LMFDB - Embedded newform 4020.2.a.i.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:	$0$
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			1.3
Root			$-2.67981$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -0.239317 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -0.239317 q^{7} +1.00000 q^{9} +4.45891 q^{11} +6.75426 q^{13} +1.00000 q^{15} +0.115680 q^{17} +0.902140 q^{19} -0.239317 q^{21} +8.04085 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.28017 q^{29} +5.30358 q^{31} +4.45891 q^{33} -0.239317 q^{35} -11.4005 q^{37} +6.75426 q^{39} +1.22063 q^{41} -3.53276 q^{43} +1.00000 q^{45} -4.40972 q^{47} -6.94273 q^{49} +0.115680 q^{51} -6.23099 q^{53} +4.45891 q^{55} +0.902140 q^{57} -8.41284 q^{59} +5.35001 q^{61} -0.239317 q^{63} +6.75426 q^{65} -1.00000 q^{67} +8.04085 q^{69} +6.97516 q^{71} +7.40880 q^{73} +1.00000 q^{75} -1.06709 q^{77} -14.9748 q^{79} +1.00000 q^{81} -1.14003 q^{83} +0.115680 q^{85} -6.28017 q^{87} +9.99024 q^{89} -1.61641 q^{91} +5.30358 q^{93} +0.902140 q^{95} +5.69513 q^{97} +4.45891 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−0.239317	−0.0904533	−0.0452266	−	0.998977i	$-0.514401\pi$
$7$	−0.239317	−0.0904533	−0.0452266	+	0.998977i	$0.514401\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.45891	1.34441	0.672205	−	0.740365i	$-0.265347\pi$
$11$	4.45891	1.34441	0.672205	+	0.740365i	$0.265347\pi$
$12$	0	0
$13$	6.75426	1.87330	0.936648	−	0.350273i	$-0.113911\pi$
$13$	6.75426	1.87330	0.936648	+	0.350273i	$0.113911\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	0.115680	0.0280565	0.0140283	−	0.999902i	$-0.495535\pi$
$17$	0.115680	0.0280565	0.0140283	+	0.999902i	$0.495535\pi$
$18$	0	0
$19$	0.902140	0.206965	0.103483	−	0.994631i	$-0.467001\pi$
$19$	0.902140	0.206965	0.103483	+	0.994631i	$0.467001\pi$
$20$	0	0
$21$	−0.239317	−0.0522232
$22$	0	0
$23$	8.04085	1.67663	0.838317	−	0.545183i	$-0.183540\pi$
$23$	8.04085	1.67663	0.838317	+	0.545183i	$0.183540\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.28017	−1.16620	−0.583099	−	0.812401i	$-0.698160\pi$
$29$	−6.28017	−1.16620	−0.583099	+	0.812401i	$0.698160\pi$
$30$	0	0
$31$	5.30358	0.952551	0.476275	−	0.879296i	$-0.341987\pi$
$31$	5.30358	0.952551	0.476275	+	0.879296i	$0.341987\pi$
$32$	0	0
$33$	4.45891	0.776196
$34$	0	0
$35$	−0.239317	−0.0404519
$36$	0	0
$37$	−11.4005	−1.87423	−0.937113	−	0.349026i	$-0.886512\pi$
$37$	−11.4005	−1.87423	−0.937113	+	0.349026i	$0.886512\pi$
$38$	0	0
$39$	6.75426	1.08155
$40$	0	0
$41$	1.22063	0.190631	0.0953156	−	0.995447i	$-0.469614\pi$
$41$	1.22063	0.190631	0.0953156	+	0.995447i	$0.469614\pi$
$42$	0	0
$43$	−3.53276	−0.538741	−0.269371	−	0.963037i	$-0.586816\pi$
$43$	−3.53276	−0.538741	−0.269371	+	0.963037i	$0.586816\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−4.40972	−0.643224	−0.321612	−	0.946872i	$-0.604225\pi$
$47$	−4.40972	−0.643224	−0.321612	+	0.946872i	$0.604225\pi$
$48$	0	0
$49$	−6.94273	−0.991818
$50$	0	0
$51$	0.115680	0.0161984
$52$	0	0
$53$	−6.23099	−0.855892	−0.427946	−	0.903804i	$-0.640763\pi$
$53$	−6.23099	−0.855892	−0.427946	+	0.903804i	$0.640763\pi$
$54$	0	0
$55$	4.45891	0.601239
$56$	0	0
$57$	0.902140	0.119491
$58$	0	0
$59$	−8.41284	−1.09526	−0.547629	−	0.836721i	$-0.684469\pi$
$59$	−8.41284	−1.09526	−0.547629	+	0.836721i	$0.684469\pi$
$60$	0	0
$61$	5.35001	0.684998	0.342499	−	0.939518i	$-0.388727\pi$
$61$	5.35001	0.684998	0.342499	+	0.939518i	$0.388727\pi$
$62$	0	0
$63$	−0.239317	−0.0301511
$64$	0	0
$65$	6.75426	0.837763
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	8.04085	0.968005
$70$	0	0
$71$	6.97516	0.827800	0.413900	−	0.910322i	$-0.364166\pi$
$71$	6.97516	0.827800	0.413900	+	0.910322i	$0.364166\pi$
$72$	0	0
$73$	7.40880	0.867134	0.433567	−	0.901121i	$-0.357255\pi$
$73$	7.40880	0.867134	0.433567	+	0.901121i	$0.357255\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−1.06709	−0.121606
$78$	0	0
$79$	−14.9748	−1.68480	−0.842398	−	0.538856i	$-0.818857\pi$
$79$	−14.9748	−1.68480	−0.842398	+	0.538856i	$0.818857\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.14003	−0.125134	−0.0625671	−	0.998041i	$-0.519929\pi$
$83$	−1.14003	−0.125134	−0.0625671	+	0.998041i	$0.519929\pi$
$84$	0	0
$85$	0.115680	0.0125473
$86$	0	0
$87$	−6.28017	−0.673305
$88$	0	0
$89$	9.99024	1.05896	0.529482	−	0.848321i	$-0.322386\pi$
$89$	9.99024	1.05896	0.529482	+	0.848321i	$0.322386\pi$
$90$	0	0
