LMFDB - Embedded newform 4020.2.a.e.1.1

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			1.1
Root			$1.14431$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -4.78538 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -4.78538 q^{7} +1.00000 q^{9} +1.53257 q^{11} -4.69056 q^{13} -1.00000 q^{15} +2.92969 q^{17} +6.94337 q^{19} +4.78538 q^{21} +2.12449 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.88251 q^{29} +2.53027 q^{31} -1.53257 q^{33} -4.78538 q^{35} -1.09898 q^{37} +4.69056 q^{39} -6.75188 q^{41} +11.1320 q^{43} +1.00000 q^{45} +0.205628 q^{47} +15.8999 q^{49} -2.92969 q^{51} -8.49807 q^{53} +1.53257 q^{55} -6.94337 q^{57} -1.17550 q^{59} +5.62256 q^{61} -4.78538 q^{63} -4.69056 q^{65} -1.00000 q^{67} -2.12449 q^{69} +6.31795 q^{71} -2.16282 q^{73} -1.00000 q^{75} -7.33394 q^{77} -13.4526 q^{79} +1.00000 q^{81} -14.7246 q^{83} +2.92969 q^{85} +4.88251 q^{87} +1.02451 q^{89} +22.4461 q^{91} -2.53027 q^{93} +6.94337 q^{95} -15.1578 q^{97} +1.53257 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−4.78538	−1.80870	−0.904352	−	0.426787i	$-0.859645\pi$
$7$	−4.78538	−1.80870	−0.904352	+	0.426787i	$0.859645\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.53257	0.462088	0.231044	−	0.972943i	$-0.425786\pi$
$11$	1.53257	0.462088	0.231044	+	0.972943i	$0.425786\pi$
$12$	0	0
$13$	−4.69056	−1.30093	−0.650464	−	0.759537i	$-0.725426\pi$
$13$	−4.69056	−1.30093	−0.650464	+	0.759537i	$0.725426\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	2.92969	0.710554	0.355277	−	0.934761i	$-0.384387\pi$
$17$	2.92969	0.710554	0.355277	+	0.934761i	$0.384387\pi$
$18$	0	0
$19$	6.94337	1.59292	0.796459	−	0.604692i	$-0.206704\pi$
$19$	6.94337	1.59292	0.796459	+	0.604692i	$0.206704\pi$
$20$	0	0
$21$	4.78538	1.04426
$22$	0	0
$23$	2.12449	0.442987	0.221494	−	0.975162i	$-0.428907\pi$
$23$	2.12449	0.442987	0.221494	+	0.975162i	$0.428907\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.88251	−0.906659	−0.453329	−	0.891343i	$-0.649764\pi$
$29$	−4.88251	−0.906659	−0.453329	+	0.891343i	$0.649764\pi$
$30$	0	0
$31$	2.53027	0.454449	0.227225	−	0.973842i	$-0.427035\pi$
$31$	2.53027	0.454449	0.227225	+	0.973842i	$0.427035\pi$
$32$	0	0
$33$	−1.53257	−0.266787
$34$	0	0
$35$	−4.78538	−0.808877
$36$	0	0
$37$	−1.09898	−0.180670	−0.0903352	−	0.995911i	$-0.528794\pi$
$37$	−1.09898	−0.180670	−0.0903352	+	0.995911i	$0.528794\pi$
$38$	0	0
$39$	4.69056	0.751091
$40$	0	0
$41$	−6.75188	−1.05447	−0.527234	−	0.849720i	$-0.676771\pi$
$41$	−6.75188	−1.05447	−0.527234	+	0.849720i	$0.676771\pi$
$42$	0	0
$43$	11.1320	1.69761	0.848807	−	0.528703i	$-0.177321\pi$
$43$	11.1320	1.69761	0.848807	+	0.528703i	$0.177321\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	0.205628	0.0299940	0.0149970	−	0.999888i	$-0.495226\pi$
$47$	0.205628	0.0299940	0.0149970	+	0.999888i	$0.495226\pi$
$48$	0	0
$49$	15.8999	2.27141
$50$	0	0
$51$	−2.92969	−0.410238
$52$	0	0
$53$	−8.49807	−1.16730	−0.583650	−	0.812006i	$-0.698376\pi$
$53$	−8.49807	−1.16730	−0.583650	+	0.812006i	$0.698376\pi$
$54$	0	0
$55$	1.53257	0.206652
$56$	0	0
$57$	−6.94337	−0.919672
$58$	0	0
$59$	−1.17550	−0.153037	−0.0765183	−	0.997068i	$-0.524380\pi$
$59$	−1.17550	−0.153037	−0.0765183	+	0.997068i	$0.524380\pi$
$60$	0	0
$61$	5.62256	0.719895	0.359947	−	0.932973i	$-0.382795\pi$
$61$	5.62256	0.719895	0.359947	+	0.932973i	$0.382795\pi$
$62$	0	0
$63$	−4.78538	−0.602901
$64$	0	0
$65$	−4.69056	−0.581793
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	−2.12449	−0.255759
$70$	0	0
$71$	6.31795	0.749803	0.374902	−	0.927065i	$-0.377676\pi$
$71$	6.31795	0.749803	0.374902	+	0.927065i	$0.377676\pi$
$72$	0	0
$73$	−2.16282	−0.253139	−0.126570	−	0.991958i	$-0.540397\pi$
$73$	−2.16282	−0.253139	−0.126570	+	0.991958i	$0.540397\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−7.33394	−0.835781
$78$	0	0
$79$	−13.4526	−1.51353	−0.756767	−	0.653684i	$-0.773222\pi$
$79$	−13.4526	−1.51353	−0.756767	+	0.653684i	$0.773222\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.7246	−1.61623	−0.808117	−	0.589022i	$-0.799513\pi$
$83$	−14.7246	−1.61623	−0.808117	+	0.589022i	$0.799513\pi$
$84$	0	0
$85$	2.92969	0.317769
$86$	0	0
$87$	4.88251	0.523460
$88$	0	0
$89$	1.02451	0.108597	0.0542987	−	0.998525i	$-0.482708\pi$
$89$	1.02451	0.108597	0.0542987	+	0.998525i	$0.482708\pi$
$90$	0	0
$91$	22.4461	2.35299
$92$	0	0
$93$	−2.53027	−0.262376
$94$	0	0
