LMFDB - Embedded newform 4020.2.a.e.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:	$5$
Coefficient field:	5.5.1257629.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 9x^{3} + 9x^{2} + 17x - 18 $ x^5 - x^4 - 9x^3 + 9x^2 + 17*x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.67454$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -0.300647 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -0.300647 q^{7} +1.00000 q^{9} -4.87305 q^{11} -3.19591 q^{13} -1.00000 q^{15} -4.37390 q^{17} +7.36960 q^{19} +0.300647 q^{21} +8.95558 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.23078 q^{29} +1.76311 q^{31} +4.87305 q^{33} -0.300647 q^{35} -1.44144 q^{37} +3.19591 q^{39} -4.18726 q^{41} -11.2042 q^{43} +1.00000 q^{45} -3.68319 q^{47} -6.90961 q^{49} +4.37390 q^{51} +12.7697 q^{53} -4.87305 q^{55} -7.36960 q^{57} -2.07500 q^{59} -8.81415 q^{61} -0.300647 q^{63} -3.19591 q^{65} -1.00000 q^{67} -8.95558 q^{69} -4.57240 q^{71} -12.1148 q^{73} -1.00000 q^{75} +1.46507 q^{77} +10.3386 q^{79} +1.00000 q^{81} -7.33355 q^{83} -4.37390 q^{85} -8.23078 q^{87} -9.26916 q^{89} +0.960839 q^{91} -1.76311 q^{93} +7.36960 q^{95} -14.3841 q^{97} -4.87305 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + 5 * q^5 - 3 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + 5 q^{5} - 3 q^{7} + 5 q^{9} + 3 q^{11} - 11 q^{13} - 5 q^{15} - 11 q^{17} + 6 q^{19} + 3 q^{21} + 3 q^{23} + 5 q^{25} - 5 q^{27} - 2 q^{29} + q^{31} - 3 q^{33} - 3 q^{35} - 13 q^{37} + 11 q^{39} - 13 q^{41} - 9 q^{43} + 5 q^{45} - 12 q^{47} - 2 q^{49} + 11 q^{51} - 19 q^{53} + 3 q^{55} - 6 q^{57} + 6 q^{59} - 3 q^{61} - 3 q^{63} - 11 q^{65} - 5 q^{67} - 3 q^{69} + 6 q^{71} - 21 q^{73} - 5 q^{75} - 15 q^{77} - q^{79} + 5 q^{81} - 8 q^{83} - 11 q^{85} + 2 q^{87} - 29 q^{89} + 23 q^{91} - q^{93} + 6 q^{95} - 13 q^{97} + 3 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 5 * q^5 - 3 * q^7 + 5 * q^9 + 3 * q^11 - 11 * q^13 - 5 * q^15 - 11 * q^17 + 6 * q^19 + 3 * q^21 + 3 * q^23 + 5 * q^25 - 5 * q^27 - 2 * q^29 + q^31 - 3 * q^33 - 3 * q^35 - 13 * q^37 + 11 * q^39 - 13 * q^41 - 9 * q^43 + 5 * q^45 - 12 * q^47 - 2 * q^49 + 11 * q^51 - 19 * q^53 + 3 * q^55 - 6 * q^57 + 6 * q^59 - 3 * q^61 - 3 * q^63 - 11 * q^65 - 5 * q^67 - 3 * q^69 + 6 * q^71 - 21 * q^73 - 5 * q^75 - 15 * q^77 - q^79 + 5 * q^81 - 8 * q^83 - 11 * q^85 + 2 * q^87 - 29 * q^89 + 23 * q^91 - q^93 + 6 * q^95 - 13 * 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
$7$	−0.300647	−0.113634	−0.0568169	−	0.998385i	$-0.518095\pi$
$7$	−0.300647	−0.113634	−0.0568169	+	0.998385i	$0.518095\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.87305	−1.46928	−0.734640	−	0.678457i	$-0.762649\pi$
$11$	−4.87305	−1.46928	−0.734640	+	0.678457i	$0.762649\pi$
$12$	0	0
$13$	−3.19591	−0.886385	−0.443193	−	0.896426i	$-0.646154\pi$
$13$	−3.19591	−0.886385	−0.443193	+	0.896426i	$0.646154\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−4.37390	−1.06083	−0.530413	−	0.847740i	$-0.677963\pi$
$17$	−4.37390	−1.06083	−0.530413	+	0.847740i	$0.677963\pi$
$18$	0	0
$19$	7.36960	1.69070	0.845352	−	0.534210i	$-0.179391\pi$
$19$	7.36960	1.69070	0.845352	+	0.534210i	$0.179391\pi$
$20$	0	0
$21$	0.300647	0.0656065
$22$	0	0
$23$	8.95558	1.86737	0.933683	−	0.358100i	$-0.116575\pi$
$23$	8.95558	1.86737	0.933683	+	0.358100i	$0.116575\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.23078	1.52842	0.764208	−	0.644969i	$-0.223130\pi$
$29$	8.23078	1.52842	0.764208	+	0.644969i	$0.223130\pi$
$30$	0	0
$31$	1.76311	0.316664	0.158332	−	0.987386i	$-0.449388\pi$
$31$	1.76311	0.316664	0.158332	+	0.987386i	$0.449388\pi$
$32$	0	0
$33$	4.87305	0.848289
$34$	0	0
$35$	−0.300647	−0.0508185
$36$	0	0
$37$	−1.44144	−0.236971	−0.118486	−	0.992956i	$-0.537804\pi$
$37$	−1.44144	−0.236971	−0.118486	+	0.992956i	$0.537804\pi$
$38$	0	0
$39$	3.19591	0.511755
$40$	0	0
$41$	−4.18726	−0.653941	−0.326970	−	0.945035i	$-0.606028\pi$
$41$	−4.18726	−0.653941	−0.326970	+	0.945035i	$0.606028\pi$
$42$	0	0
$43$	−11.2042	−1.70863	−0.854313	−	0.519758i	$-0.826022\pi$
$43$	−11.2042	−1.70863	−0.854313	+	0.519758i	$0.826022\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−3.68319	−0.537248	−0.268624	−	0.963245i	$-0.586569\pi$
$47$	−3.68319	−0.537248	−0.268624	+	0.963245i	$0.586569\pi$
$48$	0	0
$49$	−6.90961	−0.987087
$50$	0	0
$51$	4.37390	0.612468
$52$	0	0
$53$	12.7697	1.75406	0.877028	−	0.480439i	$-0.159523\pi$
$53$	12.7697	1.75406	0.877028	+	0.480439i	$0.159523\pi$
$54$	0	0
$55$	−4.87305	−0.657082
$56$	0	0
$57$	−7.36960	−0.976128
$58$	0	0
$59$	−2.07500	−0.270142	−0.135071	−	0.990836i	$-0.543126\pi$
$59$	−2.07500	−0.270142	−0.135071	+	0.990836i	$0.543126\pi$
$60$	0	0
$61$	−8.81415	−1.12854	−0.564268	−	0.825592i	$-0.690842\pi$