$91$	−1.61641	−0.169446
$92$	0	0
$93$	5.30358	0.549955
$94$	0	0
$95$	0.902140	0.0925576
$96$	0	0
$97$	5.69513	0.578252	0.289126	−	0.957291i	$-0.406635\pi$
$97$	5.69513	0.578252	0.289126	+	0.957291i	$0.406635\pi$
$98$	0	0
$99$	4.45891	0.448137
$100$	0	0
$101$	6.57117	0.653856	0.326928	−	0.945049i	$-0.393987\pi$
$101$	6.57117	0.653856	0.326928	+	0.945049i	$0.393987\pi$
$102$	0	0
$103$	−4.70783	−0.463877	−0.231938	−	0.972730i	$-0.574507\pi$
$103$	−4.70783	−0.463877	−0.231938	+	0.972730i	$0.574507\pi$
$104$	0	0
$105$	−0.239317	−0.0233549
$106$	0	0
$107$	0.768324	0.0742767	0.0371383	−	0.999310i	$-0.488176\pi$
$107$	0.768324	0.0742767	0.0371383	+	0.999310i	$0.488176\pi$
$108$	0	0
$109$	18.2215	1.74530	0.872650	−	0.488346i	$-0.162400\pi$
$109$	18.2215	1.74530	0.872650	+	0.488346i	$0.162400\pi$
$110$	0	0
$111$	−11.4005	−1.08208
$112$	0	0
$113$	−7.24829	−0.681862	−0.340931	−	0.940088i	$-0.610742\pi$
$113$	−7.24829	−0.681862	−0.340931	+	0.940088i	$0.610742\pi$
$114$	0	0
$115$	8.04085	0.749814
$116$	0	0
$117$	6.75426	0.624432
$118$	0	0
$119$	−0.0276842	−0.00253780
$120$	0	0
$121$	8.88185	0.807441
$122$	0	0
$123$	1.22063	0.110061
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.30923	−0.204911	−0.102455	−	0.994738i	$-0.532670\pi$
$127$	−2.30923	−0.204911	−0.102455	+	0.994738i	$0.532670\pi$
$128$	0	0
$129$	−3.53276	−0.311043
$130$	0	0
$131$	−10.3006	−0.899970	−0.449985	−	0.893036i	$-0.648571\pi$
$131$	−10.3006	−0.899970	−0.449985	+	0.893036i	$0.648571\pi$
$132$	0	0
$133$	−0.215897	−0.0187207
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	12.1210	1.03556	0.517782	−	0.855512i	$-0.326758\pi$
$137$	12.1210	1.03556	0.517782	+	0.855512i	$0.326758\pi$
$138$	0	0
$139$	−22.1930	−1.88239	−0.941193	−	0.337871i	$-0.890293\pi$
$139$	−22.1930	−1.88239	−0.941193	+	0.337871i	$0.890293\pi$
$140$	0	0
$141$	−4.40972	−0.371366
$142$	0	0
$143$	30.1166	2.51848
$144$	0	0
$145$	−6.28017	−0.521540
$146$	0	0
$147$	−6.94273	−0.572627
$148$	0	0
$149$	−2.68884	−0.220278	−0.110139	−	0.993916i	$-0.535130\pi$
$149$	−2.68884	−0.220278	−0.110139	+	0.993916i	$0.535130\pi$
$150$	0	0
$151$	5.41907	0.440997	0.220499	−	0.975387i	$-0.429232\pi$
$151$	5.41907	0.440997	0.220499	+	0.975387i	$0.429232\pi$
$152$	0	0
$153$	0.115680	0.00935218
$154$	0	0
$155$	5.30358	0.425994
$156$	0	0
$157$	−7.49088	−0.597837	−0.298919	−	0.954279i	$-0.596626\pi$
$157$	−7.49088	−0.597837	−0.298919	+	0.954279i	$0.596626\pi$
$158$	0	0
$159$	−6.23099	−0.494149
$160$	0	0
$161$	−1.92431	−0.151657
$162$	0	0
$163$	20.4600	1.60255	0.801276	−	0.598295i	$-0.204155\pi$
$163$	20.4600	1.60255	0.801276	+	0.598295i	$0.204155\pi$
$164$	0	0
$165$	4.45891	0.347125
$166$	0	0
$167$	19.2979	1.49332	0.746658	−	0.665208i	$-0.231657\pi$
$167$	19.2979	1.49332	0.746658	+	0.665208i	$0.231657\pi$
$168$	0	0
$169$	32.6200	2.50923
$170$	0	0
$171$	0.902140	0.0689884
$172$	0	0
$173$	−11.6054	−0.882343	−0.441172	−	0.897423i	$-0.645437\pi$
$173$	−11.6054	−0.882343	−0.441172	+	0.897423i	$0.645437\pi$
$174$	0	0
$175$	−0.239317	−0.0180907
$176$	0	0
$177$	−8.41284	−0.632348
$178$	0	0
$179$	18.5890	1.38941	0.694704	−	0.719295i	$-0.255535\pi$
$179$	18.5890	1.38941	0.694704	+	0.719295i	$0.255535\pi$
$180$	0	0
$181$	8.75881	0.651037	0.325518	−	0.945536i	$-0.394461\pi$
$181$	8.75881	0.651037	0.325518	+	0.945536i	$0.394461\pi$
$182$	0	0
$183$	5.35001	0.395484
$184$	0	0
$185$	−11.4005	−0.838179
$186$	0	0
$187$	0.515806	0.0377195
$188$	0	0
$189$	−0.239317	−0.0174077
$190$	0	0
$191$	−5.20323	−0.376492	−0.188246	−	0.982122i	$-0.560280\pi$
$191$	−5.20323	−0.376492	−0.188246	+	0.982122i	$0.560280\pi$
$192$	0	0
$193$	7.33586	0.528047	0.264024	−	0.964516i	$-0.414950\pi$
$193$	7.33586	0.528047	0.264024	+	0.964516i	$0.414950\pi$
$194$	0	0
$195$	6.75426	0.483683
$196$	0	0
$197$	17.1781	1.22389	0.611946	−	0.790900i	$-0.290387\pi$
$197$	17.1781	1.22389	0.611946	+	0.790900i	$0.290387\pi$
$198$	0	0
$199$	−6.18763	−0.438630	−0.219315	−	0.975654i	$-0.570382\pi$
$199$	−6.18763	−0.438630	−0.219315	+	0.975654i	$0.570382\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	1.50295	0.105486
$204$	0	0
$205$	1.22063	0.0852528
$206$	0	0
$207$	8.04085	0.558878
$208$	0	0
$209$	4.02256	0.278246
$210$	0	0
$211$	−12.4991	−0.860474	−0.430237	−	0.902716i	$-0.641570\pi$
$211$	−12.4991	−0.860474	−0.430237	+	0.902716i	$0.641570\pi$
$212$	0	0
$213$	6.97516	0.477930
$214$	0	0
$215$	−3.53276	−0.240933
$216$	0	0