$95$	6.94337	0.712375
$96$	0	0
$97$	−15.1578	−1.53905	−0.769523	−	0.638619i	$-0.779506\pi$
$97$	−15.1578	−1.53905	−0.769523	+	0.638619i	$0.779506\pi$
$98$	0	0
$99$	1.53257	0.154029
$100$	0	0
$101$	−16.5595	−1.64773	−0.823864	−	0.566787i	$-0.808186\pi$
$101$	−16.5595	−1.64773	−0.823864	+	0.566787i	$0.808186\pi$
$102$	0	0
$103$	−12.6187	−1.24336	−0.621680	−	0.783271i	$-0.713550\pi$
$103$	−12.6187	−1.24336	−0.621680	+	0.783271i	$0.713550\pi$
$104$	0	0
$105$	4.78538	0.467005
$106$	0	0
$107$	8.25411	0.797955	0.398977	−	0.916961i	$-0.369365\pi$
$107$	8.25411	0.797955	0.398977	+	0.916961i	$0.369365\pi$
$108$	0	0
$109$	−12.8952	−1.23513	−0.617567	−	0.786518i	$-0.711882\pi$
$109$	−12.8952	−1.23513	−0.617567	+	0.786518i	$0.711882\pi$
$110$	0	0
$111$	1.09898	0.104310
$112$	0	0
$113$	−15.4853	−1.45673	−0.728365	−	0.685189i	$-0.759719\pi$
$113$	−15.4853	−1.45673	−0.728365	+	0.685189i	$0.759719\pi$
$114$	0	0
$115$	2.12449	0.198110
$116$	0	0
$117$	−4.69056	−0.433643
$118$	0	0
$119$	−14.0197	−1.28518
$120$	0	0
$121$	−8.65122	−0.786475
$122$	0	0
$123$	6.75188	0.608797
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.9204	−1.23523	−0.617616	−	0.786480i	$-0.711902\pi$
$127$	−13.9204	−1.23523	−0.617616	+	0.786480i	$0.711902\pi$
$128$	0	0
$129$	−11.1320	−0.980118
$130$	0	0
$131$	−0.511748	−0.0447117	−0.0223558	−	0.999750i	$-0.507117\pi$
$131$	−0.511748	−0.0447117	−0.0223558	+	0.999750i	$0.507117\pi$
$132$	0	0
$133$	−33.2267	−2.88112
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	4.90886	0.419393	0.209696	−	0.977767i	$-0.432752\pi$
$137$	4.90886	0.419393	0.209696	+	0.977767i	$0.432752\pi$
$138$	0	0
$139$	9.84086	0.834691	0.417345	−	0.908748i	$-0.362961\pi$
$139$	9.84086	0.834691	0.417345	+	0.908748i	$0.362961\pi$
$140$	0	0
$141$	−0.205628	−0.0173170
$142$	0	0
$143$	−7.18863	−0.601143
$144$	0	0
$145$	−4.88251	−0.405470
$146$	0	0
$147$	−15.8999	−1.31140
$148$	0	0
$149$	23.4753	1.92317	0.961584	−	0.274512i	$-0.0885163\pi$
$149$	23.4753	1.92317	0.961584	+	0.274512i	$0.0885163\pi$
$150$	0	0
$151$	22.3338	1.81750	0.908750	−	0.417342i	$-0.137038\pi$
$151$	22.3338	1.81750	0.908750	+	0.417342i	$0.137038\pi$
$152$	0	0
$153$	2.92969	0.236851
$154$	0	0
$155$	2.53027	0.203236
$156$	0	0
$157$	−4.18264	−0.333811	−0.166905	−	0.985973i	$-0.553377\pi$
$157$	−4.18264	−0.333811	−0.166905	+	0.985973i	$0.553377\pi$
$158$	0	0
$159$	8.49807	0.673940
$160$	0	0
$161$	−10.1665	−0.801233
$162$	0	0
$163$	−7.25248	−0.568058	−0.284029	−	0.958816i	$-0.591671\pi$
$163$	−7.25248	−0.568058	−0.284029	+	0.958816i	$0.591671\pi$
$164$	0	0
$165$	−1.53257	−0.119311
$166$	0	0
$167$	23.4533	1.81487	0.907437	−	0.420188i	$-0.138036\pi$
$167$	23.4533	1.81487	0.907437	+	0.420188i	$0.138036\pi$
$168$	0	0
$169$	9.00137	0.692413
$170$	0	0
$171$	6.94337	0.530973
$172$	0	0
$173$	16.2205	1.23322	0.616610	−	0.787269i	$-0.288506\pi$
$173$	16.2205	1.23322	0.616610	+	0.787269i	$0.288506\pi$
$174$	0	0
$175$	−4.78538	−0.361741
$176$	0	0
$177$	1.17550	0.0883557
$178$	0	0
$179$	−1.03097	−0.0770584	−0.0385292	−	0.999257i	$-0.512267\pi$
$179$	−1.03097	−0.0770584	−0.0385292	+	0.999257i	$0.512267\pi$
$180$	0	0
$181$	−16.8459	−1.25215	−0.626073	−	0.779764i	$-0.715339\pi$
$181$	−16.8459	−1.25215	−0.626073	+	0.779764i	$0.715339\pi$
$182$	0	0
$183$	−5.62256	−0.415631
$184$	0	0
$185$	−1.09898	−0.0807983
$186$	0	0
$187$	4.48996	0.328338
$188$	0	0
$189$	4.78538	0.348085
$190$	0	0
$191$	6.15614	0.445443	0.222721	−	0.974882i	$-0.428506\pi$
$191$	6.15614	0.445443	0.222721	+	0.974882i	$0.428506\pi$
$192$	0	0
$193$	1.80505	0.129931	0.0649653	−	0.997888i	$-0.479306\pi$
$193$	1.80505	0.129931	0.0649653	+	0.997888i	$0.479306\pi$
$194$	0	0
$195$	4.69056	0.335898
$196$	0	0
$197$	−22.8051	−1.62479	−0.812396	−	0.583106i	$-0.801837\pi$
$197$	−22.8051	−1.62479	−0.812396	+	0.583106i	$0.801837\pi$
$198$	0	0
$199$	2.87767	0.203993	0.101996	−	0.994785i	$-0.467477\pi$
$199$	2.87767	0.203993	0.101996	+	0.994785i	$0.467477\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	23.3647	1.63988
$204$	0	0
$205$	−6.75188	−0.471572
$206$	0	0
$207$	2.12449	0.147662
$208$	0	0
$209$	10.6412	0.736069
$210$	0	0
$211$	4.44430	0.305958	0.152979	−	0.988229i	$-0.451113\pi$
$211$	4.44430	0.305958	0.152979	+	0.988229i	$0.451113\pi$
$212$	0	0
$213$	−6.31795	−0.432899
$214$	0	0
$215$	11.1320	0.759196
$216$	0	0
$217$	−12.1083	−0.821964
$218$	0	0
$219$	2.16282	0.146150
$220$	0	0