$61$	−8.81415	−1.12854	−0.564268	+	0.825592i	$0.690842\pi$
$62$	0	0
$63$	−0.300647	−0.0378779
$64$	0	0
$65$	−3.19591	−0.396404
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	−8.95558	−1.07812
$70$	0	0
$71$	−4.57240	−0.542644	−0.271322	−	0.962489i	$-0.587461\pi$
$71$	−4.57240	−0.542644	−0.271322	+	0.962489i	$0.587461\pi$
$72$	0	0
$73$	−12.1148	−1.41793	−0.708965	−	0.705244i	$-0.750838\pi$
$73$	−12.1148	−1.41793	−0.708965	+	0.705244i	$0.750838\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	1.46507	0.166960
$78$	0	0
$79$	10.3386	1.16319	0.581593	−	0.813480i	$-0.302430\pi$
$79$	10.3386	1.16319	0.581593	+	0.813480i	$0.302430\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.33355	−0.804962	−0.402481	−	0.915428i	$-0.631852\pi$
$83$	−7.33355	−0.804962	−0.402481	+	0.915428i	$0.631852\pi$
$84$	0	0
$85$	−4.37390	−0.474416
$86$	0	0
$87$	−8.23078	−0.882432
$88$	0	0
$89$	−9.26916	−0.982529	−0.491264	−	0.871011i	$-0.663465\pi$
$89$	−9.26916	−0.982529	−0.491264	+	0.871011i	$0.663465\pi$
$90$	0	0
$91$	0.960839	0.100723
$92$	0	0
$93$	−1.76311	−0.182826
$94$	0	0
$95$	7.36960	0.756105
$96$	0	0
$97$	−14.3841	−1.46048	−0.730241	−	0.683190i	$-0.760592\pi$
$97$	−14.3841	−1.46048	−0.730241	+	0.683190i	$0.760592\pi$
$98$	0	0
$99$	−4.87305	−0.489760
$100$	0	0
$101$	−4.85492	−0.483082	−0.241541	−	0.970391i	$-0.577653\pi$
$101$	−4.85492	−0.483082	−0.241541	+	0.970391i	$0.577653\pi$
$102$	0	0
$103$	−9.92330	−0.977772	−0.488886	−	0.872348i	$-0.662597\pi$
$103$	−9.92330	−0.977772	−0.488886	+	0.872348i	$0.662597\pi$
$104$	0	0
$105$	0.300647	0.0293401
$106$	0	0
$107$	−12.2458	−1.18384	−0.591921	−	0.805996i	$-0.701630\pi$
$107$	−12.2458	−1.18384	−0.591921	+	0.805996i	$0.701630\pi$
$108$	0	0
$109$	9.27057	0.887960	0.443980	−	0.896037i	$-0.353566\pi$
$109$	9.27057	0.887960	0.443980	+	0.896037i	$0.353566\pi$
$110$	0	0
$111$	1.44144	0.136815
$112$	0	0
$113$	2.41481	0.227166	0.113583	−	0.993528i	$-0.463767\pi$
$113$	2.41481	0.227166	0.113583	+	0.993528i	$0.463767\pi$
$114$	0	0
$115$	8.95558	0.835112
$116$	0	0
$117$	−3.19591	−0.295462
$118$	0	0
$119$	1.31500	0.120546
$120$	0	0
$121$	12.7466	1.15878
$122$	0	0
$123$	4.18726	0.377553
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	16.8306	1.49347	0.746737	−	0.665120i	$-0.231620\pi$
$127$	16.8306	1.49347	0.746737	+	0.665120i	$0.231620\pi$
$128$	0	0
$129$	11.2042	0.986476
$130$	0	0
$131$	13.0262	1.13811	0.569053	−	0.822301i	$-0.307310\pi$
$131$	13.0262	1.13811	0.569053	+	0.822301i	$0.307310\pi$
$132$	0	0
$133$	−2.21565	−0.192121
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−9.52707	−0.813953	−0.406976	−	0.913439i	$-0.633417\pi$
$137$	−9.52707	−0.813953	−0.406976	+	0.913439i	$0.633417\pi$
$138$	0	0
$139$	−17.5371	−1.48748	−0.743740	−	0.668469i	$-0.766950\pi$
$139$	−17.5371	−1.48748	−0.743740	+	0.668469i	$0.766950\pi$
$140$	0	0
$141$	3.68319	0.310180
$142$	0	0
$143$	15.5738	1.30235
$144$	0	0
$145$	8.23078	0.683529
$146$	0	0
$147$	6.90961	0.569895
$148$	0	0
$149$	−19.5806	−1.60411	−0.802054	−	0.597251i	$-0.796260\pi$
$149$	−19.5806	−1.60411	−0.802054	+	0.597251i	$0.796260\pi$
$150$	0	0
$151$	7.85006	0.638828	0.319414	−	0.947615i	$-0.396514\pi$
$151$	7.85006	0.638828	0.319414	+	0.947615i	$0.396514\pi$
$152$	0	0
$153$	−4.37390	−0.353609
$154$	0	0
$155$	1.76311	0.141617
$156$	0	0
$157$	−4.48468	−0.357916	−0.178958	−	0.983857i	$-0.557273\pi$
$157$	−4.48468	−0.357916	−0.178958	+	0.983857i	$0.557273\pi$
$158$	0	0
$159$	−12.7697	−1.01271
$160$	0	0
$161$	−2.69246	−0.212196
$162$	0	0
$163$	5.90046	0.462160	0.231080	−	0.972935i	$-0.425774\pi$
$163$	5.90046	0.462160	0.231080	+	0.972935i	$0.425774\pi$
$164$	0	0
$165$	4.87305	0.379366
$166$	0	0
$167$	12.0038	0.928882	0.464441	−	0.885604i	$-0.346255\pi$
$167$	12.0038	0.928882	0.464441	+	0.885604i	$0.346255\pi$
$168$	0	0
$169$	−2.78617	−0.214321
$170$	0	0
$171$	7.36960	0.563568
$172$	0	0
$173$	−8.04402	−0.611576	−0.305788	−	0.952100i	$-0.598920\pi$
$173$	−8.04402	−0.611576	−0.305788	+	0.952100i	$0.598920\pi$
$174$	0	0
$175$	−0.300647	−0.0227267
$176$	0	0
$177$	2.07500	0.155967
$178$	0	0
$179$	11.5686	0.864679	0.432340	−	0.901711i	$-0.357688\pi$
$179$	11.5686	0.864679	0.432340	+	0.901711i	$0.357688\pi$
$180$	0	0
$181$	−10.8269	−0.804755	−0.402378	−	0.915474i	$-0.631816\pi$
$181$	−10.8269	−0.804755	−0.402378	+	0.915474i	$0.631816\pi$
$182$	0	0
$183$	8.81415	0.651561
$184$	0	0
$185$	−1.44144	−0.105977
$186$	0	0
$187$	21.3142	1.55865
$188$	0	0
$189$	0.300647	0.0218688
$190$	0	0
$191$	2.09609	0.151668	0.0758340	−	0.997120i	$-0.475838\pi$
$191$	2.09609	0.151668	0.0758340	+	0.997120i	$0.475838\pi$
$192$	0	0
$193$	−18.0144	−1.29670	−0.648351	−	0.761342i	$-0.724541\pi$