$217$	−1.26924	−0.0861613
$218$	0	0
$219$	7.40880	0.500640
$220$	0	0
$221$	0.781333	0.0525582
$222$	0	0
$223$	−12.1224	−0.811776	−0.405888	−	0.913923i	$-0.633038\pi$
$223$	−12.1224	−0.811776	−0.405888	+	0.913923i	$0.633038\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−12.1770	−0.808215	−0.404108	−	0.914711i	$-0.632418\pi$
$227$	−12.1770	−0.808215	−0.404108	+	0.914711i	$0.632418\pi$
$228$	0	0
$229$	−3.40227	−0.224829	−0.112414	−	0.993661i	$-0.535858\pi$
$229$	−3.40227	−0.224829	−0.112414	+	0.993661i	$0.535858\pi$
$230$	0	0
$231$	−1.06709	−0.0702095
$232$	0	0
$233$	8.84131	0.579213	0.289607	−	0.957146i	$-0.406475\pi$
$233$	8.84131	0.579213	0.289607	+	0.957146i	$0.406475\pi$
$234$	0	0
$235$	−4.40972	−0.287659
$236$	0	0
$237$	−14.9748	−0.972717
$238$	0	0
$239$	−15.4566	−0.999804	−0.499902	−	0.866082i	$-0.666631\pi$
$239$	−15.4566	−0.999804	−0.499902	+	0.866082i	$0.666631\pi$
$240$	0	0
$241$	−22.9350	−1.47737	−0.738687	−	0.674048i	$-0.764554\pi$
$241$	−22.9350	−1.47737	−0.738687	+	0.674048i	$0.764554\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−6.94273	−0.443555
$246$	0	0
$247$	6.09329	0.387707
$248$	0	0
$249$	−1.14003	−0.0722463
$250$	0	0
$251$	25.4264	1.60490	0.802451	−	0.596718i	$-0.203529\pi$
$251$	25.4264	1.60490	0.802451	+	0.596718i	$0.203529\pi$
$252$	0	0
$253$	35.8534	2.25409
$254$	0	0
$255$	0.115680	0.00724416
$256$	0	0
$257$	−3.36553	−0.209936	−0.104968	−	0.994476i	$-0.533474\pi$
$257$	−3.36553	−0.209936	−0.104968	+	0.994476i	$0.533474\pi$
$258$	0	0
$259$	2.72833	0.169530
$260$	0	0
$261$	−6.28017	−0.388733
$262$	0	0
$263$	−1.90709	−0.117596	−0.0587980	−	0.998270i	$-0.518727\pi$
$263$	−1.90709	−0.117596	−0.0587980	+	0.998270i	$0.518727\pi$
$264$	0	0
$265$	−6.23099	−0.382767
$266$	0	0
$267$	9.99024	0.611393
$268$	0	0
$269$	−27.4740	−1.67512	−0.837561	−	0.546344i	$-0.816019\pi$
$269$	−27.4740	−1.67512	−0.837561	+	0.546344i	$0.816019\pi$
$270$	0	0
$271$	−2.64265	−0.160530	−0.0802648	−	0.996774i	$-0.525577\pi$
$271$	−2.64265	−0.160530	−0.0802648	+	0.996774i	$0.525577\pi$
$272$	0	0
$273$	−1.61641	−0.0978295
$274$	0	0
$275$	4.45891	0.268882
$276$	0	0
$277$	14.8423	0.891787	0.445894	−	0.895086i	$-0.352886\pi$
$277$	14.8423	0.891787	0.445894	+	0.895086i	$0.352886\pi$
$278$	0	0
$279$	5.30358	0.317517
$280$	0	0
$281$	−10.2335	−0.610479	−0.305239	−	0.952276i	$-0.598736\pi$
$281$	−10.2335	−0.610479	−0.305239	+	0.952276i	$0.598736\pi$
$282$	0	0
$283$	−24.4994	−1.45634	−0.728169	−	0.685398i	$-0.759628\pi$
$283$	−24.4994	−1.45634	−0.728169	+	0.685398i	$0.759628\pi$
$284$	0	0
$285$	0.902140	0.0534382
$286$	0	0
$287$	−0.292119	−0.0172432
$288$	0	0
$289$	−16.9866	−0.999213
$290$	0	0
$291$	5.69513	0.333854
$292$	0	0
$293$	10.0102	0.584804	0.292402	−	0.956295i	$-0.405545\pi$
$293$	10.0102	0.584804	0.292402	+	0.956295i	$0.405545\pi$
$294$	0	0
$295$	−8.41284	−0.489814
$296$	0	0
$297$	4.45891	0.258732
$298$	0	0
$299$	54.3100	3.14083
$300$	0	0
$301$	0.845450	0.0487309
$302$	0	0
$303$	6.57117	0.377504
$304$	0	0
$305$	5.35001	0.306340
$306$	0	0
$307$	−0.0918408	−0.00524163	−0.00262081	−	0.999997i	$-0.500834\pi$
$307$	−0.0918408	−0.00524163	−0.00262081	+	0.999997i	$0.500834\pi$
$308$	0	0
$309$	−4.70783	−0.267819
$310$	0	0
$311$	−23.2066	−1.31593	−0.657964	−	0.753049i	$-0.728582\pi$
$311$	−23.2066	−1.31593	−0.657964	+	0.753049i	$0.728582\pi$
$312$	0	0
$313$	−31.7123	−1.79248	−0.896241	−	0.443567i	$-0.853713\pi$
$313$	−31.7123	−1.79248	−0.896241	+	0.443567i	$0.853713\pi$
$314$	0	0
$315$	−0.239317	−0.0134840
$316$	0	0
$317$	21.1237	1.18642	0.593212	−	0.805046i	$-0.297859\pi$
$317$	21.1237	1.18642	0.593212	+	0.805046i	$0.297859\pi$
$318$	0	0
$319$	−28.0027	−1.56785
$320$	0	0
$321$	0.768324	0.0428837
$322$	0	0
$323$	0.104360	0.00580672
$324$	0	0
$325$	6.75426	0.374659
$326$	0	0
$327$	18.2215	1.00765
$328$	0	0
$329$	1.05532	0.0581817
$330$	0	0
$331$	−28.9892	−1.59339	−0.796694	−	0.604383i	$-0.793420\pi$
$331$	−28.9892	−1.59339	−0.796694	+	0.604383i	$0.793420\pi$
$332$	0	0
$333$	−11.4005	−0.624742
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−16.0674	−0.875249	−0.437625	−	0.899158i	$-0.644180\pi$
$337$	−16.0674	−0.875249	−0.437625	+	0.899158i	$0.644180\pi$
$338$	0	0
$339$	−7.24829	−0.393673
$340$	0	0
$341$	23.6482	1.28062
$342$	0	0
$343$	3.33673	0.180166
$344$	0	0
$345$	8.04085	0.432905
$346$	0	0
$347$	−5.15119	−0.276530	−0.138265	−	0.990395i	$-0.544153\pi$