$221$	−13.7419	−0.924379
$222$	0	0
$223$	−19.0375	−1.27485	−0.637423	−	0.770514i	$-0.720000\pi$
$223$	−19.0375	−1.27485	−0.637423	+	0.770514i	$0.720000\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−13.2889	−0.882017	−0.441009	−	0.897503i	$-0.645379\pi$
$227$	−13.2889	−0.882017	−0.441009	+	0.897503i	$0.645379\pi$
$228$	0	0
$229$	−2.47688	−0.163677	−0.0818383	−	0.996646i	$-0.526079\pi$
$229$	−2.47688	−0.163677	−0.0818383	+	0.996646i	$0.526079\pi$
$230$	0	0
$231$	7.33394	0.482538
$232$	0	0
$233$	13.4123	0.878669	0.439334	−	0.898324i	$-0.355214\pi$
$233$	13.4123	0.878669	0.439334	+	0.898324i	$0.355214\pi$
$234$	0	0
$235$	0.205628	0.0134137
$236$	0	0
$237$	13.4526	0.873840
$238$	0	0
$239$	−20.2142	−1.30755	−0.653775	−	0.756689i	$-0.726816\pi$
$239$	−20.2142	−1.30755	−0.653775	+	0.756689i	$0.726816\pi$
$240$	0	0
$241$	−4.87660	−0.314129	−0.157065	−	0.987588i	$-0.550203\pi$
$241$	−4.87660	−0.314129	−0.157065	+	0.987588i	$0.550203\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	15.8999	1.01581
$246$	0	0
$247$	−32.5683	−2.07227
$248$	0	0
$249$	14.7246	0.933133
$250$	0	0
$251$	−19.6513	−1.24038	−0.620189	−	0.784452i	$-0.712944\pi$
$251$	−19.6513	−1.24038	−0.620189	+	0.784452i	$0.712944\pi$
$252$	0	0
$253$	3.25594	0.204699
$254$	0	0
$255$	−2.92969	−0.183464
$256$	0	0
$257$	23.5859	1.47125	0.735625	−	0.677389i	$-0.236888\pi$
$257$	23.5859	1.47125	0.735625	+	0.677389i	$0.236888\pi$
$258$	0	0
$259$	5.25902	0.326779
$260$	0	0
$261$	−4.88251	−0.302220
$262$	0	0
$263$	−16.5077	−1.01791	−0.508953	−	0.860794i	$-0.669967\pi$
$263$	−16.5077	−1.01791	−0.508953	+	0.860794i	$0.669967\pi$
$264$	0	0
$265$	−8.49807	−0.522032
$266$	0	0
$267$	−1.02451	−0.0626988
$268$	0	0
$269$	−28.2582	−1.72294	−0.861468	−	0.507812i	$-0.830454\pi$
$269$	−28.2582	−1.72294	−0.861468	+	0.507812i	$0.830454\pi$
$270$	0	0
$271$	−7.02833	−0.426941	−0.213470	−	0.976950i	$-0.568477\pi$
$271$	−7.02833	−0.426941	−0.213470	+	0.976950i	$0.568477\pi$
$272$	0	0
$273$	−22.4461	−1.35850
$274$	0	0
$275$	1.53257	0.0924176
$276$	0	0
$277$	9.01934	0.541920	0.270960	−	0.962591i	$-0.412659\pi$
$277$	9.01934	0.541920	0.270960	+	0.962591i	$0.412659\pi$
$278$	0	0
$279$	2.53027	0.151483
$280$	0	0
$281$	−15.1424	−0.903318	−0.451659	−	0.892191i	$-0.649168\pi$
$281$	−15.1424	−0.903318	−0.451659	+	0.892191i	$0.649168\pi$
$282$	0	0
$283$	−15.7022	−0.933400	−0.466700	−	0.884416i	$-0.654557\pi$
$283$	−15.7022	−0.933400	−0.466700	+	0.884416i	$0.654557\pi$
$284$	0	0
$285$	−6.94337	−0.411290
$286$	0	0
$287$	32.3103	1.90722
$288$	0	0
$289$	−8.41693	−0.495113
$290$	0	0
$291$	15.1578	0.888569
$292$	0	0
$293$	−28.6649	−1.67462	−0.837310	−	0.546728i	$-0.815873\pi$
$293$	−28.6649	−1.67462	−0.837310	+	0.546728i	$0.815873\pi$
$294$	0	0
$295$	−1.17550	−0.0684400
$296$	0	0
$297$	−1.53257	−0.0889289
$298$	0	0
$299$	−9.96506	−0.576295
$300$	0	0
$301$	−53.2709	−3.07048
$302$	0	0
$303$	16.5595	0.951316
$304$	0	0
$305$	5.62256	0.321947
$306$	0	0
$307$	−2.88666	−0.164751	−0.0823753	−	0.996601i	$-0.526251\pi$
$307$	−2.88666	−0.164751	−0.0823753	+	0.996601i	$0.526251\pi$
$308$	0	0
$309$	12.6187	0.717855
$310$	0	0
$311$	29.5640	1.67642	0.838210	−	0.545348i	$-0.183602\pi$
$311$	29.5640	1.67642	0.838210	+	0.545348i	$0.183602\pi$
$312$	0	0
$313$	23.2104	1.31193	0.655963	−	0.754793i	$-0.272263\pi$
$313$	23.2104	1.31193	0.655963	+	0.754793i	$0.272263\pi$
$314$	0	0
$315$	−4.78538	−0.269626
$316$	0	0
$317$	−21.2623	−1.19421	−0.597105	−	0.802163i	$-0.703682\pi$
$317$	−21.2623	−1.19421	−0.597105	+	0.802163i	$0.703682\pi$
$318$	0	0
$319$	−7.48280	−0.418956
$320$	0	0
$321$	−8.25411	−0.460699
$322$	0	0
$323$	20.3419	1.13185
$324$	0	0
$325$	−4.69056	−0.260186
$326$	0	0
$327$	12.8952	0.713105
$328$	0	0
$329$	−0.984009	−0.0542502
$330$	0	0
$331$	−10.5647	−0.580691	−0.290345	−	0.956922i	$-0.593770\pi$
$331$	−10.5647	−0.580691	−0.290345	+	0.956922i	$0.593770\pi$
$332$	0	0
$333$	−1.09898	−0.0602235
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−6.24035	−0.339934	−0.169967	−	0.985450i	$-0.554366\pi$
$337$	−6.24035	−0.339934	−0.169967	+	0.985450i	$0.554366\pi$
$338$	0	0
$339$	15.4853	0.841043
$340$	0	0
$341$	3.87782	0.209995
$342$	0	0
$343$	−42.5893	−2.29961
$344$	0	0
$345$	−2.12449	−0.114379
$346$	0	0
$347$	21.8055	1.17058	0.585291	−	0.810823i	$-0.300980\pi$
$347$	21.8055	1.17058	0.585291	+	0.810823i	$0.300980\pi$
$348$	0	0