$193$	−18.0144	−1.29670	−0.648351	+	0.761342i	$0.724541\pi$
$194$	0	0
$195$	3.19591	0.228864
$196$	0	0
$197$	−2.98565	−0.212719	−0.106359	−	0.994328i	$-0.533919\pi$
$197$	−2.98565	−0.212719	−0.106359	+	0.994328i	$0.533919\pi$
$198$	0	0
$199$	−15.2766	−1.08293	−0.541465	−	0.840723i	$-0.682130\pi$
$199$	−15.2766	−1.08293	−0.541465	+	0.840723i	$0.682130\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	−2.47455	−0.173680
$204$	0	0
$205$	−4.18726	−0.292451
$206$	0	0
$207$	8.95558	0.622455
$208$	0	0
$209$	−35.9124	−2.48412
$210$	0	0
$211$	9.35603	0.644096	0.322048	−	0.946723i	$-0.395629\pi$
$211$	9.35603	0.644096	0.322048	+	0.946723i	$0.395629\pi$
$212$	0	0
$213$	4.57240	0.313296
$214$	0	0
$215$	−11.2042	−0.764121
$216$	0	0
$217$	−0.530074	−0.0359838
$218$	0	0
$219$	12.1148	0.818642
$220$	0	0
$221$	13.9786	0.940300
$222$	0	0
$223$	−14.7652	−0.988751	−0.494376	−	0.869248i	$-0.664603\pi$
$223$	−14.7652	−0.988751	−0.494376	+	0.869248i	$0.664603\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	12.6430	0.839145	0.419572	−	0.907722i	$-0.362180\pi$
$227$	12.6430	0.839145	0.419572	+	0.907722i	$0.362180\pi$
$228$	0	0
$229$	−2.64656	−0.174890	−0.0874449	−	0.996169i	$-0.527870\pi$
$229$	−2.64656	−0.174890	−0.0874449	+	0.996169i	$0.527870\pi$
$230$	0	0
$231$	−1.46507	−0.0963942
$232$	0	0
$233$	−21.2267	−1.39061	−0.695304	−	0.718716i	$-0.744730\pi$
$233$	−21.2267	−1.39061	−0.695304	+	0.718716i	$0.744730\pi$
$234$	0	0
$235$	−3.68319	−0.240265
$236$	0	0
$237$	−10.3386	−0.671566
$238$	0	0
$239$	−14.5736	−0.942688	−0.471344	−	0.881949i	$-0.656231\pi$
$239$	−14.5736	−0.942688	−0.471344	+	0.881949i	$0.656231\pi$
$240$	0	0
$241$	−25.4610	−1.64009	−0.820044	−	0.572301i	$-0.806051\pi$
$241$	−25.4610	−1.64009	−0.820044	+	0.572301i	$0.806051\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−6.90961	−0.441439
$246$	0	0
$247$	−23.5526	−1.49861
$248$	0	0
$249$	7.33355	0.464745
$250$	0	0
$251$	−8.88668	−0.560922	−0.280461	−	0.959865i	$-0.590487\pi$
$251$	−8.88668	−0.560922	−0.280461	+	0.959865i	$0.590487\pi$
$252$	0	0
$253$	−43.6410	−2.74368
$254$	0	0
$255$	4.37390	0.273904
$256$	0	0
$257$	−19.1246	−1.19296	−0.596479	−	0.802629i	$-0.703434\pi$
$257$	−19.1246	−1.19296	−0.596479	+	0.802629i	$0.703434\pi$
$258$	0	0
$259$	0.433364	0.0269279
$260$	0	0
$261$	8.23078	0.509472
$262$	0	0
$263$	−10.2704	−0.633297	−0.316649	−	0.948543i	$-0.602558\pi$
$263$	−10.2704	−0.633297	−0.316649	+	0.948543i	$0.602558\pi$
$264$	0	0
$265$	12.7697	0.784438
$266$	0	0
$267$	9.26916	0.567263
$268$	0	0
$269$	24.2771	1.48020	0.740101	−	0.672496i	$-0.234778\pi$
$269$	24.2771	1.48020	0.740101	+	0.672496i	$0.234778\pi$
$270$	0	0
$271$	15.0066	0.911587	0.455793	−	0.890086i	$-0.349356\pi$
$271$	15.0066	0.911587	0.455793	+	0.890086i	$0.349356\pi$
$272$	0	0
$273$	−0.960839	−0.0581526
$274$	0	0
$275$	−4.87305	−0.293856
$276$	0	0
$277$	−21.3892	−1.28515	−0.642575	−	0.766223i	$-0.722134\pi$
$277$	−21.3892	−1.28515	−0.642575	+	0.766223i	$0.722134\pi$
$278$	0	0
$279$	1.76311	0.105555
$280$	0	0
$281$	10.6078	0.632811	0.316405	−	0.948624i	$-0.397524\pi$
$281$	10.6078	0.632811	0.316405	+	0.948624i	$0.397524\pi$
$282$	0	0
$283$	11.6272	0.691164	0.345582	−	0.938389i	$-0.387681\pi$
$283$	11.6272	0.691164	0.345582	+	0.938389i	$0.387681\pi$
$284$	0	0
$285$	−7.36960	−0.436538
$286$	0	0
$287$	1.25889	0.0743097
$288$	0	0
$289$	2.13096	0.125351
$290$	0	0
$291$	14.3841	0.843210
$292$	0	0
$293$	20.3712	1.19010	0.595048	−	0.803690i	$-0.297133\pi$
$293$	20.3712	1.19010	0.595048	+	0.803690i	$0.297133\pi$
$294$	0	0
$295$	−2.07500	−0.120811
$296$	0	0
$297$	4.87305	0.282763
$298$	0	0
$299$	−28.6212	−1.65521
$300$	0	0
$301$	3.36851	0.194158
$302$	0	0
$303$	4.85492	0.278908
$304$	0	0
$305$	−8.81415	−0.504697
$306$	0	0
$307$	6.89408	0.393466	0.196733	−	0.980457i	$-0.436967\pi$
$307$	6.89408	0.393466	0.196733	+	0.980457i	$0.436967\pi$
$308$	0	0
$309$	9.92330	0.564517
$310$	0	0
$311$	−7.37785	−0.418359	−0.209180	−	0.977877i	$-0.567079\pi$
$311$	−7.37785	−0.418359	−0.209180	+	0.977877i	$0.567079\pi$
$312$	0	0
$313$	−24.9658	−1.41115	−0.705577	−	0.708634i	$-0.749312\pi$
$313$	−24.9658	−1.41115	−0.705577	+	0.708634i	$0.749312\pi$
$314$	0	0
$315$	−0.300647	−0.0169395
$316$	0	0
$317$	26.6964	1.49942	0.749710	−	0.661767i	$-0.230193\pi$
$317$	26.6964	1.49942	0.749710	+	0.661767i	$0.230193\pi$
$318$	0	0
$319$	−40.1090	−2.24567
$320$	0	0
$321$	12.2458	0.683492
$322$	0	0
$323$	−32.2339	−1.79354
$324$	0	0
$325$	−3.19591	−0.177277
$326$	0	0
$327$	−9.27057	−0.512664
$328$	0	0
$329$	1.10734	0.0610495
$330$	0	0
$331$	−17.7309	−0.974578	−0.487289	−	0.873241i	$-0.662014\pi$
$331$	−17.7309	−0.974578	−0.487289	+	0.873241i	$0.662014\pi$
$332$	0	0
$333$	−1.44144	−0.0789903