$347$	−5.15119	−0.276530	−0.138265	+	0.990395i	$0.544153\pi$
$348$	0	0
$349$	−23.1721	−1.24037	−0.620186	−	0.784455i	$-0.712943\pi$
$349$	−23.1721	−1.24037	−0.620186	+	0.784455i	$0.712943\pi$
$350$	0	0
$351$	6.75426	0.360516
$352$	0	0
$353$	11.1246	0.592104	0.296052	−	0.955172i	$-0.404330\pi$
$353$	11.1246	0.592104	0.296052	+	0.955172i	$0.404330\pi$
$354$	0	0
$355$	6.97516	0.370203
$356$	0	0
$357$	−0.0276842	−0.00146520
$358$	0	0
$359$	−7.96592	−0.420425	−0.210213	−	0.977656i	$-0.567416\pi$
$359$	−7.96592	−0.420425	−0.210213	+	0.977656i	$0.567416\pi$
$360$	0	0
$361$	−18.1861	−0.957165
$362$	0	0
$363$	8.88185	0.466176
$364$	0	0
$365$	7.40880	0.387794
$366$	0	0
$367$	−19.8621	−1.03679	−0.518397	−	0.855140i	$-0.673471\pi$
$367$	−19.8621	−1.03679	−0.518397	+	0.855140i	$0.673471\pi$
$368$	0	0
$369$	1.22063	0.0635437
$370$	0	0
$371$	1.49118	0.0774182
$372$	0	0
$373$	33.6756	1.74366	0.871829	−	0.489811i	$-0.162934\pi$
$373$	33.6756	1.74366	0.871829	+	0.489811i	$0.162934\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−42.4179	−2.18463
$378$	0	0
$379$	−29.1785	−1.49880	−0.749398	−	0.662120i	$-0.769657\pi$
$379$	−29.1785	−1.49880	−0.749398	+	0.662120i	$0.769657\pi$
$380$	0	0
$381$	−2.30923	−0.118305
$382$	0	0
$383$	−24.1456	−1.23378	−0.616891	−	0.787048i	$-0.711608\pi$
$383$	−24.1456	−1.23378	−0.616891	+	0.787048i	$0.711608\pi$
$384$	0	0
$385$	−1.06709	−0.0543840
$386$	0	0
$387$	−3.53276	−0.179580
$388$	0	0
$389$	12.2758	0.622406	0.311203	−	0.950344i	$-0.399268\pi$
$389$	12.2758	0.622406	0.311203	+	0.950344i	$0.399268\pi$
$390$	0	0
$391$	0.930166	0.0470405
$392$	0	0
$393$	−10.3006	−0.519598
$394$	0	0
$395$	−14.9748	−0.753463
$396$	0	0
$397$	−11.9043	−0.597462	−0.298731	−	0.954337i	$-0.596563\pi$
$397$	−11.9043	−0.597462	−0.298731	+	0.954337i	$0.596563\pi$
$398$	0	0
$399$	−0.215897	−0.0108084
$400$	0	0
$401$	22.7618	1.13667	0.568335	−	0.822797i	$-0.307588\pi$
$401$	22.7618	1.13667	0.568335	+	0.822797i	$0.307588\pi$
$402$	0	0
$403$	35.8218	1.78441
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−50.8336	−2.51973
$408$	0	0
$409$	8.83216	0.436722	0.218361	−	0.975868i	$-0.429929\pi$
$409$	8.83216	0.436722	0.218361	+	0.975868i	$0.429929\pi$
$410$	0	0
$411$	12.1210	0.597883
$412$	0	0
$413$	2.01333	0.0990697
$414$	0	0
$415$	−1.14003	−0.0559617
$416$	0	0
$417$	−22.1930	−1.08680
$418$	0	0
$419$	21.2224	1.03678	0.518390	−	0.855144i	$-0.326532\pi$
$419$	21.2224	1.03678	0.518390	+	0.855144i	$0.326532\pi$
$420$	0	0
$421$	33.1120	1.61378	0.806891	−	0.590700i	$-0.201148\pi$
$421$	33.1120	1.61378	0.806891	+	0.590700i	$0.201148\pi$
$422$	0	0
$423$	−4.40972	−0.214408
$424$	0	0
$425$	0.115680	0.00561131
$426$	0	0
$427$	−1.28035	−0.0619603
$428$	0	0
$429$	30.1166	1.45404
$430$	0	0
$431$	28.0250	1.34991	0.674957	−	0.737857i	$-0.264162\pi$
$431$	28.0250	1.34991	0.674957	+	0.737857i	$0.264162\pi$
$432$	0	0
$433$	19.2755	0.926322	0.463161	−	0.886274i	$-0.346715\pi$
$433$	19.2755	0.926322	0.463161	+	0.886274i	$0.346715\pi$
$434$	0	0
$435$	−6.28017	−0.301111
$436$	0	0
$437$	7.25397	0.347005
$438$	0	0
$439$	32.3477	1.54387	0.771935	−	0.635702i	$-0.219289\pi$
$439$	32.3477	1.54387	0.771935	+	0.635702i	$0.219289\pi$
$440$	0	0
$441$	−6.94273	−0.330606
$442$	0	0
$443$	−12.7791	−0.607153	−0.303576	−	0.952807i	$-0.598181\pi$
$443$	−12.7791	−0.607153	−0.303576	+	0.952807i	$0.598181\pi$
$444$	0	0
$445$	9.99024	0.473583
$446$	0	0
$447$	−2.68884	−0.127178
$448$	0	0
$449$	1.36200	0.0642768	0.0321384	−	0.999483i	$-0.489768\pi$
$449$	1.36200	0.0642768	0.0321384	+	0.999483i	$0.489768\pi$
$450$	0	0
$451$	5.44270	0.256287
$452$	0	0
$453$	5.41907	0.254610
$454$	0	0
$455$	−1.61641	−0.0757784
$456$	0	0
$457$	3.53203	0.165222	0.0826108	−	0.996582i	$-0.473674\pi$
$457$	3.53203	0.165222	0.0826108	+	0.996582i	$0.473674\pi$
$458$	0	0
$459$	0.115680	0.00539948
$460$	0	0
$461$	7.62815	0.355278	0.177639	−	0.984096i	$-0.443154\pi$
$461$	7.62815	0.355278	0.177639	+	0.984096i	$0.443154\pi$
$462$	0	0
$463$	−5.20163	−0.241740	−0.120870	−	0.992668i	$-0.538568\pi$
$463$	−5.20163	−0.241740	−0.120870	+	0.992668i	$0.538568\pi$
$464$	0	0
$465$	5.30358	0.245948
$466$	0	0
$467$	40.9172	1.89342	0.946712	−	0.322083i	$-0.104383\pi$
$467$	40.9172	1.89342	0.946712	+	0.322083i	$0.104383\pi$
$468$	0	0
$469$	0.239317	0.0110506
$470$	0	0
$471$	−7.49088	−0.345162
$472$	0	0
$473$	−15.7523	−0.724290
$474$	0	0
$475$	0.902140	0.0413930
$476$	0	0
