230\pi$
$600$	0	0
$601$	24.5242	1.00036	0.500182	−	0.865920i	$-0.333266\pi$
$601$	24.5242	1.00036	0.500182	+	0.865920i	$0.333266\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−6.34237	−0.257854
$606$	0	0
$607$	−38.8895	−1.57848	−0.789238	−	0.614087i	$-0.789524\pi$
$607$	−38.8895	−1.57848	−0.789238	+	0.614087i	$0.789524\pi$
$608$	0	0
$609$	−1.11533	−0.0451956
$610$	0	0
$611$	−23.8877	−0.966393
$612$	0	0
$613$	46.6134	1.88270	0.941349	−	0.337434i	$-0.109559\pi$
$613$	46.6134	1.88270	0.941349	+	0.337434i	$0.109559\pi$
$614$	0	0
$615$	0.446858	0.0180191
$616$	0	0
$617$	−15.2761	−0.614992	−0.307496	−	0.951549i	$-0.599491\pi$
$617$	−15.2761	−0.614992	−0.307496	+	0.951549i	$0.599491\pi$
$618$	0	0
$619$	33.6757	1.35354	0.676770	−	0.736194i	$-0.263379\pi$
$619$	33.6757	1.35354	0.676770	+	0.736194i	$0.263379\pi$
$620$	0	0
$621$	−3.00000	−0.120386
$622$	0	0
$623$	−10.7910	−0.432334
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	10.6323	0.424615
$628$	0	0
$629$	−26.9828	−1.07588
$630$	0	0
$631$	10.3563	0.412276	0.206138	−	0.978523i	$-0.433910\pi$
$631$	10.3563	0.412276	0.206138	+	0.978523i	$0.433910\pi$
$632$	0	0
$633$	−7.46887	−0.296861
$634$	0	0
$635$	1.61128	0.0639415
$636$	0	0
$637$	−33.0874	−1.31097
$638$	0	0
$639$	5.93918	0.234950
$640$	0	0
$641$	41.8009	1.65104	0.825519	−	0.564374i	$-0.190883\pi$
$641$	41.8009	1.65104	0.825519	+	0.564374i	$0.190883\pi$
$642$	0	0
$643$	−12.9148	−0.509311	−0.254656	−	0.967032i	$-0.581962\pi$
$643$	−12.9148	−0.509311	−0.254656	+	0.967032i	$0.581962\pi$
$644$	0	0
$645$	0.942801	0.0371228
$646$	0	0
$647$	25.5441	1.00424	0.502122	−	0.864797i	$-0.332553\pi$
$647$	25.5441	1.00424	0.502122	+	0.864797i	$0.332553\pi$
$648$	0	0
$649$	43.2066	1.69601
$650$	0	0
$651$	5.71213	0.223876
$652$	0	0
$653$	12.2053	0.477632	0.238816	−	0.971065i	$-0.423241\pi$
$653$	12.2053	0.477632	0.238816	+	0.971065i	$0.423241\pi$
$654$	0	0
$655$	−15.1864	−0.593383
$656$	0	0
$657$	−10.5505	−0.411612
$658$	0	0
$659$	−31.3563	−1.22147	−0.610733	−	0.791836i	$-0.709125\pi$
$659$	−31.3563	−1.22147	−0.610733	+	0.791836i	$0.709125\pi$
$660$	0	0
$661$	−7.55134	−0.293713	−0.146857	−	0.989158i	$-0.546916\pi$
$661$	−7.55134	−0.293713	−0.146857	+	0.989158i	$0.546916\pi$
$662$	0	0
$663$	−31.4135	−1.22000
$664$	0	0
$665$	1.70671	0.0661834
$666$	0	0
$667$	5.00542	0.193811
$668$	0	0
$669$	14.8630	0.574637
$670$	0	0
$671$	43.9477	1.69658
$672$	0	0
$673$	26.6802	1.02845	0.514223	−	0.857657i	$-0.328080\pi$