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−33.5317	−1.82659	−0.913293	−	0.407304i	$-0.866469\pi$
$337$	−33.5317	−1.82659	−0.913293	+	0.407304i	$0.866469\pi$
$338$	0	0
$339$	−2.41481	−0.131154
$340$	0	0
$341$	−8.59174	−0.465269
$342$	0	0
$343$	4.18188	0.225800
$344$	0	0
$345$	−8.95558	−0.482152
$346$	0	0
$347$	22.7447	1.22100	0.610499	−	0.792017i	$-0.290969\pi$
$347$	22.7447	1.22100	0.610499	+	0.792017i	$0.290969\pi$
$348$	0	0
$349$	20.9895	1.12354	0.561772	−	0.827292i	$-0.310120\pi$
$349$	20.9895	1.12354	0.561772	+	0.827292i	$0.310120\pi$
$350$	0	0
$351$	3.19591	0.170585
$352$	0	0
$353$	−0.185002	−0.00984669	−0.00492334	−	0.999988i	$-0.501567\pi$
$353$	−0.185002	−0.00984669	−0.00492334	+	0.999988i	$0.501567\pi$
$354$	0	0
$355$	−4.57240	−0.242678
$356$	0	0
$357$	−1.31500	−0.0695970
$358$	0	0
$359$	0.117610	0.00620720	0.00310360	−	0.999995i	$-0.499012\pi$
$359$	0.117610	0.00620720	0.00310360	+	0.999995i	$0.499012\pi$
$360$	0	0
$361$	35.3111	1.85848
$362$	0	0
$363$	−12.7466	−0.669023
$364$	0	0
$365$	−12.1148	−0.634117
$366$	0	0
$367$	5.02889	0.262506	0.131253	−	0.991349i	$-0.458100\pi$
$367$	5.02889	0.262506	0.131253	+	0.991349i	$0.458100\pi$
$368$	0	0
$369$	−4.18726	−0.217980
$370$	0	0
$371$	−3.83917	−0.199320
$372$	0	0
$373$	−21.2427	−1.09990	−0.549951	−	0.835197i	$-0.685354\pi$
$373$	−21.2427	−1.09990	−0.549951	+	0.835197i	$0.685354\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−26.3048	−1.35477
$378$	0	0
$379$	9.47625	0.486762	0.243381	−	0.969931i	$-0.421743\pi$
$379$	9.47625	0.486762	0.243381	+	0.969931i	$0.421743\pi$
$380$	0	0
$381$	−16.8306	−0.862257
$382$	0	0
$383$	18.1826	0.929088	0.464544	−	0.885550i	$-0.346218\pi$
$383$	18.1826	0.929088	0.464544	+	0.885550i	$0.346218\pi$
$384$	0	0
$385$	1.46507	0.0746666
$386$	0	0
$387$	−11.2042	−0.569542
$388$	0	0
$389$	15.7449	0.798296	0.399148	−	0.916887i	$-0.369306\pi$
$389$	15.7449	0.798296	0.399148	+	0.916887i	$0.369306\pi$
$390$	0	0
$391$	−39.1708	−1.98095
$392$	0	0
$393$	−13.0262	−0.657086
$394$	0	0
$395$	10.3386	0.520193
$396$	0	0
$397$	−18.4986	−0.928419	−0.464210	−	0.885725i	$-0.653662\pi$
$397$	−18.4986	−0.928419	−0.464210	+	0.885725i	$0.653662\pi$
$398$	0	0
$399$	2.21565	0.110921
$400$	0	0
$401$	33.0185	1.64886	0.824432	−	0.565961i	$-0.191495\pi$
$401$	33.0185	1.64886	0.824432	+	0.565961i	$0.191495\pi$
$402$	0	0
$403$	−5.63475	−0.280687
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	7.02420	0.348177
$408$	0	0
$409$	27.4222	1.35594	0.677970	−	0.735090i	$-0.262860\pi$
$409$	27.4222	1.35594	0.677970	+	0.735090i	$0.262860\pi$
$410$	0	0
$411$	9.52707	0.469936
$412$	0	0
$413$	0.623842	0.0306973
$414$	0	0
$415$	−7.33355	−0.359990
$416$	0	0
$417$	17.5371	0.858797
$418$	0	0
$419$	10.4221	0.509154	0.254577	−	0.967053i	$-0.418064\pi$
$419$	10.4221	0.509154	0.254577	+	0.967053i	$0.418064\pi$
$420$	0	0
$421$	5.04696	0.245974	0.122987	−	0.992408i	$-0.460753\pi$
$421$	5.04696	0.245974	0.122987	+	0.992408i	$0.460753\pi$
$422$	0	0
$423$	−3.68319	−0.179083
$424$	0	0
$425$	−4.37390	−0.212165
$426$	0	0
$427$	2.64994	0.128240
$428$	0	0
$429$	−15.5738	−0.751911
$430$	0	0
$431$	25.2632	1.21688	0.608442	−	0.793598i	$-0.291795\pi$
$431$	25.2632	1.21688	0.608442	+	0.793598i	$0.291795\pi$
$432$	0	0
$433$	18.1035	0.870001	0.435000	−	0.900430i	$-0.356748\pi$
$433$	18.1035	0.870001	0.435000	+	0.900430i	$0.356748\pi$
$434$	0	0
$435$	−8.23078	−0.394636
$436$	0	0
$437$	65.9990	3.15716
$438$	0	0
$439$	−25.9923	−1.24055	−0.620273	−	0.784386i	$-0.712978\pi$
$439$	−25.9923	−1.24055	−0.620273	+	0.784386i	$0.712978\pi$
$440$	0	0
$441$	−6.90961	−0.329029
$442$	0	0
$443$	−27.2589	−1.29511	−0.647554	−	0.762019i	$-0.724208\pi$
$443$	−27.2589	−1.29511	−0.647554	+	0.762019i	$0.724208\pi$
$444$	0	0
$445$	−9.26916	−0.439400
$446$	0	0
$447$	19.5806	0.926133
$448$	0	0
$449$	8.12135	0.383270	0.191635	−	0.981466i	$-0.438621\pi$
$449$	8.12135	0.383270	0.191635	+	0.981466i	$0.438621\pi$
$450$	0	0
$451$	20.4047	0.960822
$452$	0	0
$453$	−7.85006	−0.368828
$454$	0	0
$455$	0.960839	0.0450448
$456$	0	0
$457$	−7.72851	−0.361525	−0.180762	−	0.983527i	$-0.557856\pi$
$457$	−7.72851	−0.361525	−0.180762	+	0.983527i	$0.557856\pi$
$458$	0	0
$459$	4.37390	0.204156
$460$	0	0
$461$	1.71009	0.0796467	0.0398233	−	0.999207i	$-0.487320\pi$
$461$	1.71009	0.0796467	0.0398233	+	0.999207i	$0.487320\pi$
$462$	0	0
$463$	5.66469	0.263261	0.131630	−	0.991299i	$-0.457979\pi$
$463$	5.66469	0.263261	0.131630	+	0.991299i	$0.457979\pi$
$464$	0	0
$465$	−1.76311	−0.0817624
$466$	0	0
$467$	25.3139	1.17139	0.585693	−	0.810533i	$-0.300822\pi$
$467$	25.3139	1.17139	0.585693	+	0.810533i	$0.300822\pi$
$468$	0	0
$469$	0.300647	0.0138826
$470$	0	0
$471$	4.48468	0.206643