$349$	−22.9585	−1.22894	−0.614470	−	0.788941i	$-0.710630\pi$
$349$	−22.9585	−1.22894	−0.614470	+	0.788941i	$0.710630\pi$
$350$	0	0
$351$	4.69056	0.250364
$352$	0	0
$353$	−0.447394	−0.0238124	−0.0119062	−	0.999929i	$-0.503790\pi$
$353$	−0.447394	−0.0238124	−0.0119062	+	0.999929i	$0.503790\pi$
$354$	0	0
$355$	6.31795	0.335322
$356$	0	0
$357$	14.0197	0.742000
$358$	0	0
$359$	−13.1743	−0.695313	−0.347656	−	0.937622i	$-0.613022\pi$
$359$	−13.1743	−0.695313	−0.347656	+	0.937622i	$0.613022\pi$
$360$	0	0
$361$	29.2104	1.53739
$362$	0	0
$363$	8.65122	0.454071
$364$	0	0
$365$	−2.16282	−0.113207
$366$	0	0
$367$	−5.25584	−0.274353	−0.137176	−	0.990547i	$-0.543803\pi$
$367$	−5.25584	−0.274353	−0.137176	+	0.990547i	$0.543803\pi$
$368$	0	0
$369$	−6.75188	−0.351489
$370$	0	0
$371$	40.6665	2.11130
$372$	0	0
$373$	−14.4108	−0.746163	−0.373081	−	0.927799i	$-0.621699\pi$
$373$	−14.4108	−0.746163	−0.373081	+	0.927799i	$0.621699\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	22.9017	1.17950
$378$	0	0
$379$	−18.1589	−0.932760	−0.466380	−	0.884585i	$-0.654442\pi$
$379$	−18.1589	−0.932760	−0.466380	+	0.884585i	$0.654442\pi$
$380$	0	0
$381$	13.9204	0.713162
$382$	0	0
$383$	−0.526567	−0.0269063	−0.0134532	−	0.999910i	$-0.504282\pi$
$383$	−0.526567	−0.0269063	−0.0134532	+	0.999910i	$0.504282\pi$
$384$	0	0
$385$	−7.33394	−0.373772
$386$	0	0
$387$	11.1320	0.565871
$388$	0	0
$389$	−16.5418	−0.838702	−0.419351	−	0.907824i	$-0.637742\pi$
$389$	−16.5418	−0.838702	−0.419351	+	0.907824i	$0.637742\pi$
$390$	0	0
$391$	6.22410	0.314766
$392$	0	0
$393$	0.511748	0.0258143
$394$	0	0
$395$	−13.4526	−0.676873
$396$	0	0
$397$	32.7184	1.64209	0.821044	−	0.570864i	$-0.193392\pi$
$397$	32.7184	1.64209	0.821044	+	0.570864i	$0.193392\pi$
$398$	0	0
$399$	33.2267	1.66341
$400$	0	0
$401$	17.2650	0.862172	0.431086	−	0.902311i	$-0.358131\pi$
$401$	17.2650	0.862172	0.431086	+	0.902311i	$0.358131\pi$
$402$	0	0
$403$	−11.8684	−0.591205
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−1.68426	−0.0834857
$408$	0	0
$409$	20.9394	1.03539	0.517693	−	0.855566i	$-0.326791\pi$
$409$	20.9394	1.03539	0.517693	+	0.855566i	$0.326791\pi$
$410$	0	0
$411$	−4.90886	−0.242136
$412$	0	0
$413$	5.62520	0.276798
$414$	0	0
$415$	−14.7246	−0.722802
$416$	0	0
$417$	−9.84086	−0.481909
$418$	0	0
$419$	21.3992	1.04542	0.522709	−	0.852511i	$-0.324921\pi$
$419$	21.3992	1.04542	0.522709	+	0.852511i	$0.324921\pi$
$420$	0	0
$421$	−1.53806	−0.0749603	−0.0374801	−	0.999297i	$-0.511933\pi$
$421$	−1.53806	−0.0749603	−0.0374801	+	0.999297i	$0.511933\pi$
$422$	0	0
$423$	0.205628	0.00999799
$424$	0	0
$425$	2.92969	0.142111
$426$	0	0
$427$	−26.9061	−1.30208
$428$	0	0
$429$	7.18863	0.347070
$430$	0	0
$431$	23.6428	1.13883	0.569416	−	0.822049i	$-0.307169\pi$
$431$	23.6428	1.13883	0.569416	+	0.822049i	$0.307169\pi$
$432$	0	0
$433$	−11.4915	−0.552247	−0.276124	−	0.961122i	$-0.589050\pi$
$433$	−11.4915	−0.552247	−0.276124	+	0.961122i	$0.589050\pi$
$434$	0	0
$435$	4.88251	0.234098
$436$	0	0
$437$	14.7511	0.705643
$438$	0	0
$439$	−36.4615	−1.74021	−0.870107	−	0.492863i	$-0.835950\pi$
$439$	−36.4615	−1.74021	−0.870107	+	0.492863i	$0.835950\pi$
$440$	0	0
$441$	15.8999	0.757137
$442$	0	0
$443$	−32.5158	−1.54487	−0.772437	−	0.635091i	$-0.780962\pi$
$443$	−32.5158	−1.54487	−0.772437	+	0.635091i	$0.780962\pi$
$444$	0	0
$445$	1.02451	0.0485663
$446$	0	0
$447$	−23.4753	−1.11034
$448$	0	0
$449$	−36.5503	−1.72492	−0.862458	−	0.506129i	$-0.831076\pi$
$449$	−36.5503	−1.72492	−0.862458	+	0.506129i	$0.831076\pi$
$450$	0	0
$451$	−10.3478	−0.487257
$452$	0	0
$453$	−22.3338	−1.04933
$454$	0	0
$455$	22.4461	1.05229
$456$	0	0
$457$	22.7593	1.06463	0.532317	−	0.846545i	$-0.321321\pi$
$457$	22.7593	1.06463	0.532317	+	0.846545i	$0.321321\pi$
$458$	0	0
$459$	−2.92969	−0.136746
$460$	0	0
$461$	−0.339455	−0.0158100	−0.00790501	−	0.999969i	$-0.502516\pi$
$461$	−0.339455	−0.0158100	−0.00790501	+	0.999969i	$0.502516\pi$
$462$	0	0
$463$	−8.26510	−0.384112	−0.192056	−	0.981384i	$-0.561515\pi$
$463$	−8.26510	−0.384112	−0.192056	+	0.981384i	$0.561515\pi$
$464$	0	0
$465$	−2.53027	−0.117338
$466$	0	0
$467$	15.8733	0.734528	0.367264	−	0.930117i	$-0.380295\pi$
$467$	15.8733	0.734528	0.367264	+	0.930117i	$0.380295\pi$
$468$	0	0
$469$	4.78538	0.220968
$470$	0	0
$471$	4.18264	0.192726
$472$	0	0
$473$	17.0606	0.784447
$474$	0	0
$475$	6.94337	0.318584
$476$	0	0
$477$	−8.49807	−0.389100
$478$	0	0