$673$	26.6802	1.02845	0.514223	+	0.857657i	$0.328080\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	5.08334	0.195369	0.0976843	−	0.995217i	$-0.468856\pi$
$677$	5.08334	0.195369	0.0976843	+	0.995217i	$0.468856\pi$
$678$	0	0
$679$	−5.44503	−0.208961
$680$	0	0
$681$	−29.1990	−1.11891
$682$	0	0
$683$	33.6504	1.28760	0.643798	−	0.765196i	$-0.277358\pi$
$683$	33.6504	1.28760	0.643798	+	0.765196i	$0.277358\pi$
$684$	0	0
$685$	5.59680	0.213843
$686$	0	0
$687$	−26.5725	−1.01380
$688$	0	0
$689$	−48.1936	−1.83603
$690$	0	0
$691$	34.5014	1.31249	0.656247	−	0.754546i	$-0.272143\pi$
$691$	34.5014	1.31249	0.656247	+	0.754546i	$0.272143\pi$
$692$	0	0
$693$	−2.78381	−0.105748
$694$	0	0
$695$	−4.33426	−0.164408
$696$	0	0
$697$	2.78018	0.105307
$698$	0	0
$699$	10.9428	0.413895
$700$	0	0
$701$	−4.23422	−0.159924	−0.0799621	−	0.996798i	$-0.525480\pi$
$701$	−4.23422	−0.159924	−0.0799621	+	0.996798i	$0.525480\pi$
$702$	0	0
$703$	−11.0728	−0.417620
$704$	0	0
$705$	4.73110	0.178183
$706$	0	0
$707$	11.6495	0.438125
$708$	0	0
$709$	−44.3544	−1.66576	−0.832882	−	0.553450i	$-0.813311\pi$
$709$	−44.3544	−1.66576	−0.832882	+	0.553450i	$0.813311\pi$
$710$	0	0
$711$	2.04908	0.0768467
$712$	0	0
$713$	−25.6351	−0.960041
$714$	0	0
$715$	21.0265	0.786346
$716$	0	0
$717$	−2.77296	−0.103558
$718$	0	0
$719$	−5.38783	−0.200932	−0.100466	−	0.994940i	$-0.532033\pi$
$719$	−5.38783	−0.200932	−0.100466	+	0.994940i	$0.532033\pi$
$720$	0	0
$721$	−0.187006	−0.00696448
$722$	0	0
$723$	20.1729	0.750238
$724$	0	0
$725$	−1.66847	−0.0619656
$726$	0	0
$727$	7.71034	0.285961	0.142980	−	0.989726i	$-0.454331\pi$
$727$	7.71034	0.285961	0.142980	+	0.989726i	$0.454331\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.86575	0.216953
$732$	0	0
$733$	12.2653	0.453028	0.226514	−	0.974008i	$-0.427267\pi$
$733$	12.2653	0.453028	0.226514	+	0.974008i	$0.427267\pi$
$734$	0	0
$735$	6.55314	0.241716
$736$	0	0
$737$	4.16442	0.153398
$738$	0	0
$739$	45.9175	1.68910	0.844552	−	0.535474i	$-0.179867\pi$
$739$	45.9175	1.68910	0.844552	+	0.535474i	$0.179867\pi$
$740$	0	0
$741$	−12.8910	−0.473564
$742$	0	0
$743$	−26.8329	−0.984405	−0.492202	−	0.870481i	$-0.663808\pi$
$743$	−26.8329	−0.984405	−0.492202	+	0.870481i	$0.663808\pi$
$744$	0	0
$745$	−2.88198	−0.105587
$746$	0	0
$747$	9.15088	0.334813
$748$	0	0
$749$	−11.6131	−0.424333
$750$	0	0
$751$	−31.4315	−1.14695	−0.573476	−	0.819223i	$-0.694405\pi$
$751$	−31.4315	−1.14695	−0.573476	+	0.819223i	$0.694405\pi$