$472$	0	0
$473$	54.5987	2.51045
$474$	0	0
$475$	7.36960	0.338141
$476$	0	0
$477$	12.7697	0.584686
$478$	0	0
$479$	26.3073	1.20201	0.601006	−	0.799245i	$-0.294767\pi$
$479$	26.3073	1.20201	0.601006	+	0.799245i	$0.294767\pi$
$480$	0	0
$481$	4.60670	0.210048
$482$	0	0
$483$	2.69246	0.122511
$484$	0	0
$485$	−14.3841	−0.653147
$486$	0	0
$487$	0.809415	0.0366781	0.0183391	−	0.999832i	$-0.494162\pi$
$487$	0.809415	0.0366781	0.0183391	+	0.999832i	$0.494162\pi$
$488$	0	0
$489$	−5.90046	−0.266828
$490$	0	0
$491$	15.7894	0.712566	0.356283	−	0.934378i	$-0.384044\pi$
$491$	15.7894	0.712566	0.356283	+	0.934378i	$0.384044\pi$
$492$	0	0
$493$	−36.0006	−1.62138
$494$	0	0
$495$	−4.87305	−0.219027
$496$	0	0
$497$	1.37468	0.0616627
$498$	0	0
$499$	−35.4536	−1.58712	−0.793561	−	0.608490i	$-0.791775\pi$
$499$	−35.4536	−1.58712	−0.793561	+	0.608490i	$0.791775\pi$
$500$	0	0
$501$	−12.0038	−0.536290
$502$	0	0
$503$	−32.4367	−1.44628	−0.723141	−	0.690700i	$-0.757303\pi$
$503$	−32.4367	−1.44628	−0.723141	+	0.690700i	$0.757303\pi$
$504$	0	0
$505$	−4.85492	−0.216041
$506$	0	0
$507$	2.78617	0.123738
$508$	0	0
$509$	22.4154	0.993544	0.496772	−	0.867881i	$-0.334518\pi$
$509$	22.4154	0.993544	0.496772	+	0.867881i	$0.334518\pi$
$510$	0	0
$511$	3.64227	0.161125
$512$	0	0
$513$	−7.36960	−0.325376
$514$	0	0
$515$	−9.92330	−0.437273
$516$	0	0
$517$	17.9483	0.789367
$518$	0	0
$519$	8.04402	0.353093
$520$	0	0
$521$	−15.1435	−0.663449	−0.331724	−	0.943376i	$-0.607630\pi$
$521$	−15.1435	−0.663449	−0.331724	+	0.943376i	$0.607630\pi$
$522$	0	0
$523$	4.11519	0.179945	0.0899724	−	0.995944i	$-0.471322\pi$
$523$	4.11519	0.179945	0.0899724	+	0.995944i	$0.471322\pi$
$524$	0	0
$525$	0.300647	0.0131213
$526$	0	0
$527$	−7.71167	−0.335926
$528$	0	0
$529$	57.2023	2.48706
$530$	0	0
$531$	−2.07500	−0.0900474
$532$	0	0
$533$	13.3821	0.579644
$534$	0	0
$535$	−12.2458	−0.529430
$536$	0	0
$537$	−11.5686	−0.499223
$538$	0	0
$539$	33.6709	1.45031
$540$	0	0
$541$	−18.9395	−0.814273	−0.407136	−	0.913367i	$-0.633473\pi$
$541$	−18.9395	−0.814273	−0.407136	+	0.913367i	$0.633473\pi$
$542$	0	0
$543$	10.8269	0.464626
$544$	0	0
$545$	9.27057	0.397108
$546$	0	0
$547$	11.6929	0.499954	0.249977	−	0.968252i	$-0.419577\pi$
$547$	11.6929	0.499954	0.249977	+	0.968252i	$0.419577\pi$
$548$	0	0
$549$	−8.81415	−0.376179
$550$	0	0
$551$	60.6576	2.58410
$552$	0	0
$553$	−3.10827	−0.132177
$554$	0	0
$555$	1.44144	0.0611857
$556$	0	0
$557$	−15.1220	−0.640741	−0.320371	−	0.947292i	$-0.603807\pi$
$557$	−15.1220	−0.640741	−0.320371	+	0.947292i	$0.603807\pi$
$558$	0	0
$559$	35.8076	1.51450
$560$	0	0
$561$	−21.3142	−0.899887
$562$	0	0
$563$	−30.7372	−1.29542	−0.647709	−	0.761888i	$-0.724273\pi$
$563$	−30.7372	−1.29542	−0.647709	+	0.761888i	$0.724273\pi$
$564$	0	0
$565$	2.41481	0.101592
$566$	0	0
$567$	−0.300647	−0.0126260
$568$	0	0
$569$	−45.9608	−1.92678	−0.963388	−	0.268110i	$-0.913601\pi$
$569$	−45.9608	−1.92678	−0.963388	+	0.268110i	$0.913601\pi$
$570$	0	0
$571$	−39.3967	−1.64870	−0.824350	−	0.566080i	$-0.808459\pi$
$571$	−39.3967	−1.64870	−0.824350	+	0.566080i	$0.808459\pi$
$572$	0	0
$573$	−2.09609	−0.0875656
$574$	0	0
$575$	8.95558	0.373473
$576$	0	0
$577$	−34.3629	−1.43055	−0.715273	−	0.698846i	$-0.753697\pi$
$577$	−34.3629	−1.43055	−0.715273	+	0.698846i	$0.753697\pi$
$578$	0	0
$579$	18.0144	0.748651
$580$	0	0
$581$	2.20481	0.0914708
$582$	0	0
$583$	−62.2275	−2.57720
$584$	0	0
$585$	−3.19591	−0.132135
$586$	0	0
$587$	−11.6256	−0.479840	−0.239920	−	0.970793i	$-0.577121\pi$
$587$	−11.6256	−0.479840	−0.239920	+	0.970793i	$0.577121\pi$
$588$	0	0
$589$	12.9934	0.535386
$590$	0	0
$591$	2.98565	0.122813
$592$	0	0
$593$	5.90549	0.242509	0.121255	−	0.992621i	$-0.461308\pi$
$593$	5.90549	0.242509	0.121255	+	0.992621i	$0.461308\pi$
$594$	0	0
$595$	1.31500	0.0539096
$596$	0	0
$597$	15.2766	0.625230
$598$	0	0
$599$	−17.2604	−0.705243	−0.352621	−	0.935766i	$-0.614710\pi$
$599$	−17.2604	−0.705243	−0.352621	+	0.935766i	$0.614710\pi$
$600$	0	0
$601$	25.6448	1.04607	0.523037	−	0.852310i	$-0.324799\pi$
$601$	25.6448	1.04607	0.523037	+	0.852310i	$0.324799\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	12.7466	0.518223
$606$	0	0
$607$	−12.8451	−0.521368	−0.260684	−	0.965424i	$-0.583948\pi$
$607$	−12.8451	−0.521368	−0.260684	+	0.965424i	$0.583948\pi$
$608$	0	0
$609$	2.47455	0.100274
$610$	0	0
$611$	11.7711	0.476209
$612$	0	0
$613$	35.1214	1.41854	0.709270	−	0.704937i	$-0.249025\pi$
$613$	35.1214	1.41854	0.709270	+	0.704937i	$0.249025\pi$
$614$	0	0
$615$	4.18726	0.168847
$616$	0	0
$617$	−20.6291	−0.830498	−0.415249	−	0.909708i	$-0.636306\pi$
$617$	−20.6291	−0.830498	−0.415249	+	0.909708i	$0.636306\pi$
$618$	0	0
$619$	−4.43155	−0.178119	−0.0890594	−	0.996026i	$-0.528386\pi$