$479$	−8.70271	−0.397637	−0.198819	−	0.980036i	$-0.563710\pi$
$479$	−8.70271	−0.397637	−0.198819	+	0.980036i	$0.563710\pi$
$480$	0	0
$481$	5.15481	0.235039
$482$	0	0
$483$	10.1665	0.462592
$484$	0	0
$485$	−15.1578	−0.688282
$486$	0	0
$487$	40.3159	1.82689	0.913443	−	0.406966i	$-0.133413\pi$
$487$	40.3159	1.82689	0.913443	+	0.406966i	$0.133413\pi$
$488$	0	0
$489$	7.25248	0.327968
$490$	0	0
$491$	11.3533	0.512365	0.256183	−	0.966628i	$-0.417535\pi$
$491$	11.3533	0.512365	0.256183	+	0.966628i	$0.417535\pi$
$492$	0	0
$493$	−14.3042	−0.644230
$494$	0	0
$495$	1.53257	0.0688840
$496$	0	0
$497$	−30.2338	−1.35617
$498$	0	0
$499$	22.3426	1.00019	0.500095	−	0.865970i	$-0.333298\pi$
$499$	22.3426	1.00019	0.500095	+	0.865970i	$0.333298\pi$
$500$	0	0
$501$	−23.4533	−1.04782
$502$	0	0
$503$	−7.87606	−0.351176	−0.175588	−	0.984464i	$-0.556183\pi$
$503$	−7.87606	−0.351176	−0.175588	+	0.984464i	$0.556183\pi$
$504$	0	0
$505$	−16.5595	−0.736886
$506$	0	0
$507$	−9.00137	−0.399765
$508$	0	0
$509$	8.61347	0.381785	0.190893	−	0.981611i	$-0.438862\pi$
$509$	8.61347	0.381785	0.190893	+	0.981611i	$0.438862\pi$
$510$	0	0
$511$	10.3499	0.457854
$512$	0	0
$513$	−6.94337	−0.306557
$514$	0	0
$515$	−12.6187	−0.556048
$516$	0	0
$517$	0.315140	0.0138599
$518$	0	0
$519$	−16.2205	−0.711999
$520$	0	0
$521$	34.4473	1.50916	0.754581	−	0.656206i	$-0.227840\pi$
$521$	34.4473	1.50916	0.754581	+	0.656206i	$0.227840\pi$
$522$	0	0
$523$	29.6089	1.29470	0.647352	−	0.762191i	$-0.275876\pi$
$523$	29.6089	1.29470	0.647352	+	0.762191i	$0.275876\pi$
$524$	0	0
$525$	4.78538	0.208851
$526$	0	0
$527$	7.41289	0.322910
$528$	0	0
$529$	−18.4865	−0.803762
$530$	0	0
$531$	−1.17550	−0.0510122
$532$	0	0
$533$	31.6701	1.37179
$534$	0	0
$535$	8.25411	0.356856
$536$	0	0
$537$	1.03097	0.0444897
$538$	0	0
$539$	24.3677	1.04959
$540$	0	0
$541$	36.7181	1.57864	0.789318	−	0.613985i	$-0.210434\pi$
$541$	36.7181	1.57864	0.789318	+	0.613985i	$0.210434\pi$
$542$	0	0
$543$	16.8459	0.722927
$544$	0	0
$545$	−12.8952	−0.552369
$546$	0	0
$547$	−24.9544	−1.06697	−0.533486	−	0.845809i	$-0.679118\pi$
$547$	−24.9544	−1.06697	−0.533486	+	0.845809i	$0.679118\pi$
$548$	0	0
$549$	5.62256	0.239965
$550$	0	0
$551$	−33.9011	−1.44423
$552$	0	0
$553$	64.3758	2.73754
$554$	0	0
$555$	1.09898	0.0466489
$556$	0	0
$557$	4.43205	0.187792	0.0938960	−	0.995582i	$-0.470068\pi$
$557$	4.43205	0.187792	0.0938960	+	0.995582i	$0.470068\pi$
$558$	0	0
$559$	−52.2153	−2.20847
$560$	0	0
$561$	−4.48996	−0.189566
$562$	0	0
$563$	−8.91973	−0.375922	−0.187961	−	0.982177i	$-0.560188\pi$
$563$	−8.91973	−0.375922	−0.187961	+	0.982177i	$0.560188\pi$
$564$	0	0
$565$	−15.4853	−0.651469
$566$	0	0
$567$	−4.78538	−0.200967
$568$	0	0
$569$	−38.7072	−1.62269	−0.811344	−	0.584568i	$-0.801264\pi$
$569$	−38.7072	−1.62269	−0.811344	+	0.584568i	$0.801264\pi$
$570$	0	0
$571$	−19.2026	−0.803602	−0.401801	−	0.915727i	$-0.631616\pi$
$571$	−19.2026	−0.803602	−0.401801	+	0.915727i	$0.631616\pi$
$572$	0	0
$573$	−6.15614	−0.257176
$574$	0	0
$575$	2.12449	0.0885975
$576$	0	0
$577$	11.2466	0.468200	0.234100	−	0.972213i	$-0.424786\pi$
$577$	11.2466	0.468200	0.234100	+	0.972213i	$0.424786\pi$
$578$	0	0
$579$	−1.80505	−0.0750155
$580$	0	0
$581$	70.4628	2.92329
$582$	0	0
$583$	−13.0239	−0.539395
$584$	0	0
$585$	−4.69056	−0.193931
$586$	0	0
$587$	−37.2561	−1.53772	−0.768862	−	0.639415i	$-0.779177\pi$
$587$	−37.2561	−1.53772	−0.768862	+	0.639415i	$0.779177\pi$
$588$	0	0
$589$	17.5686	0.723900
$590$	0	0
$591$	22.8051	0.938075
$592$	0	0
$593$	−41.3741	−1.69903	−0.849516	−	0.527563i	$-0.823106\pi$
$593$	−41.3741	−1.69903	−0.849516	+	0.527563i	$0.823106\pi$
$594$	0	0
$595$	−14.0197	−0.574751
$596$	0	0
$597$	−2.87767	−0.117775
$598$	0	0
$599$	0.297921	0.0121727	0.00608636	−	0.999981i	$-0.498063\pi$
$599$	0.297921	0.0121727	0.00608636	+	0.999981i	$0.498063\pi$
$600$	0	0
$601$	−11.2082	−0.457191	−0.228596	−	0.973521i	$-0.573413\pi$
$601$	−11.2082	−0.457191	−0.228596	+	0.973521i	$0.573413\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−8.65122	−0.351722
$606$	0	0
$607$	−21.7734	−0.883755	−0.441878	−	0.897075i	$-0.645687\pi$
$607$	−21.7734	−0.883755	−0.441878	+	0.897075i	$0.645687\pi$
$608$	0	0
$609$	−23.3647	−0.946784
$610$	0	0
$611$	−0.964512	−0.0390200
$612$	0	0
$613$	3.13449	0.126601	0.0633005	−	0.997995i	$-0.479837\pi$
$613$	3.13449	0.126601	0.0633005	+	0.997995i	$0.479837\pi$
$614$	0	0