$752$	0	0
$753$	3.74464	0.136462
$754$	0	0
$755$	18.9121	0.688282
$756$	0	0
$757$	15.6022	0.567071	0.283535	−	0.958962i	$-0.408493\pi$
$757$	15.6022	0.567071	0.283535	+	0.958962i	$0.408493\pi$
$758$	0	0
$759$	12.4933	0.453476
$760$	0	0
$761$	−5.54865	−0.201139	−0.100569	−	0.994930i	$-0.532066\pi$
$761$	−5.54865	−0.201139	−0.100569	+	0.994930i	$0.532066\pi$
$762$	0	0
$763$	2.10628	0.0762526
$764$	0	0
$765$	6.22162	0.224943
$766$	0	0
$767$	−52.3852	−1.89152
$768$	0	0
$769$	−43.0693	−1.55312	−0.776559	−	0.630045i	$-0.783036\pi$
$769$	−43.0693	−1.55312	−0.776559	+	0.630045i	$0.783036\pi$
$770$	0	0
$771$	15.7450	0.567042
$772$	0	0
$773$	21.0338	0.756533	0.378266	−	0.925697i	$-0.376520\pi$
$773$	21.0338	0.756533	0.378266	+	0.925697i	$0.376520\pi$
$774$	0	0
$775$	8.54503	0.306946
$776$	0	0
$777$	2.89914	0.104006
$778$	0	0
$779$	1.14089	0.0408767
$780$	0	0
$781$	−24.7332	−0.885024
$782$	0	0
$783$	−1.66847	−0.0596264
$784$	0	0
$785$	5.66847	0.202317
$786$	0	0
$787$	28.9446	1.03177	0.515883	−	0.856659i	$-0.327464\pi$
$787$	28.9446	1.03177	0.515883	+	0.856659i	$0.327464\pi$
$788$	0	0
$789$	−2.56126	−0.0911831
$790$	0	0
$791$	6.08369	0.216311
$792$	0	0
$793$	−53.2837	−1.89216
$794$	0	0
$795$	9.54503	0.338527
$796$	0	0
$797$	−39.1716	−1.38753	−0.693765	−	0.720201i	$-0.744050\pi$
$797$	−39.1716	−1.38753	−0.693765	+	0.720201i	$0.744050\pi$
$798$	0	0
$799$	29.4351	1.04134
$800$	0	0
$801$	−16.1428	−0.570377
$802$	0	0
$803$	43.9365	1.55048
$804$	0	0
$805$	2.00542	0.0706819
$806$	0	0
$807$	0.123492	0.00434711
$808$	0	0
$809$	3.69501	0.129910	0.0649549	−	0.997888i	$-0.479310\pi$
$809$	3.69501	0.129910	0.0649549	+	0.997888i	$0.479310\pi$
$810$	0	0
$811$	−43.6261	−1.53192	−0.765959	−	0.642889i	$-0.777736\pi$
$811$	−43.6261	−1.53192	−0.765959	+	0.642889i	$0.777736\pi$
$812$	0	0
$813$	−22.4460	−0.787214
$814$	0	0
$815$	1.39146	0.0487407
$816$	0	0
$817$	2.40711	0.0842139
$818$	0	0
$819$	3.37519	0.117939
$820$	0	0
$821$	7.44050	0.259675	0.129838	−	0.991535i	$-0.458554\pi$
$821$	7.44050	0.259675	0.129838	+	0.991535i	$0.458554\pi$
$822$	0	0
$823$	−28.3981	−0.989896	−0.494948	−	0.868923i	$-0.664813\pi$
$823$	−28.3981	−0.989896	−0.494948	+	0.868923i	$0.664813\pi$
$824$	0	0
$825$	−4.16442	−0.144986
$826$	0	0
$827$	54.7887	1.90519	0.952595	−	0.304242i	$-0.0984031\pi$
$827$	54.7887	1.90519	0.952595	+	0.304242i	$0.0984031\pi$
$828$	0	0
$829$	−11.9085	−0.413601	−0.206800	−	0.978383i	$-0.566305\pi$
$829$	−11.9085	−0.413601	−0.206800	+	0.978383i	$0.566305\pi$