$619$	−4.43155	−0.178119	−0.0890594	+	0.996026i	$0.528386\pi$
$620$	0	0
$621$	−8.95558	−0.359375
$622$	0	0
$623$	2.78674	0.111648
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	35.9124	1.43420
$628$	0	0
$629$	6.30470	0.251385
$630$	0	0
$631$	−5.09824	−0.202958	−0.101479	−	0.994838i	$-0.532357\pi$
$631$	−5.09824	−0.202958	−0.101479	+	0.994838i	$0.532357\pi$
$632$	0	0
$633$	−9.35603	−0.371869
$634$	0	0
$635$	16.8306	0.667902
$636$	0	0
$637$	22.0825	0.874940
$638$	0	0
$639$	−4.57240	−0.180881
$640$	0	0
$641$	28.8509	1.13954	0.569772	−	0.821803i	$-0.307032\pi$
$641$	28.8509	1.13954	0.569772	+	0.821803i	$0.307032\pi$
$642$	0	0
$643$	−5.54933	−0.218844	−0.109422	−	0.993995i	$-0.534900\pi$
$643$	−5.54933	−0.218844	−0.109422	+	0.993995i	$0.534900\pi$
$644$	0	0
$645$	11.2042	0.441165
$646$	0	0
$647$	17.9145	0.704293	0.352147	−	0.935945i	$-0.385452\pi$
$647$	17.9145	0.704293	0.352147	+	0.935945i	$0.385452\pi$
$648$	0	0
$649$	10.1116	0.396915
$650$	0	0
$651$	0.530074	0.0207752
$652$	0	0
$653$	−41.4457	−1.62189	−0.810947	−	0.585119i	$-0.801048\pi$
$653$	−41.4457	−1.62189	−0.810947	+	0.585119i	$0.801048\pi$
$654$	0	0
$655$	13.0262	0.508977
$656$	0	0
$657$	−12.1148	−0.472643
$658$	0	0
$659$	−26.4743	−1.03129	−0.515647	−	0.856801i	$-0.672448\pi$
$659$	−26.4743	−1.03129	−0.515647	+	0.856801i	$0.672448\pi$
$660$	0	0
$661$	42.9486	1.67051	0.835254	−	0.549864i	$-0.185320\pi$
$661$	42.9486	1.67051	0.835254	+	0.549864i	$0.185320\pi$
$662$	0	0
$663$	−13.9786	−0.542883
$664$	0	0
$665$	−2.21565	−0.0859191
$666$	0	0
$667$	73.7113	2.85411
$668$	0	0
$669$	14.7652	0.570856
$670$	0	0
$671$	42.9518	1.65814
$672$	0	0
$673$	9.80060	0.377786	0.188893	−	0.981998i	$-0.439510\pi$
$673$	9.80060	0.377786	0.188893	+	0.981998i	$0.439510\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−40.6682	−1.56301	−0.781503	−	0.623901i	$-0.785547\pi$
$677$	−40.6682	−1.56301	−0.781503	+	0.623901i	$0.785547\pi$
$678$	0	0
$679$	4.32452	0.165960
$680$	0	0
$681$	−12.6430	−0.484480
$682$	0	0
$683$	16.5683	0.633968	0.316984	−	0.948431i	$-0.397330\pi$
$683$	16.5683	0.633968	0.316984	+	0.948431i	$0.397330\pi$
$684$	0	0
$685$	−9.52707	−0.364011
$686$	0	0
$687$	2.64656	0.100973
$688$	0	0
$689$	−40.8109	−1.55477
$690$	0	0
$691$	10.7883	0.410405	0.205202	−	0.978720i	$-0.434215\pi$
$691$	10.7883	0.410405	0.205202	+	0.978720i	$0.434215\pi$
$692$	0	0
$693$	1.46507	0.0556532
$694$	0	0
$695$	−17.5371	−0.665221
$696$	0	0
$697$	18.3147	0.693717
$698$	0	0
$699$	21.2267	0.802867
$700$	0	0
$701$	−5.22388	−0.197303	−0.0986517	−	0.995122i	$-0.531453\pi$
$701$	−5.22388	−0.197303	−0.0986517	+	0.995122i	$0.531453\pi$
$702$	0	0
$703$	−10.6228	−0.400648
$704$	0	0
$705$	3.68319	0.138717
$706$	0	0
$707$	1.45961	0.0548944
$708$	0	0
$709$	−48.7309	−1.83013	−0.915064	−	0.403308i	$-0.867860\pi$
$709$	−48.7309	−1.83013	−0.915064	+	0.403308i	$0.867860\pi$
$710$	0	0
$711$	10.3386	0.387729
$712$	0	0
$713$	15.7897	0.591329
$714$	0	0
$715$	15.5738	0.582428
$716$	0	0
$717$	14.5736	0.544261
$718$	0	0
$719$	30.8200	1.14939	0.574697	−	0.818366i	$-0.305120\pi$
$719$	30.8200	1.14939	0.574697	+	0.818366i	$0.305120\pi$
$720$	0	0
$721$	2.98341	0.111108
$722$	0	0
$723$	25.4610	0.946905
$724$	0	0
$725$	8.23078	0.305683
$726$	0	0
$727$	−36.0006	−1.33519	−0.667594	−	0.744526i	$-0.732676\pi$
$727$	−36.0006	−1.33519	−0.667594	+	0.744526i	$0.732676\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	49.0061	1.81255
$732$	0	0
$733$	−28.7354	−1.06136	−0.530682	−	0.847571i	$-0.678064\pi$
$733$	−28.7354	−1.06136	−0.530682	+	0.847571i	$0.678064\pi$
$734$	0	0
$735$	6.90961	0.254865
$736$	0	0
$737$	4.87305	0.179501
$738$	0	0
$739$	19.1371	0.703970	0.351985	−	0.936006i	$-0.385507\pi$
$739$	19.1371	0.703970	0.351985	+	0.936006i	$0.385507\pi$
$740$	0	0
$741$	23.5526	0.865226
$742$	0	0
$743$	−10.9206	−0.400636	−0.200318	−	0.979731i	$-0.564198\pi$
$743$	−10.9206	−0.400636	−0.200318	+	0.979731i	$0.564198\pi$
$744$	0	0
$745$	−19.5806	−0.717379
$746$	0	0
$747$	−7.33355	−0.268321
$748$	0	0
$749$	3.68165	0.134524
$750$	0	0
$751$	−0.317301	−0.0115785	−0.00578923	−	0.999983i	$-0.501843\pi$
$751$	−0.317301	−0.0115785	−0.00578923	+	0.999983i	$0.501843\pi$
$752$	0	0
$753$	8.88668	0.323849
$754$	0	0
$755$	7.85006	0.285693
$756$	0	0
$757$	9.12735	0.331739	0.165870	−	0.986148i	$-0.446957\pi$
$757$	9.12735	0.331739	0.165870	+	0.986148i	$0.446957\pi$
$758$	0	0
$759$	43.6410	1.58407
$760$	0	0
$761$	10.9948	0.398561	0.199281	−	0.979942i	$-0.436139\pi$
$761$	10.9948	0.398561	0.199281	+	0.979942i	$0.436139\pi$
$762$	0	0
$763$	−2.78717	−0.100902
$764$	0	0
$765$	−4.37390	−0.158139
$766$	0	0
$767$	6.63152	0.239450
$768$	0	0
$769$	−35.0771	−1.26491	−0.632456	−	0.774596i	$-0.717953\pi$