$615$	6.75188	0.272262
$616$	0	0
$617$	24.2145	0.974839	0.487420	−	0.873168i	$-0.337938\pi$
$617$	24.2145	0.974839	0.487420	+	0.873168i	$0.337938\pi$
$618$	0	0
$619$	9.96628	0.400579	0.200289	−	0.979737i	$-0.435812\pi$
$619$	9.96628	0.400579	0.200289	+	0.979737i	$0.435812\pi$
$620$	0	0
$621$	−2.12449	−0.0852529
$622$	0	0
$623$	−4.90265	−0.196421
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−10.6412	−0.424969
$628$	0	0
$629$	−3.21966	−0.128376
$630$	0	0
$631$	−25.3482	−1.00910	−0.504548	−	0.863384i	$-0.668341\pi$
$631$	−25.3482	−1.00910	−0.504548	+	0.863384i	$0.668341\pi$
$632$	0	0
$633$	−4.44430	−0.176645
$634$	0	0
$635$	−13.9204	−0.552413
$636$	0	0
$637$	−74.5793	−2.95494
$638$	0	0
$639$	6.31795	0.249934
$640$	0	0
$641$	−30.3927	−1.20044	−0.600220	−	0.799835i	$-0.704920\pi$
$641$	−30.3927	−1.20044	−0.600220	+	0.799835i	$0.704920\pi$
$642$	0	0
$643$	2.52664	0.0996411	0.0498206	−	0.998758i	$-0.484135\pi$
$643$	2.52664	0.0996411	0.0498206	+	0.998758i	$0.484135\pi$
$644$	0	0
$645$	−11.1320	−0.438322
$646$	0	0
$647$	0.660524	0.0259679	0.0129839	−	0.999916i	$-0.495867\pi$
$647$	0.660524	0.0259679	0.0129839	+	0.999916i	$0.495867\pi$
$648$	0	0
$649$	−1.80153	−0.0707164
$650$	0	0
$651$	12.1083	0.474561
$652$	0	0
$653$	20.3075	0.794696	0.397348	−	0.917668i	$-0.369931\pi$
$653$	20.3075	0.794696	0.397348	+	0.917668i	$0.369931\pi$
$654$	0	0
$655$	−0.511748	−0.0199957
$656$	0	0
$657$	−2.16282	−0.0843798
$658$	0	0
$659$	−3.24369	−0.126356	−0.0631782	−	0.998002i	$-0.520124\pi$
$659$	−3.24369	−0.126356	−0.0631782	+	0.998002i	$0.520124\pi$
$660$	0	0
$661$	19.3374	0.752138	0.376069	−	0.926592i	$-0.377276\pi$
$661$	19.3374	0.752138	0.376069	+	0.926592i	$0.377276\pi$
$662$	0	0
$663$	13.7419	0.533691
$664$	0	0
$665$	−33.2267	−1.28848
$666$	0	0
$667$	−10.3729	−0.401638
$668$	0	0
$669$	19.0375	0.736033
$670$	0	0
$671$	8.61698	0.332655
$672$	0	0
$673$	−3.73136	−0.143833	−0.0719167	−	0.997411i	$-0.522912\pi$
$673$	−3.73136	−0.143833	−0.0719167	+	0.997411i	$0.522912\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	2.03275	0.0781251	0.0390625	−	0.999237i	$-0.487563\pi$
$677$	2.03275	0.0781251	0.0390625	+	0.999237i	$0.487563\pi$
$678$	0	0
$679$	72.5361	2.78368
$680$	0	0
$681$	13.2889	0.509233
$682$	0	0
$683$	41.9094	1.60362	0.801809	−	0.597581i	$-0.203871\pi$
$683$	41.9094	1.60362	0.801809	+	0.597581i	$0.203871\pi$
$684$	0	0
$685$	4.90886	0.187558
$686$	0	0
$687$	2.47688	0.0944987
$688$	0	0
$689$	39.8607	1.51857
$690$	0	0
$691$	2.50587	0.0953279	0.0476639	−	0.998863i	$-0.484822\pi$
$691$	2.50587	0.0953279	0.0476639	+	0.998863i	$0.484822\pi$
$692$	0	0
$693$	−7.33394	−0.278594
$694$	0	0
$695$	9.84086	0.373285
$696$	0	0
$697$	−19.7809	−0.749255
$698$	0	0
$699$	−13.4123	−0.507300
$700$	0	0
$701$	26.0752	0.984846	0.492423	−	0.870356i	$-0.336111\pi$
$701$	26.0752	0.984846	0.492423	+	0.870356i	$0.336111\pi$
$702$	0	0
$703$	−7.63059	−0.287793
$704$	0	0
$705$	−0.205628	−0.00774441
$706$	0	0
$707$	79.2433	2.98025
$708$	0	0
$709$	32.2465	1.21104	0.605522	−	0.795829i	$-0.292964\pi$
$709$	32.2465	1.21104	0.605522	+	0.795829i	$0.292964\pi$
$710$	0	0
$711$	−13.4526	−0.504512
$712$	0	0
$713$	5.37553	0.201315
$714$	0	0
$715$	−7.18863	−0.268839
$716$	0	0
$717$	20.2142	0.754914
$718$	0	0
$719$	1.19348	0.0445093	0.0222546	−	0.999752i	$-0.492916\pi$
$719$	1.19348	0.0445093	0.0222546	+	0.999752i	$0.492916\pi$
$720$	0	0
$721$	60.3855	2.24887
$722$	0	0
$723$	4.87660	0.181363
$724$	0	0
$725$	−4.88251	−0.181332
$726$	0	0
$727$	−9.95416	−0.369179	−0.184590	−	0.982816i	$-0.559096\pi$
$727$	−9.95416	−0.369179	−0.184590	+	0.982816i	$0.559096\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	32.6133	1.20625
$732$	0	0
$733$	−13.4889	−0.498225	−0.249112	−	0.968475i	$-0.580139\pi$
$733$	−13.4889	−0.498225	−0.249112	+	0.968475i	$0.580139\pi$
$734$	0	0
$735$	−15.8999	−0.586476
$736$	0	0
$737$	−1.53257	−0.0564530
$738$	0	0
$739$	−36.8218	−1.35451	−0.677256	−	0.735747i	$-0.736831\pi$
$739$	−36.8218	−1.35451	−0.677256	+	0.735747i	$0.736831\pi$
$740$	0	0
$741$	32.5683	1.19643
$742$	0	0
$743$	−29.6407	−1.08741	−0.543705	−	0.839276i	$-0.682979\pi$
$743$	−29.6407	−1.08741	−0.543705	+	0.839276i	$0.682979\pi$
$744$	0	0
$745$	23.4753	0.860067
$746$	0	0
$747$	−14.7246	−0.538745
$748$	0	0
$749$	−39.4990	−1.44326
$750$	0	0
$751$	9.77072	0.356539	0.178269	−	0.983982i	$-0.442950\pi$
$751$	9.77072	0.356539	0.178269	+	0.983982i	$0.442950\pi$