$477$	−6.23099	−0.285297
$478$	0	0
$479$	12.9892	0.593491	0.296745	−	0.954957i	$-0.404099\pi$
$479$	12.9892	0.593491	0.296745	+	0.954957i	$0.404099\pi$
$480$	0	0
$481$	−77.0018	−3.51098
$482$	0	0
$483$	−1.92431	−0.0875592
$484$	0	0
$485$	5.69513	0.258602
$486$	0	0
$487$	−10.2186	−0.463049	−0.231524	−	0.972829i	$-0.574371\pi$
$487$	−10.2186	−0.463049	−0.231524	+	0.972829i	$0.574371\pi$
$488$	0	0
$489$	20.4600	0.925234
$490$	0	0
$491$	−11.2180	−0.506260	−0.253130	−	0.967432i	$-0.581460\pi$
$491$	−11.2180	−0.506260	−0.253130	+	0.967432i	$0.581460\pi$
$492$	0	0
$493$	−0.726490	−0.0327195
$494$	0	0
$495$	4.45891	0.200413
$496$	0	0
$497$	−1.66927	−0.0748772
$498$	0	0
$499$	−11.9028	−0.532841	−0.266421	−	0.963857i	$-0.585841\pi$
$499$	−11.9028	−0.532841	−0.266421	+	0.963857i	$0.585841\pi$
$500$	0	0
$501$	19.2979	0.862166
$502$	0	0
$503$	26.7653	1.19341	0.596703	−	0.802462i	$-0.296477\pi$
$503$	26.7653	1.19341	0.596703	+	0.802462i	$0.296477\pi$
$504$	0	0
$505$	6.57117	0.292413
$506$	0	0
$507$	32.6200	1.44871
$508$	0	0
$509$	21.5523	0.955290	0.477645	−	0.878553i	$-0.341490\pi$
$509$	21.5523	0.955290	0.477645	+	0.878553i	$0.341490\pi$
$510$	0	0
$511$	−1.77305	−0.0784352
$512$	0	0
$513$	0.902140	0.0398304
$514$	0	0
$515$	−4.70783	−0.207452
$516$	0	0
$517$	−19.6625	−0.864758
$518$	0	0
$519$	−11.6054	−0.509421
$520$	0	0
$521$	39.2527	1.71969	0.859845	−	0.510555i	$-0.170560\pi$
$521$	39.2527	1.71969	0.859845	+	0.510555i	$0.170560\pi$
$522$	0	0
$523$	−31.4056	−1.37327	−0.686635	−	0.727003i	$-0.740913\pi$
$523$	−31.4056	−1.37327	−0.686635	+	0.727003i	$0.740913\pi$
$524$	0	0
$525$	−0.239317	−0.0104446
$526$	0	0
$527$	0.613518	0.0267253
$528$	0	0
$529$	41.6553	1.81110
$530$	0	0
$531$	−8.41284	−0.365086
$532$	0	0
$533$	8.24449	0.357108
$534$	0	0
$535$	0.768324	0.0332175
$536$	0	0
$537$	18.5890	0.802176
$538$	0	0
$539$	−30.9570	−1.33341
$540$	0	0
$541$	−41.3701	−1.77864	−0.889319	−	0.457287i	$-0.848821\pi$
$541$	−41.3701	−1.77864	−0.889319	+	0.457287i	$0.848821\pi$
$542$	0	0
$543$	8.75881	0.375876
$544$	0	0
$545$	18.2215	0.780522
$546$	0	0
$547$	−1.20609	−0.0515688	−0.0257844	−	0.999668i	$-0.508208\pi$
$547$	−1.20609	−0.0515688	−0.0257844	+	0.999668i	$0.508208\pi$
$548$	0	0
$549$	5.35001	0.228333
$550$	0	0
$551$	−5.66559	−0.241362
$552$	0	0
$553$	3.58372	0.152395
$554$	0	0
$555$	−11.4005	−0.483923
$556$	0	0
$557$	27.8437	1.17978	0.589888	−	0.807485i	$-0.299172\pi$
$557$	27.8437	1.17978	0.589888	+	0.807485i	$0.299172\pi$
$558$	0	0
$559$	−23.8612	−1.00922
$560$	0	0
$561$	0.515806	0.0217774
$562$	0	0
$563$	17.6031	0.741882	0.370941	−	0.928656i	$-0.379035\pi$
$563$	17.6031	0.741882	0.370941	+	0.928656i	$0.379035\pi$
$564$	0	0
$565$	−7.24829	−0.304938
$566$	0	0
$567$	−0.239317	−0.0100504
$568$	0	0
$569$	−29.4036	−1.23266	−0.616332	−	0.787487i	$-0.711382\pi$
$569$	−29.4036	−1.23266	−0.616332	+	0.787487i	$0.711382\pi$
$570$	0	0
$571$	6.74286	0.282180	0.141090	−	0.989997i	$-0.454939\pi$
$571$	6.74286	0.282180	0.141090	+	0.989997i	$0.454939\pi$
$572$	0	0
$573$	−5.20323	−0.217368
$574$	0	0
$575$	8.04085	0.335327
$576$	0	0
$577$	−12.4610	−0.518758	−0.259379	−	0.965776i	$-0.583518\pi$
$577$	−12.4610	−0.518758	−0.259379	+	0.965776i	$0.583518\pi$
$578$	0	0
$579$	7.33586	0.304868
$580$	0	0
$581$	0.272828	0.0113188
$582$	0	0
$583$	−27.7834	−1.15067
$584$	0	0
$585$	6.75426	0.279254
$586$	0	0
$587$	−24.5791	−1.01449	−0.507243	−	0.861803i	$-0.669335\pi$
$587$	−24.5791	−1.01449	−0.507243	+	0.861803i	$0.669335\pi$
$588$	0	0
$589$	4.78457	0.197145
$590$	0	0
$591$	17.1781	0.706614
$592$	0	0
$593$	22.3842	0.919210	0.459605	−	0.888123i	$-0.347991\pi$
$593$	22.3842	0.919210	0.459605	+	0.888123i	$0.347991\pi$
$594$	0	0
$595$	−0.0276842	−0.00113494
$596$	0	0
$597$	−6.18763	−0.253243
$598$	0	0
$599$	2.14694	0.0877215	0.0438607	−	0.999038i	$-0.486034\pi$
$599$	2.14694	0.0877215	0.0438607	+	0.999038i	$0.486034\pi$
$600$	0	0
$601$	−5.99848	−0.244683	−0.122342	−	0.992488i	$-0.539040\pi$
$601$	−5.99848	−0.244683	−0.122342	+	0.992488i	$0.539040\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	8.88185	0.361098
$606$	0	0
$607$	38.0671	1.54510	0.772548	−	0.634956i	$-0.218982\pi$
$607$	38.0671	1.54510	0.772548	+	0.634956i	$0.218982\pi$
$608$	0	0
$609$	1.50295	0.0609026
$610$	0	0
$611$	−29.7844	−1.20495
$612$	0	0
$613$	−2.17802	−0.0879694	−0.0439847	−	0.999032i	$-0.514005\pi$
$613$	−2.17802	−0.0879694	−0.0439847	+	0.999032i	$0.514005\pi$