$769$	−35.0771	−1.26491	−0.632456	+	0.774596i	$0.717953\pi$
$770$	0	0
$771$	19.1246	0.688754
$772$	0	0
$773$	−22.2406	−0.799937	−0.399969	−	0.916529i	$-0.630979\pi$
$773$	−22.2406	−0.799937	−0.399969	+	0.916529i	$0.630979\pi$
$774$	0	0
$775$	1.76311	0.0633329
$776$	0	0
$777$	−0.433364	−0.0155468
$778$	0	0
$779$	−30.8585	−1.10562
$780$	0	0
$781$	22.2815	0.797296
$782$	0	0
$783$	−8.23078	−0.294144
$784$	0	0
$785$	−4.48468	−0.160065
$786$	0	0
$787$	−16.4528	−0.586478	−0.293239	−	0.956039i	$-0.594733\pi$
$787$	−16.4528	−0.586478	−0.293239	+	0.956039i	$0.594733\pi$
$788$	0	0
$789$	10.2704	0.365634
$790$	0	0
$791$	−0.726004	−0.0258137
$792$	0	0
$793$	28.1692	1.00032
$794$	0	0
$795$	−12.7697	−0.452895
$796$	0	0
$797$	38.3769	1.35938	0.679690	−	0.733499i	$-0.262114\pi$
$797$	38.3769	1.35938	0.679690	+	0.733499i	$0.262114\pi$
$798$	0	0
$799$	16.1099	0.569926
$800$	0	0
$801$	−9.26916	−0.327510
$802$	0	0
$803$	59.0360	2.08334
$804$	0	0
$805$	−2.69246	−0.0948968
$806$	0	0
$807$	−24.2771	−0.854595
$808$	0	0
$809$	−7.14979	−0.251373	−0.125687	−	0.992070i	$-0.540113\pi$
$809$	−7.14979	−0.251373	−0.125687	+	0.992070i	$0.540113\pi$
$810$	0	0
$811$	7.89079	0.277083	0.138541	−	0.990357i	$-0.455759\pi$
$811$	7.89079	0.277083	0.138541	+	0.990357i	$0.455759\pi$
$812$	0	0
$813$	−15.0066	−0.526305
$814$	0	0
$815$	5.90046	0.206684
$816$	0	0
$817$	−82.5706	−2.88878
$818$	0	0
$819$	0.960839	0.0335744
$820$	0	0
$821$	44.2160	1.54315	0.771574	−	0.636140i	$-0.219470\pi$
$821$	44.2160	1.54315	0.771574	+	0.636140i	$0.219470\pi$
$822$	0	0
$823$	24.3383	0.848381	0.424190	−	0.905573i	$-0.360559\pi$
$823$	24.3383	0.848381	0.424190	+	0.905573i	$0.360559\pi$
$824$	0	0
$825$	4.87305	0.169658
$826$	0	0
$827$	−39.3105	−1.36696	−0.683481	−	0.729969i	$-0.739535\pi$
$827$	−39.3105	−1.36696	−0.683481	+	0.729969i	$0.739535\pi$
$828$	0	0
$829$	−10.9651	−0.380834	−0.190417	−	0.981703i	$-0.560984\pi$
$829$	−10.9651	−0.380834	−0.190417	+	0.981703i	$0.560984\pi$
$830$	0	0
$831$	21.3892	0.741981
$832$	0	0
$833$	30.2219	1.04713
$834$	0	0
$835$	12.0038	0.415409
$836$	0	0
$837$	−1.76311	−0.0609421
$838$	0	0
$839$	33.2287	1.14718	0.573592	−	0.819141i	$-0.305550\pi$
$839$	33.2287	1.14718	0.573592	+	0.819141i	$0.305550\pi$
$840$	0	0
$841$	38.7457	1.33606
$842$	0	0
$843$	−10.6078	−0.365354
$844$	0	0
$845$	−2.78617	−0.0958472
$846$	0	0
$847$	−3.83222	−0.131677
$848$	0	0
$849$	−11.6272	−0.399044
$850$	0	0
$851$	−12.9089	−0.442512
$852$	0	0
$853$	6.52988	0.223579	0.111789	−	0.993732i	$-0.464342\pi$
$853$	6.52988	0.223579	0.111789	+	0.993732i	$0.464342\pi$
$854$	0	0
$855$	7.36960	0.252035
$856$	0	0
$857$	−46.1426	−1.57620	−0.788100	−	0.615547i	$-0.788935\pi$
$857$	−46.1426	−1.57620	−0.788100	+	0.615547i	$0.788935\pi$
$858$	0	0
$859$	30.6397	1.04541	0.522707	−	0.852513i	$-0.324922\pi$
$859$	30.6397	1.04541	0.522707	+	0.852513i	$0.324922\pi$
$860$	0	0
$861$	−1.25889	−0.0429027
$862$	0	0
$863$	49.5617	1.68710	0.843550	−	0.537051i	$-0.180462\pi$
$863$	49.5617	1.68710	0.843550	+	0.537051i	$0.180462\pi$
$864$	0	0
$865$	−8.04402	−0.273505
$866$	0	0
$867$	−2.13096	−0.0723713
$868$	0	0
$869$	−50.3806	−1.70905
$870$	0	0
$871$	3.19591	0.108289
$872$	0	0
$873$	−14.3841	−0.486827
$874$	0	0
$875$	−0.300647	−0.0101637
$876$	0	0
$877$	34.9045	1.17864	0.589321	−	0.807899i	$-0.299395\pi$
$877$	34.9045	1.17864	0.589321	+	0.807899i	$0.299395\pi$
$878$	0	0
$879$	−20.3712	−0.687102
$880$	0	0
$881$	12.1679	0.409949	0.204974	−	0.978767i	$-0.434289\pi$
$881$	12.1679	0.409949	0.204974	+	0.978767i	$0.434289\pi$
$882$	0	0
$883$	11.6283	0.391323	0.195661	−	0.980672i	$-0.437315\pi$
$883$	11.6283	0.391323	0.195661	+	0.980672i	$0.437315\pi$
$884$	0	0
$885$	2.07500	0.0697504
$886$	0	0
$887$	10.3005	0.345858	0.172929	−	0.984934i	$-0.444677\pi$
$887$	10.3005	0.345858	0.172929	+	0.984934i	$0.444677\pi$
$888$	0	0
$889$	−5.06006	−0.169709
$890$	0	0
$891$	−4.87305	−0.163253
$892$	0	0
$893$	−27.1436	−0.908327
$894$	0	0
$895$	11.5686	0.386696
$896$	0	0
$897$	28.6212	0.955634
$898$	0	0
$899$	14.5118	0.483995
$900$	0	0
$901$	−55.8534	−1.86075
$902$	0	0
$903$	−3.36851	−0.112097
$904$	0	0
$905$	−10.8269	−0.359897
$906$	0	0
$907$	−2.62405	−0.0871303	−0.0435652	−	0.999051i	$-0.513872\pi$
$907$	−2.62405	−0.0871303	−0.0435652	+	0.999051i	$0.513872\pi$
$908$	0	0
$909$	−4.85492	−0.161027
$910$	0	0
$911$	30.6048	1.01398	0.506991	−	0.861951i	$-0.330758\pi$
$911$	30.6048	1.01398	0.506991	+	0.861951i	$0.330758\pi$
$912$	0	0
$913$	35.7368	1.18271
$914$	0	0
$915$	8.81415	0.291387
$916$	0	0
$917$	−3.91629	−0.129327
$918$	0	0
$919$	−26.1497	−0.862601	−0.431300	−	0.902208i	$-0.641945\pi$
$919$	−26.1497	−0.862601	−0.431300	+	0.902208i	$0.641945\pi$
$920$	0	0
$921$	−6.89408	−0.227168