$830$	0	0
$831$	−20.9220	−0.725777
$832$	0	0
$833$	40.7711	1.41264
$834$	0	0
$835$	2.28244	0.0789871
$836$	0	0
$837$	8.54503	0.295359
$838$	0	0
$839$	−0.252630	−0.00872174	−0.00436087	−	0.999990i	$-0.501388\pi$
$839$	−0.252630	−0.00872174	−0.00436087	+	0.999990i	$0.501388\pi$
$840$	0	0
$841$	−26.2162	−0.904007
$842$	0	0
$843$	−25.5968	−0.881602
$844$	0	0
$845$	−12.4933	−0.429781
$846$	0	0
$847$	4.23972	0.145678
$848$	0	0
$849$	1.96445	0.0674199
$850$	0	0
$851$	−13.0108	−0.446006
$852$	0	0
$853$	−14.4414	−0.494463	−0.247231	−	0.968956i	$-0.579521\pi$
$853$	−14.4414	−0.494463	−0.247231	+	0.968956i	$0.579521\pi$
$854$	0	0
$855$	2.55314	0.0873156
$856$	0	0
$857$	23.2108	0.792866	0.396433	−	0.918064i	$-0.370248\pi$
$857$	23.2108	0.792866	0.396433	+	0.918064i	$0.370248\pi$
$858$	0	0
$859$	−32.8378	−1.12041	−0.560205	−	0.828354i	$-0.689278\pi$
$859$	−32.8378	−1.12041	−0.560205	+	0.828354i	$0.689278\pi$
$860$	0	0
$861$	−0.298714	−0.0101801
$862$	0	0
$863$	49.1437	1.67287	0.836435	−	0.548066i	$-0.184636\pi$
$863$	49.1437	1.67287	0.836435	+	0.548066i	$0.184636\pi$
$864$	0	0
$865$	15.8630	0.539359
$866$	0	0
$867$	21.7085	0.737260
$868$	0	0
$869$	−8.53324	−0.289470
$870$	0	0
$871$	−5.04908	−0.171082
$872$	0	0
$873$	−8.14545	−0.275682
$874$	0	0
$875$	−0.668475	−0.0225986
$876$	0	0
$877$	−14.7167	−0.496947	−0.248473	−	0.968639i	$-0.579929\pi$
$877$	−14.7167	−0.496947	−0.248473	+	0.968639i	$0.579929\pi$
$878$	0	0
$879$	26.2981	0.887014
$880$	0	0
$881$	36.5896	1.23274	0.616368	−	0.787458i	$-0.288603\pi$
$881$	36.5896	1.23274	0.616368	+	0.787458i	$0.288603\pi$
$882$	0	0
$883$	11.4639	0.385793	0.192896	−	0.981219i	$-0.438212\pi$
$883$	11.4639	0.385793	0.192896	+	0.981219i	$0.438212\pi$
$884$	0	0
$885$	10.3752	0.348758
$886$	0	0
$887$	−45.8422	−1.53923	−0.769616	−	0.638507i	$-0.779552\pi$
$887$	−45.8422	−1.53923	−0.769616	+	0.638507i	$0.779552\pi$
$888$	0	0
$889$	−1.07710	−0.0361247
$890$	0	0
$891$	−4.16442	−0.139513
$892$	0	0
$893$	12.0792	0.404214
$894$	0	0
$895$	3.62930	0.121314
$896$	0	0
$897$	−15.1473	−0.505752
$898$	0	0
$899$	−14.2572	−0.475503
$900$	0	0
$901$	59.3855	1.97842
$902$	0	0
$903$	−0.630239	−0.0209730
$904$	0	0
$905$	24.5279	0.815334
$906$	0	0
$907$	−34.9413	−1.16021	−0.580104	−	0.814543i	$-0.696988\pi$
$907$	−34.9413	−1.16021	−0.580104	+	0.814543i	$0.696988\pi$
$908$	0	0
$909$	17.4270	0.578017
$910$	0	0
$911$	−10.5441	−0.349343	−0.174671	−	0.984627i	$-0.555886\pi$