$614$	0	0
$615$	1.22063	0.0492208
$616$	0	0
$617$	39.7907	1.60191	0.800956	−	0.598724i	$-0.204325\pi$
$617$	39.7907	1.60191	0.800956	+	0.598724i	$0.204325\pi$
$618$	0	0
$619$	23.0047	0.924639	0.462319	−	0.886713i	$-0.347017\pi$
$619$	23.0047	0.924639	0.462319	+	0.886713i	$0.347017\pi$
$620$	0	0
$621$	8.04085	0.322668
$622$	0	0
$623$	−2.39083	−0.0957867
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.02256	0.160645
$628$	0	0
$629$	−1.31881	−0.0525843
$630$	0	0
$631$	−2.26900	−0.0903273	−0.0451636	−	0.998980i	$-0.514381\pi$
$631$	−2.26900	−0.0903273	−0.0451636	+	0.998980i	$0.514381\pi$
$632$	0	0
$633$	−12.4991	−0.496795
$634$	0	0
$635$	−2.30923	−0.0916388
$636$	0	0
$637$	−46.8930	−1.85797
$638$	0	0
$639$	6.97516	0.275933
$640$	0	0
$641$	−16.6825	−0.658918	−0.329459	−	0.944170i	$-0.606866\pi$
$641$	−16.6825	−0.658918	−0.329459	+	0.944170i	$0.606866\pi$
$642$	0	0
$643$	−17.2730	−0.681181	−0.340591	−	0.940212i	$-0.610627\pi$
$643$	−17.2730	−0.681181	−0.340591	+	0.940212i	$0.610627\pi$
$644$	0	0
$645$	−3.53276	−0.139102
$646$	0	0
$647$	−38.8271	−1.52645	−0.763225	−	0.646133i	$-0.776385\pi$
$647$	−38.8271	−1.52645	−0.763225	+	0.646133i	$0.776385\pi$
$648$	0	0
$649$	−37.5121	−1.47248
$650$	0	0
$651$	−1.26924	−0.0497453
$652$	0	0
$653$	35.8562	1.40316	0.701581	−	0.712590i	$-0.252478\pi$
$653$	35.8562	1.40316	0.701581	+	0.712590i	$0.252478\pi$
$654$	0	0
$655$	−10.3006	−0.402479
$656$	0	0
$657$	7.40880	0.289045
$658$	0	0
$659$	−49.7583	−1.93831	−0.969153	−	0.246458i	$-0.920733\pi$
$659$	−49.7583	−1.93831	−0.969153	+	0.246458i	$0.920733\pi$
$660$	0	0
$661$	4.34972	0.169185	0.0845923	−	0.996416i	$-0.473041\pi$
$661$	4.34972	0.169185	0.0845923	+	0.996416i	$0.473041\pi$
$662$	0	0
$663$	0.781333	0.0303445
$664$	0	0
$665$	−0.215897	−0.00837214
$666$	0	0
$667$	−50.4979	−1.95529
$668$	0	0
$669$	−12.1224	−0.468679
$670$	0	0
$671$	23.8552	0.920919
$672$	0	0
$673$	35.8360	1.38138	0.690688	−	0.723153i	$-0.257308\pi$
$673$	35.8360	1.38138	0.690688	+	0.723153i	$0.257308\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	5.41671	0.208181	0.104090	−	0.994568i	$-0.466807\pi$
$677$	5.41671	0.208181	0.104090	+	0.994568i	$0.466807\pi$
$678$	0	0
$679$	−1.36294	−0.0523048
$680$	0	0
$681$	−12.1770	−0.466623
$682$	0	0
$683$	−19.7744	−0.756645	−0.378323	−	0.925674i	$-0.623499\pi$
$683$	−19.7744	−0.756645	−0.378323	+	0.925674i	$0.623499\pi$
$684$	0	0
$685$	12.1210	0.463119
$686$	0	0
$687$	−3.40227	−0.129805
$688$	0	0
$689$	−42.0857	−1.60334
$690$	0	0
$691$	−19.8496	−0.755114	−0.377557	−	0.925986i	$-0.623236\pi$
$691$	−19.8496	−0.755114	−0.377557	+	0.925986i	$0.623236\pi$
$692$	0	0
$693$	−1.06709	−0.0405355
$694$	0	0
$695$	−22.1930	−0.841828
$696$	0	0
$697$	0.141203	0.00534845
$698$	0	0
$699$	8.84131	0.334409
$700$	0	0
$701$	−23.4238	−0.884704	−0.442352	−	0.896842i	$-0.645856\pi$
$701$	−23.4238	−0.884704	−0.442352	+	0.896842i	$0.645856\pi$
$702$	0	0
$703$	−10.2848	−0.387899
$704$	0	0
$705$	−4.40972	−0.166080
$706$	0	0
$707$	−1.57259	−0.0591434
$708$	0	0
$709$	−35.9921	−1.35171	−0.675856	−	0.737034i	$-0.736226\pi$
$709$	−35.9921	−1.35171	−0.675856	+	0.737034i	$0.736226\pi$
$710$	0	0
$711$	−14.9748	−0.561599
$712$	0	0
$713$	42.6453	1.59708
$714$	0	0
$715$	30.1166	1.12630
$716$	0	0
$717$	−15.4566	−0.577237
$718$	0	0
$719$	0.224912	0.00838780	0.00419390	−	0.999991i	$-0.498665\pi$
$719$	0.224912	0.00838780	0.00419390	+	0.999991i	$0.498665\pi$
$720$	0	0
$721$	1.12666	0.0419592
$722$	0	0
$723$	−22.9350	−0.852963
$724$	0	0
$725$	−6.28017	−0.233240
$726$	0	0
$727$	−10.8524	−0.402495	−0.201247	−	0.979540i	$-0.564499\pi$
$727$	−10.8524	−0.402495	−0.201247	+	0.979540i	$0.564499\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.408670	−0.0151152
$732$	0	0
$733$	−52.4969	−1.93902	−0.969508	−	0.245058i	$-0.921193\pi$
$733$	−52.4969	−1.93902	−0.969508	+	0.245058i	$0.921193\pi$
$734$	0	0
$735$	−6.94273	−0.256086
$736$	0	0
$737$	−4.45891	−0.164246
$738$	0	0
$739$	4.27646	0.157312	0.0786561	−	0.996902i	$-0.474937\pi$
$739$	4.27646	0.157312	0.0786561	+	0.996902i	$0.474937\pi$
$740$	0	0
$741$	6.09329	0.223843
$742$	0	0
$743$	20.7605	0.761629	0.380815	−	0.924651i	$-0.375644\pi$
$743$	20.7605	0.761629	0.380815	+	0.924651i	$0.375644\pi$
$744$	0	0
$745$	−2.68884	−0.0985113
$746$	0	0
$747$	−1.14003	−0.0417114
$748$	0	0
$749$	−0.183873	−0.00671857
$750$	0	0
$751$	−7.15079	−0.260936	−0.130468	−	0.991453i	$-0.541648\pi$