$922$	0	0
$923$	14.6130	0.480992
$924$	0	0
$925$	−1.44144	−0.0473942
$926$	0	0
$927$	−9.92330	−0.325924
$928$	0	0
$929$	15.8856	0.521188	0.260594	−	0.965448i	$-0.416082\pi$
$929$	15.8856	0.521188	0.260594	+	0.965448i	$0.416082\pi$
$930$	0	0
$931$	−50.9211	−1.66887
$932$	0	0
$933$	7.37785	0.241540
$934$	0	0
$935$	21.3142	0.697049
$936$	0	0
$937$	0.387170	0.0126483	0.00632415	−	0.999980i	$-0.497987\pi$
$937$	0.387170	0.0126483	0.00632415	+	0.999980i	$0.497987\pi$
$938$	0	0
$939$	24.9658	0.814730
$940$	0	0
$941$	52.2678	1.70388	0.851940	−	0.523639i	$-0.175426\pi$
$941$	52.2678	1.70388	0.851940	+	0.523639i	$0.175426\pi$
$942$	0	0
$943$	−37.4994	−1.22115
$944$	0	0
$945$	0.300647	0.00978003
$946$	0	0
$947$	32.0907	1.04281	0.521404	−	0.853310i	$-0.325408\pi$
$947$	32.0907	1.04281	0.521404	+	0.853310i	$0.325408\pi$
$948$	0	0
$949$	38.7178	1.25683
$950$	0	0
$951$	−26.6964	−0.865690
$952$	0	0
$953$	−30.5676	−0.990180	−0.495090	−	0.868842i	$-0.664865\pi$
$953$	−30.5676	−0.990180	−0.495090	+	0.868842i	$0.664865\pi$
$954$	0	0
$955$	2.09609	0.0678280
$956$	0	0
$957$	40.1090	1.29654
$958$	0	0
$959$	2.86428	0.0924925
$960$	0	0
$961$	−27.8914	−0.899724
$962$	0	0
$963$	−12.2458	−0.394614
$964$	0	0
$965$	−18.0144	−0.579902
$966$	0	0
$967$	39.2849	1.26332	0.631659	−	0.775246i	$-0.282374\pi$
$967$	39.2849	1.26332	0.631659	+	0.775246i	$0.282374\pi$
$968$	0	0
$969$	32.2339	1.03550
$970$	0	0
$971$	−44.2150	−1.41893	−0.709464	−	0.704742i	$-0.751063\pi$
$971$	−44.2150	−1.41893	−0.709464	+	0.704742i	$0.751063\pi$
$972$	0	0
$973$	5.27248	0.169028
$974$	0	0
$975$	3.19591	0.102351
$976$	0	0
$977$	19.9359	0.637805	0.318902	−	0.947788i	$-0.396686\pi$
$977$	19.9359	0.637805	0.318902	+	0.947788i	$0.396686\pi$
$978$	0	0
$979$	45.1691	1.44361
$980$	0	0
$981$	9.27057	0.295987
$982$	0	0
$983$	−19.8434	−0.632907	−0.316454	−	0.948608i	$-0.602492\pi$
$983$	−19.8434	−0.632907	−0.316454	+	0.948608i	$0.602492\pi$
$984$	0	0
$985$	−2.98565	−0.0951307
$986$	0	0
$987$	−1.10734	−0.0352469
$988$	0	0
$989$	−100.340	−3.19063
$990$	0	0
$991$	−0.664632	−0.0211127	−0.0105564	−	0.999944i	$-0.503360\pi$
$991$	−0.664632	−0.0211127	−0.0105564	+	0.999944i	$0.503360\pi$
$992$	0	0
$993$	17.7309	0.562673
$994$	0	0
$995$	−15.2766	−0.484301
$996$	0	0
$997$	14.8267	0.469567	0.234783	−	0.972048i	$-0.424562\pi$
$997$	14.8267	0.469567	0.234783	+	0.972048i	$0.424562\pi$
$998$	0	0
$999$	1.44144	0.0456051

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.e.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.e.1.3	✓	5	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -0.300647 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -0.300647 q^{7} +1.00000 q^{9} -4.87305 q^{11} -3.19591 q^{13} -1.00000 q^{15} -4.37390 q^{17} +7.36960 q^{19} +0.300647 q^{21} +8.95558 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.23078 q^{29} +1.76311 q^{31} +4.87305 q^{33} -0.300647 q^{35} -1.44144 q^{37} +3.19591 q^{39} -4.18726 q^{41} -11.2042 q^{43} +1.00000 q^{45} -3.68319 q^{47} -6.90961 q^{49} +4.37390 q^{51} +12.7697 q^{53} -4.87305 q^{55} -7.36960 q^{57} -2.07500 q^{59} -8.81415 q^{61} -0.300647 q^{63} -3.19591 q^{65} -1.00000 q^{67} -8.95558 q^{69} -4.57240 q^{71} -12.1148 q^{73} -1.00000 q^{75} +1.46507 q^{77} +10.3386 q^{79} +1.00000 q^{81} -7.33355 q^{83} -4.37390 q^{85} -8.23078 q^{87} -9.26916 q^{89} +0.960839 q^{91} -1.76311 q^{93} +7.36960 q^{95} -14.3841 q^{97} -4.87305 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + 5 * q^5 - 3 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + 5 q^{5} - 3 q^{7} + 5 q^{9} + 3 q^{11} - 11 q^{13} - 5 q^{15} - 11 q^{17} + 6 q^{19} + 3 q^{21} + 3 q^{23} + 5 q^{25} - 5 q^{27} - 2 q^{29} + q^{31} - 3 q^{33} - 3 q^{35} - 13 q^{37} + 11 q^{39} - 13 q^{41} - 9 q^{43} + 5 q^{45} - 12 q^{47} - 2 q^{49} + 11 q^{51} - 19 q^{53} + 3 q^{55} - 6 q^{57} + 6 q^{59} - 3 q^{61} - 3 q^{63} - 11 q^{65} - 5 q^{67} - 3 q^{69} + 6 q^{71} - 21 q^{73} - 5 q^{75} - 15 q^{77} - q^{79} + 5 q^{81} - 8 q^{83} - 11 q^{85} + 2 q^{87} - 29 q^{89} + 23 q^{91} - q^{93} + 6 q^{95} - 13 q^{97} + 3 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 5 * q^5 - 3 * q^7 + 5 * q^9 + 3 * q^11 - 11 * q^13 - 5 * q^15 - 11 * q^17 + 6 * q^19 + 3 * q^21 + 3 * q^23 + 5 * q^25 - 5 * q^27 - 2 * q^29 + q^31 - 3 * q^33 - 3 * q^35 - 13 * q^37 + 11 * q^39 - 13 * q^41 - 9 * q^43 + 5 * q^45 - 12 * q^47 - 2 * q^49 + 11 * q^51 - 19 * q^53 + 3 * q^55 - 6 * q^57 + 6 * q^59 - 3 * q^61 - 3 * q^63 - 11 * q^65 - 5 * q^67 - 3 * q^69 + 6 * q^71 - 21 * q^73 - 5 * q^75 - 15 * q^77 - q^79 + 5 * q^81 - 8 * q^83 - 11 * q^85 + 2 * q^87 - 29 * q^89 + 23 * q^91 - q^93 + 6 * q^95 - 13 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more