$911$	−10.5441	−0.349343	−0.174671	+	0.984627i	$0.555886\pi$
$912$	0	0
$913$	−38.1081	−1.26119
$914$	0	0
$915$	10.5531	0.348876
$916$	0	0
$917$	10.1517	0.335240
$918$	0	0
$919$	30.1725	0.995300	0.497650	−	0.867378i	$-0.334196\pi$
$919$	30.1725	0.995300	0.497650	+	0.867378i	$0.334196\pi$
$920$	0	0
$921$	−0.290599	−0.00957557
$922$	0	0
$923$	29.9874	0.987047
$924$	0	0
$925$	4.33695	0.142598
$926$	0	0
$927$	−0.279751	−0.00918822
$928$	0	0
$929$	−23.2171	−0.761727	−0.380863	−	0.924631i	$-0.624373\pi$
$929$	−23.2171	−0.761727	−0.380863	+	0.924631i	$0.624373\pi$
$930$	0	0
$931$	16.7311	0.548340
$932$	0	0
$933$	1.03012	0.0337247
$934$	0	0
$935$	−25.9094	−0.847328
$936$	0	0
$937$	−28.6730	−0.936706	−0.468353	−	0.883541i	$-0.655152\pi$
$937$	−28.6730	−0.936706	−0.468353	+	0.883541i	$0.655152\pi$
$938$	0	0
$939$	−18.6513	−0.608663
$940$	0	0
$941$	−46.4233	−1.51336	−0.756679	−	0.653787i	$-0.773179\pi$
$941$	−46.4233	−1.51336	−0.756679	+	0.653787i	$0.773179\pi$
$942$	0	0
$943$	1.34058	0.0436551
$944$	0	0
$945$	−0.668475	−0.0217455
$946$	0	0
$947$	−28.1382	−0.914370	−0.457185	−	0.889372i	$-0.651142\pi$
$947$	−28.1382	−0.914370	−0.457185	+	0.889372i	$0.651142\pi$
$948$	0	0
$949$	−53.2701	−1.72922
$950$	0	0
$951$	14.1473	0.458756
$952$	0	0
$953$	14.6950	0.476017	0.238008	−	0.971263i	$-0.423505\pi$
$953$	14.6950	0.476017	0.238008	+	0.971263i	$0.423505\pi$
$954$	0	0
$955$	−25.3225	−0.819417
$956$	0	0
$957$	6.94823	0.224604
$958$	0	0
$959$	−3.74132	−0.120814
$960$	0	0
$961$	42.0175	1.35540
$962$	0	0
$963$	−17.3725	−0.559821
$964$	0	0
$965$	21.6185	0.695924
$966$	0	0
$967$	19.8910	0.639652	0.319826	−	0.947476i	$-0.396375\pi$
$967$	19.8910	0.639652	0.319826	+	0.947476i	$0.396375\pi$
$968$	0	0
$969$	15.8847	0.510289
$970$	0	0
$971$	−46.9612	−1.50706	−0.753529	−	0.657415i	$-0.771650\pi$
$971$	−46.9612	−1.50706	−0.753529	+	0.657415i	$0.771650\pi$
$972$	0	0
$973$	2.89734	0.0928846
$974$	0	0
$975$	5.04908	0.161700
$976$	0	0
$977$	20.0022	0.639926	0.319963	−	0.947430i	$-0.396330\pi$
$977$	20.0022	0.639926	0.319963	+	0.947430i	$0.396330\pi$
$978$	0	0
$979$	67.2252	2.14853
$980$	0	0
$981$	3.15088	0.100600
$982$	0	0
$983$	6.26617	0.199860	0.0999299	−	0.994994i	$-0.468138\pi$
$983$	6.26617	0.199860	0.0999299	+	0.994994i	$0.468138\pi$
$984$	0	0
$985$	−24.3608	−0.776198
$986$	0	0
$987$	−3.16262	−0.100667
$988$	0	0
$989$	2.82840	0.0899380
$990$	0	0
$991$	49.0194	1.55715	0.778576	−	0.627551i	$-0.215943\pi$