$751$	−7.15079	−0.260936	−0.130468	+	0.991453i	$0.541648\pi$
$752$	0	0
$753$	25.4264	0.926590
$754$	0	0
$755$	5.41907	0.197220
$756$	0	0
$757$	8.75854	0.318335	0.159167	−	0.987252i	$-0.449119\pi$
$757$	8.75854	0.318335	0.159167	+	0.987252i	$0.449119\pi$
$758$	0	0
$759$	35.8534	1.30140
$760$	0	0
$761$	−12.1938	−0.442025	−0.221013	−	0.975271i	$-0.570936\pi$
$761$	−12.1938	−0.442025	−0.221013	+	0.975271i	$0.570936\pi$
$762$	0	0
$763$	−4.36071	−0.157868
$764$	0	0
$765$	0.115680	0.00418242
$766$	0	0
$767$	−56.8225	−2.05174
$768$	0	0
$769$	−21.9163	−0.790321	−0.395161	−	0.918612i	$-0.629311\pi$
$769$	−21.9163	−0.790321	−0.395161	+	0.918612i	$0.629311\pi$
$770$	0	0
$771$	−3.36553	−0.121207
$772$	0	0
$773$	−1.32399	−0.0476206	−0.0238103	−	0.999716i	$-0.507580\pi$
$773$	−1.32399	−0.0476206	−0.0238103	+	0.999716i	$0.507580\pi$
$774$	0	0
$775$	5.30358	0.190510
$776$	0	0
$777$	2.72833	0.0978781
$778$	0	0
$779$	1.10118	0.0394540
$780$	0	0
$781$	31.1016	1.11290
$782$	0	0
$783$	−6.28017	−0.224435
$784$	0	0
$785$	−7.49088	−0.267361
$786$	0	0
$787$	−8.39980	−0.299421	−0.149710	−	0.988730i	$-0.547834\pi$
$787$	−8.39980	−0.299421	−0.149710	+	0.988730i	$0.547834\pi$
$788$	0	0
$789$	−1.90709	−0.0678941
$790$	0	0
$791$	1.73464	0.0616766
$792$	0	0
$793$	36.1353	1.28320
$794$	0	0
$795$	−6.23099	−0.220990
$796$	0	0
$797$	5.68112	0.201236	0.100618	−	0.994925i	$-0.467918\pi$
$797$	5.68112	0.201236	0.100618	+	0.994925i	$0.467918\pi$
$798$	0	0
$799$	−0.510117	−0.0180466
$800$	0	0
$801$	9.99024	0.352988
$802$	0	0
$803$	33.0351	1.16578
$804$	0	0
$805$	−1.92431	−0.0678231
$806$	0	0
$807$	−27.4740	−0.967132
$808$	0	0
$809$	−22.0998	−0.776989	−0.388495	−	0.921451i	$-0.627005\pi$
$809$	−22.0998	−0.776989	−0.388495	+	0.921451i	$0.627005\pi$
$810$	0	0
$811$	−39.0314	−1.37058	−0.685289	−	0.728271i	$-0.740324\pi$
$811$	−39.0314	−1.37058	−0.685289	+	0.728271i	$0.740324\pi$
$812$	0	0
$813$	−2.64265	−0.0926818
$814$	0	0
$815$	20.4600	0.716683
$816$	0	0
$817$	−3.18705	−0.111501
$818$	0	0
$819$	−1.61641	−0.0564819
$820$	0	0
$821$	−32.9534	−1.15008	−0.575041	−	0.818125i	$-0.695014\pi$
$821$	−32.9534	−1.15008	−0.575041	+	0.818125i	$0.695014\pi$
$822$	0	0
$823$	22.3729	0.779871	0.389935	−	0.920842i	$-0.372497\pi$
$823$	22.3729	0.779871	0.389935	+	0.920842i	$0.372497\pi$
$824$	0	0
$825$	4.45891	0.155239
$826$	0	0
$827$	27.4237	0.953615	0.476807	−	0.879008i	$-0.341794\pi$
$827$	27.4237	0.953615	0.476807	+	0.879008i	$0.341794\pi$
$828$	0	0
$829$	28.5949	0.993142	0.496571	−	0.867996i	$-0.334592\pi$
$829$	28.5949	0.993142	0.496571	+	0.867996i	$0.334592\pi$
$830$	0	0
$831$	14.8423	0.514874
$832$	0	0
$833$	−0.803135	−0.0278270
$834$	0	0
$835$	19.2979	0.667831
$836$	0	0
$837$	5.30358	0.183318
$838$	0	0
$839$	−30.0409	−1.03713	−0.518564	−	0.855039i	$-0.673533\pi$
$839$	−30.0409	−1.03713	−0.518564	+	0.855039i	$0.673533\pi$
$840$	0	0
$841$	10.4406	0.360019
$842$	0	0
$843$	−10.2335	−0.352460
$844$	0	0
$845$	32.6200	1.12216
$846$	0	0
$847$	−2.12558	−0.0730357
$848$	0	0
$849$	−24.4994	−0.840817
$850$	0	0
$851$	−91.6695	−3.14239
$852$	0	0
$853$	24.5470	0.840472	0.420236	−	0.907415i	$-0.361947\pi$
$853$	24.5470	0.840472	0.420236	+	0.907415i	$0.361947\pi$
$854$	0	0
$855$	0.902140	0.0308525
$856$	0	0
$857$	−45.1994	−1.54398	−0.771991	−	0.635634i	$-0.780739\pi$
$857$	−45.1994	−1.54398	−0.771991	+	0.635634i	$0.780739\pi$
$858$	0	0
$859$	−36.5442	−1.24687	−0.623435	−	0.781875i	$-0.714264\pi$
$859$	−36.5442	−1.24687	−0.623435	+	0.781875i	$0.714264\pi$
$860$	0	0
$861$	−0.292119	−0.00995537
$862$	0	0
$863$	27.8988	0.949685	0.474842	−	0.880071i	$-0.342505\pi$
$863$	27.8988	0.949685	0.474842	+	0.880071i	$0.342505\pi$
$864$	0	0
$865$	−11.6054	−0.394596
$866$	0	0
$867$	−16.9866	−0.576896
$868$	0	0
$869$	−66.7712	−2.26506
$870$	0	0
$871$	−6.75426	−0.228859
$872$	0	0
$873$	5.69513	0.192751
$874$	0	0
$875$	−0.239317	−0.00809039
$876$	0	0
$877$	37.1941	1.25596	0.627978	−	0.778231i	$-0.283883\pi$
$877$	37.1941	1.25596	0.627978	+	0.778231i	$0.283883\pi$
$878$	0	0
$879$	10.0102	0.337637
$880$	0	0
$881$	−38.5905	−1.30015	−0.650074	−	0.759871i	$-0.725262\pi$
$881$	−38.5905	−1.30015	−0.650074	+	0.759871i	$0.725262\pi$
$882$	0	0
$883$	1.07033	0.0360194	0.0180097	−	0.999838i	$-0.494267\pi$
$883$	1.07033	0.0360194	0.0180097	+	0.999838i	$0.494267\pi$
$884$	0	0
$885$	−8.41284	−0.282794
$886$	0	0
$887$	23.9337	0.803616	0.401808	−	0.915724i	$-0.368382\pi$