$752$	0	0
$753$	19.6513	0.716133
$754$	0	0
$755$	22.3338	0.812810
$756$	0	0
$757$	50.9786	1.85285	0.926424	−	0.376481i	$-0.122866\pi$
$757$	50.9786	1.85285	0.926424	+	0.376481i	$0.122866\pi$
$758$	0	0
$759$	−3.25594	−0.118183
$760$	0	0
$761$	21.1577	0.766964	0.383482	−	0.923548i	$-0.374725\pi$
$761$	21.1577	0.766964	0.383482	+	0.923548i	$0.374725\pi$
$762$	0	0
$763$	61.7084	2.23399
$764$	0	0
$765$	2.92969	0.105923
$766$	0	0
$767$	5.51374	0.199090
$768$	0	0
$769$	−17.2785	−0.623079	−0.311540	−	0.950233i	$-0.600845\pi$
$769$	−17.2785	−0.623079	−0.311540	+	0.950233i	$0.600845\pi$
$770$	0	0
$771$	−23.5859	−0.849427
$772$	0	0
$773$	13.8909	0.499623	0.249811	−	0.968295i	$-0.419631\pi$
$773$	13.8909	0.499623	0.249811	+	0.968295i	$0.419631\pi$
$774$	0	0
$775$	2.53027	0.0908898
$776$	0	0
$777$	−5.25902	−0.188666
$778$	0	0
$779$	−46.8808	−1.67968
$780$	0	0
$781$	9.68272	0.346475
$782$	0	0
$783$	4.88251	0.174487
$784$	0	0
$785$	−4.18264	−0.149285
$786$	0	0
$787$	−28.0339	−0.999300	−0.499650	−	0.866227i	$-0.666538\pi$
$787$	−28.0339	−0.999300	−0.499650	+	0.866227i	$0.666538\pi$
$788$	0	0
$789$	16.5077	0.587689
$790$	0	0
$791$	74.1028	2.63479
$792$	0	0
$793$	−26.3730	−0.936531
$794$	0	0
$795$	8.49807	0.301395
$796$	0	0
$797$	36.6832	1.29939	0.649693	−	0.760196i	$-0.274897\pi$
$797$	36.6832	1.29939	0.649693	+	0.760196i	$0.274897\pi$
$798$	0	0
$799$	0.602427	0.0213123
$800$	0	0
$801$	1.02451	0.0361992
$802$	0	0
$803$	−3.31468	−0.116973
$804$	0	0
$805$	−10.1665	−0.358322
$806$	0	0
$807$	28.2582	0.994737
$808$	0	0
$809$	0.740775	0.0260443	0.0130221	−	0.999915i	$-0.495855\pi$
$809$	0.740775	0.0260443	0.0130221	+	0.999915i	$0.495855\pi$
$810$	0	0
$811$	−21.9261	−0.769929	−0.384965	−	0.922931i	$-0.625786\pi$
$811$	−21.9261	−0.769929	−0.384965	+	0.922931i	$0.625786\pi$
$812$	0	0
$813$	7.02833	0.246494
$814$	0	0
$815$	−7.25248	−0.254043
$816$	0	0
$817$	77.2936	2.70416
$818$	0	0
$819$	22.4461	0.784331
$820$	0	0
$821$	11.4918	0.401068	0.200534	−	0.979687i	$-0.435732\pi$
$821$	11.4918	0.401068	0.200534	+	0.979687i	$0.435732\pi$
$822$	0	0
$823$	−19.2694	−0.671688	−0.335844	−	0.941918i	$-0.609021\pi$
$823$	−19.2694	−0.671688	−0.335844	+	0.941918i	$0.609021\pi$
$824$	0	0
$825$	−1.53257	−0.0533573
$826$	0	0
$827$	49.0497	1.70562	0.852812	−	0.522218i	$-0.174895\pi$
$827$	49.0497	1.70562	0.852812	+	0.522218i	$0.174895\pi$
$828$	0	0
$829$	0.221427	0.00769049	0.00384524	−	0.999993i	$-0.498776\pi$
$829$	0.221427	0.00769049	0.00384524	+	0.999993i	$0.498776\pi$
$830$	0	0
$831$	−9.01934	−0.312877
$832$	0	0
$833$	46.5817	1.61396
$834$	0	0
$835$	23.4533	0.811637
$836$	0	0
$837$	−2.53027	−0.0874588
$838$	0	0
$839$	3.50054	0.120852	0.0604261	−	0.998173i	$-0.480754\pi$
$839$	3.50054	0.120852	0.0604261	+	0.998173i	$0.480754\pi$
$840$	0	0
$841$	−5.16112	−0.177970
$842$	0	0
$843$	15.1424	0.521531
$844$	0	0
$845$	9.00137	0.309657
$846$	0	0
$847$	41.3994	1.42250
$848$	0	0
$849$	15.7022	0.538899
$850$	0	0
$851$	−2.33477	−0.0800347
$852$	0	0
$853$	−37.0405	−1.26824	−0.634121	−	0.773234i	$-0.718638\pi$
$853$	−37.0405	−1.26824	−0.634121	+	0.773234i	$0.718638\pi$
$854$	0	0
$855$	6.94337	0.237458
$856$	0	0
$857$	−31.7999	−1.08627	−0.543133	−	0.839647i	$-0.682762\pi$
$857$	−31.7999	−1.08627	−0.543133	+	0.839647i	$0.682762\pi$
$858$	0	0
$859$	−5.58802	−0.190661	−0.0953304	−	0.995446i	$-0.530391\pi$
$859$	−5.58802	−0.190661	−0.0953304	+	0.995446i	$0.530391\pi$
$860$	0	0
$861$	−32.3103	−1.10113
$862$	0	0
$863$	20.1820	0.687002	0.343501	−	0.939152i	$-0.388387\pi$
$863$	20.1820	0.687002	0.343501	+	0.939152i	$0.388387\pi$
$864$	0	0
$865$	16.2205	0.551512
$866$	0	0
$867$	8.41693	0.285854
$868$	0	0
$869$	−20.6171	−0.699386
$870$	0	0
$871$	4.69056	0.158934
$872$	0	0
$873$	−15.1578	−0.513015
$874$	0	0
$875$	−4.78538	−0.161775
$876$	0	0
$877$	27.2231	0.919260	0.459630	−	0.888110i	$-0.347982\pi$
$877$	27.2231	0.919260	0.459630	+	0.888110i	$0.347982\pi$
$878$	0	0
$879$	28.6649	0.966843
$880$	0	0
$881$	26.0521	0.877717	0.438858	−	0.898556i	$-0.355383\pi$
$881$	26.0521	0.877717	0.438858	+	0.898556i	$0.355383\pi$
$882$	0	0
$883$	40.5120	1.36334	0.681669	−	0.731661i	$-0.261255\pi$
$883$	40.5120	1.36334	0.681669	+	0.731661i	$0.261255\pi$
$884$	0	0
$885$	1.17550	0.0395139
$886$	0	0
$887$	−6.23415	−0.209322	−0.104661	−	0.994508i	$-0.533376\pi$
$887$	−6.23415	−0.209322	−0.104661	+	0.994508i	$0.533376\pi$