$991$	49.0194	1.55715	0.778576	+	0.627551i	$0.215943\pi$
$992$	0	0
$993$	2.31525	0.0734723
$994$	0	0
$995$	−3.00269	−0.0951917
$996$	0	0
$997$	−12.1126	−0.383610	−0.191805	−	0.981433i	$-0.561434\pi$
$997$	−12.1126	−0.383610	−0.191805	+	0.981433i	$0.561434\pi$
$998$	0	0
$999$	4.33695	0.137215

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.c.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.c.1.3	✓	4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +0.668475 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +0.668475 q^{7} +1.00000 q^{9} -4.16442 q^{11} +5.04908 q^{13} -1.00000 q^{15} -6.22162 q^{17} -2.55314 q^{19} +0.668475 q^{21} -3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.66847 q^{29} +8.54503 q^{31} -4.16442 q^{33} -0.668475 q^{35} +4.33695 q^{37} +5.04908 q^{39} -0.446858 q^{41} -0.942801 q^{43} -1.00000 q^{45} -4.73110 q^{47} -6.55314 q^{49} -6.22162 q^{51} -9.54503 q^{53} +4.16442 q^{55} -2.55314 q^{57} -10.3752 q^{59} -10.5531 q^{61} +0.668475 q^{63} -5.04908 q^{65} -1.00000 q^{67} -3.00000 q^{69} +5.93918 q^{71} -10.5505 q^{73} +1.00000 q^{75} -2.78381 q^{77} +2.04908 q^{79} +1.00000 q^{81} +9.15088 q^{83} +6.22162 q^{85} -1.66847 q^{87} -16.1428 q^{89} +3.37519 q^{91} +8.54503 q^{93} +2.55314 q^{95} -8.14545 q^{97} -4.16442 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} - 5 q^{11} - 5 q^{13} - 4 q^{15} - 8 q^{17} + 5 q^{19} + q^{21} - 12 q^{23} + 4 q^{25} + 4 q^{27} - 5 q^{29} - q^{31} - 5 q^{33} - q^{35} + 14 q^{37} - 5 q^{39} - 17 q^{41} - 9 q^{43} - 4 q^{45} - 7 q^{47} - 11 q^{49} - 8 q^{51} - 3 q^{53} + 5 q^{55} + 5 q^{57} - 23 q^{59} - 27 q^{61} + q^{63} + 5 q^{65} - 4 q^{67} - 12 q^{69} - 20 q^{71} - 2 q^{73} + 4 q^{75} - 23 q^{77} - 17 q^{79} + 4 q^{81} + 10 q^{83} + 8 q^{85} - 5 q^{87} - 18 q^{89} - 5 q^{91} - q^{93} - 5 q^{95} - 11 q^{97} - 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 - 5 * q^11 - 5 * q^13 - 4 * q^15 - 8 * q^17 + 5 * q^19 + q^21 - 12 * q^23 + 4 * q^25 + 4 * q^27 - 5 * q^29 - q^31 - 5 * q^33 - q^35 + 14 * q^37 - 5 * q^39 - 17 * q^41 - 9 * q^43 - 4 * q^45 - 7 * q^47 - 11 * q^49 - 8 * q^51 - 3 * q^53 + 5 * q^55 + 5 * q^57 - 23 * q^59 - 27 * q^61 + q^63 + 5 * q^65 - 4 * q^67 - 12 * q^69 - 20 * q^71 - 2 * q^73 + 4 * q^75 - 23 * q^77 - 17 * q^79 + 4 * q^81 + 10 * q^83 + 8 * q^85 - 5 * q^87 - 18 * q^89 - 5 * q^91 - q^93 - 5 * q^95 - 11 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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