$887$	23.9337	0.803616	0.401808	+	0.915724i	$0.368382\pi$
$888$	0	0
$889$	0.552637	0.0185348
$890$	0	0
$891$	4.45891	0.149379
$892$	0	0
$893$	−3.97819	−0.133125
$894$	0	0
$895$	18.5890	0.621363
$896$	0	0
$897$	54.3100	1.81336
$898$	0	0
$899$	−33.3074	−1.11086
$900$	0	0
$901$	−0.720801	−0.0240134
$902$	0	0
$903$	0.845450	0.0281348
$904$	0	0
$905$	8.75881	0.291153
$906$	0	0
$907$	25.1414	0.834806	0.417403	−	0.908721i	$-0.362940\pi$
$907$	25.1414	0.834806	0.417403	+	0.908721i	$0.362940\pi$
$908$	0	0
$909$	6.57117	0.217952
$910$	0	0
$911$	−2.07413	−0.0687189	−0.0343595	−	0.999410i	$-0.510939\pi$
$911$	−2.07413	−0.0687189	−0.0343595	+	0.999410i	$0.510939\pi$
$912$	0	0
$913$	−5.08327	−0.168232
$914$	0	0
$915$	5.35001	0.176866
$916$	0	0
$917$	2.46511	0.0814052
$918$	0	0
$919$	37.0355	1.22169	0.610844	−	0.791751i	$-0.290830\pi$
$919$	37.0355	1.22169	0.610844	+	0.791751i	$0.290830\pi$
$920$	0	0
$921$	−0.0918408	−0.00302626
$922$	0	0
$923$	47.1121	1.55071
$924$	0	0
$925$	−11.4005	−0.374845
$926$	0	0
$927$	−4.70783	−0.154626
$928$	0	0
$929$	4.94564	0.162261	0.0811306	−	0.996703i	$-0.474147\pi$
$929$	4.94564	0.162261	0.0811306	+	0.996703i	$0.474147\pi$
$930$	0	0
$931$	−6.26331	−0.205272
$932$	0	0
$933$	−23.2066	−0.759751
$934$	0	0
$935$	0.515806	0.0168687
$936$	0	0
$937$	7.22904	0.236163	0.118081	−	0.993004i	$-0.462326\pi$
$937$	7.22904	0.236163	0.118081	+	0.993004i	$0.462326\pi$
$938$	0	0
$939$	−31.7123	−1.03489
$940$	0	0
$941$	31.5780	1.02941	0.514707	−	0.857366i	$-0.327901\pi$
$941$	31.5780	1.02941	0.514707	+	0.857366i	$0.327901\pi$
$942$	0	0
$943$	9.81495	0.319619
$944$	0	0
$945$	−0.239317	−0.00778498
$946$	0	0
$947$	−8.49839	−0.276161	−0.138080	−	0.990421i	$-0.544093\pi$
$947$	−8.49839	−0.276161	−0.138080	+	0.990421i	$0.544093\pi$
$948$	0	0
$949$	50.0410	1.62440
$950$	0	0
$951$	21.1237	0.684983
$952$	0	0
$953$	6.31235	0.204477	0.102238	−	0.994760i	$-0.467400\pi$
$953$	6.31235	0.204477	0.102238	+	0.994760i	$0.467400\pi$
$954$	0	0
$955$	−5.20323	−0.168372
$956$	0	0
$957$	−28.0027	−0.905199
$958$	0	0
$959$	−2.90075	−0.0936702
$960$	0	0
$961$	−2.87206	−0.0926470
$962$	0	0
$963$	0.768324	0.0247589
$964$	0	0
$965$	7.33586	0.236150
$966$	0	0
$967$	−33.2633	−1.06968	−0.534838	−	0.844955i	$-0.679627\pi$
$967$	−33.2633	−1.06968	−0.534838	+	0.844955i	$0.679627\pi$
$968$	0	0
$969$	0.104360	0.00335251
$970$	0	0
$971$	−32.6905	−1.04909	−0.524544	−	0.851383i	$-0.675764\pi$
$971$	−32.6905	−1.04909	−0.524544	+	0.851383i	$0.675764\pi$
$972$	0	0
$973$	5.31116	0.170268
$974$	0	0
$975$	6.75426	0.216309
$976$	0	0
$977$	−3.84714	−0.123081	−0.0615405	−	0.998105i	$-0.519601\pi$
$977$	−3.84714	−0.123081	−0.0615405	+	0.998105i	$0.519601\pi$
$978$	0	0
$979$	44.5455	1.42368
$980$	0	0
$981$	18.2215	0.581767
$982$	0	0
$983$	49.2286	1.57015	0.785075	−	0.619401i	$-0.212624\pi$
$983$	49.2286	1.57015	0.785075	+	0.619401i	$0.212624\pi$
$984$	0	0
$985$	17.1781	0.547341
$986$	0	0
$987$	1.05532	0.0335912
$988$	0	0
$989$	−28.4064	−0.903272
$990$	0	0
$991$	−49.3727	−1.56838	−0.784188	−	0.620523i	$-0.786920\pi$
$991$	−49.3727	−1.56838	−0.784188	+	0.620523i	$0.786920\pi$
$992$	0	0
$993$	−28.9892	−0.919943
$994$	0	0
$995$	−6.18763	−0.196161
$996$	0	0
$997$	−12.4895	−0.395546	−0.197773	−	0.980248i	$-0.563371\pi$
$997$	−12.4895	−0.395546	−0.197773	+	0.980248i	$0.563371\pi$
$998$	0	0
$999$	−11.4005	−0.360695

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.i.1.3	✓	7

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -0.239317 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -0.239317 q^{7} +1.00000 q^{9} +4.45891 q^{11} +6.75426 q^{13} +1.00000 q^{15} +0.115680 q^{17} +0.902140 q^{19} -0.239317 q^{21} +8.04085 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.28017 q^{29} +5.30358 q^{31} +4.45891 q^{33} -0.239317 q^{35} -11.4005 q^{37} +6.75426 q^{39} +1.22063 q^{41} -3.53276 q^{43} +1.00000 q^{45} -4.40972 q^{47} -6.94273 q^{49} +0.115680 q^{51} -6.23099 q^{53} +4.45891 q^{55} +0.902140 q^{57} -8.41284 q^{59} +5.35001 q^{61} -0.239317 q^{63} +6.75426 q^{65} -1.00000 q^{67} +8.04085 q^{69} +6.97516 q^{71} +7.40880 q^{73} +1.00000 q^{75} -1.06709 q^{77} -14.9748 q^{79} +1.00000 q^{81} -1.14003 q^{83} +0.115680 q^{85} -6.28017 q^{87} +9.99024 q^{89} -1.61641 q^{91} +5.30358 q^{93} +0.902140 q^{95} +5.69513 q^{97} +4.45891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more