$888$	0	0
$889$	66.6142	2.23417
$890$	0	0
$891$	1.53257	0.0513431
$892$	0	0
$893$	1.42775	0.0477779
$894$	0	0
$895$	−1.03097	−0.0344616
$896$	0	0
$897$	9.96506	0.332724
$898$	0	0
$899$	−12.3540	−0.412030
$900$	0	0
$901$	−24.8967	−0.829429
$902$	0	0
$903$	53.2709	1.77274
$904$	0	0
$905$	−16.8459	−0.559977
$906$	0	0
$907$	20.7777	0.689911	0.344955	−	0.938619i	$-0.387894\pi$
$907$	20.7777	0.689911	0.344955	+	0.938619i	$0.387894\pi$
$908$	0	0
$909$	−16.5595	−0.549243
$910$	0	0
$911$	31.2073	1.03395	0.516973	−	0.856002i	$-0.327059\pi$
$911$	31.2073	1.03395	0.516973	+	0.856002i	$0.327059\pi$
$912$	0	0
$913$	−22.5665	−0.746843
$914$	0	0
$915$	−5.62256	−0.185876
$916$	0	0
$917$	2.44891	0.0808701
$918$	0	0
$919$	−19.6648	−0.648683	−0.324341	−	0.945940i	$-0.605143\pi$
$919$	−19.6648	−0.648683	−0.324341	+	0.945940i	$0.605143\pi$
$920$	0	0
$921$	2.88666	0.0951188
$922$	0	0
$923$	−29.6348	−0.975440
$924$	0	0
$925$	−1.09898	−0.0361341
$926$	0	0
$927$	−12.6187	−0.414454
$928$	0	0
$929$	2.39969	0.0787313	0.0393657	−	0.999225i	$-0.487466\pi$
$929$	2.39969	0.0787313	0.0393657	+	0.999225i	$0.487466\pi$
$930$	0	0
$931$	110.399	3.61817
$932$	0	0
$933$	−29.5640	−0.967881
$934$	0	0
$935$	4.48996	0.146837
$936$	0	0
$937$	7.89717	0.257989	0.128995	−	0.991645i	$-0.458825\pi$
$937$	7.89717	0.257989	0.128995	+	0.991645i	$0.458825\pi$
$938$	0	0
$939$	−23.2104	−0.757441
$940$	0	0
$941$	−19.9651	−0.650844	−0.325422	−	0.945569i	$-0.605506\pi$
$941$	−19.9651	−0.650844	−0.325422	+	0.945569i	$0.605506\pi$
$942$	0	0
$943$	−14.3443	−0.467116
$944$	0	0
$945$	4.78538	0.155668
$946$	0	0
$947$	51.8666	1.68544	0.842719	−	0.538353i	$-0.180953\pi$
$947$	51.8666	1.68544	0.842719	+	0.538353i	$0.180953\pi$
$948$	0	0
$949$	10.1449	0.329316
$950$	0	0
$951$	21.2623	0.689477
$952$	0	0
$953$	−6.40222	−0.207388	−0.103694	−	0.994609i	$-0.533066\pi$
$953$	−6.40222	−0.207388	−0.103694	+	0.994609i	$0.533066\pi$
$954$	0	0
$955$	6.15614	0.199208
$956$	0	0
$957$	7.48280	0.241884
$958$	0	0
$959$	−23.4908	−0.758557
$960$	0	0
$961$	−24.5978	−0.793476
$962$	0	0
$963$	8.25411	0.265985
$964$	0	0
$965$	1.80505	0.0581067
$966$	0	0
$967$	−32.8448	−1.05622	−0.528109	−	0.849177i	$-0.677099\pi$
$967$	−32.8448	−1.05622	−0.528109	+	0.849177i	$0.677099\pi$
$968$	0	0
$969$	−20.3419	−0.653476
$970$	0	0
$971$	41.1626	1.32097	0.660486	−	0.750839i	$-0.270350\pi$
$971$	41.1626	1.32097	0.660486	+	0.750839i	$0.270350\pi$
$972$	0	0
$973$	−47.0923	−1.50971
$974$	0	0
$975$	4.69056	0.150218
$976$	0	0
$977$	−11.0214	−0.352604	−0.176302	−	0.984336i	$-0.556414\pi$
$977$	−11.0214	−0.352604	−0.176302	+	0.984336i	$0.556414\pi$
$978$	0	0
$979$	1.57013	0.0501816
$980$	0	0
$981$	−12.8952	−0.411711
$982$	0	0
$983$	−30.5676	−0.974956	−0.487478	−	0.873135i	$-0.662083\pi$
$983$	−30.5676	−0.974956	−0.487478	+	0.873135i	$0.662083\pi$
$984$	0	0
$985$	−22.8051	−0.726629
$986$	0	0
$987$	0.984009	0.0313214
$988$	0	0
$989$	23.6498	0.752021
$990$	0	0
$991$	25.9499	0.824326	0.412163	−	0.911110i	$-0.364773\pi$
$991$	25.9499	0.824326	0.412163	+	0.911110i	$0.364773\pi$
$992$	0	0
$993$	10.5647	0.335262
$994$	0	0
$995$	2.87767	0.0912284
$996$	0	0
$997$	31.7890	1.00677	0.503384	−	0.864063i	$-0.332088\pi$
$997$	31.7890	1.00677	0.503384	+	0.864063i	$0.332088\pi$
$998$	0	0
$999$	1.09898	0.0347700

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.e.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -4.78538 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -4.78538 q^{7} +1.00000 q^{9} +1.53257 q^{11} -4.69056 q^{13} -1.00000 q^{15} +2.92969 q^{17} +6.94337 q^{19} +4.78538 q^{21} +2.12449 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.88251 q^{29} +2.53027 q^{31} -1.53257 q^{33} -4.78538 q^{35} -1.09898 q^{37} +4.69056 q^{39} -6.75188 q^{41} +11.1320 q^{43} +1.00000 q^{45} +0.205628 q^{47} +15.8999 q^{49} -2.92969 q^{51} -8.49807 q^{53} +1.53257 q^{55} -6.94337 q^{57} -1.17550 q^{59} +5.62256 q^{61} -4.78538 q^{63} -4.69056 q^{65} -1.00000 q^{67} -2.12449 q^{69} +6.31795 q^{71} -2.16282 q^{73} -1.00000 q^{75} -7.33394 q^{77} -13.4526 q^{79} +1.00000 q^{81} -14.7246 q^{83} +2.92969 q^{85} +4.88251 q^{87} +1.02451 q^{89} +22.4461 q^{91} -2.53027 q^{93} +6.94337 q^{95} -15.1578 q^{97} +1.53257 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) 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

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