LMFDB - Embedded newform 4020.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0998616126$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.98117.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 4x + 1 $ x^4 - x^3 - 8x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.184077$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -0.184077 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -0.184077 q^{7} +1.00000 q^{9} +3.61657 q^{11} -1.46638 q^{13} -1.00000 q^{15} -5.78204 q^{17} -2.96612 q^{19} -0.184077 q^{21} -3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.815923 q^{29} -4.89888 q^{31} +3.61657 q^{33} +0.184077 q^{35} +2.63185 q^{37} -1.46638 q^{39} -0.0338845 q^{41} +6.39861 q^{43} -1.00000 q^{45} -8.66230 q^{47} -6.96612 q^{49} -5.78204 q^{51} +3.89888 q^{53} -3.61657 q^{55} -2.96612 q^{57} -7.26993 q^{59} -10.9661 q^{61} -0.184077 q^{63} +1.46638 q^{65} -1.00000 q^{67} -3.00000 q^{69} -1.86842 q^{71} +5.45111 q^{73} +1.00000 q^{75} -0.665730 q^{77} -4.46638 q^{79} +1.00000 q^{81} -9.92934 q^{83} +5.78204 q^{85} -0.815923 q^{87} +16.7943 q^{89} +0.269928 q^{91} -4.89888 q^{93} +2.96612 q^{95} +8.37711 q^{97} +3.61657 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} - 5 q^{11} - 5 q^{13} - 4 q^{15} - 8 q^{17} + 5 q^{19} + q^{21} - 12 q^{23} + 4 q^{25} + 4 q^{27} - 5 q^{29} - q^{31} - 5 q^{33} - q^{35} + 14 q^{37} - 5 q^{39} - 17 q^{41} - 9 q^{43} - 4 q^{45} - 7 q^{47} - 11 q^{49} - 8 q^{51} - 3 q^{53} + 5 q^{55} + 5 q^{57} - 23 q^{59} - 27 q^{61} + q^{63} + 5 q^{65} - 4 q^{67} - 12 q^{69} - 20 q^{71} - 2 q^{73} + 4 q^{75} - 23 q^{77} - 17 q^{79} + 4 q^{81} + 10 q^{83} + 8 q^{85} - 5 q^{87} - 18 q^{89} - 5 q^{91} - q^{93} - 5 q^{95} - 11 q^{97} - 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 - 5 * q^11 - 5 * q^13 - 4 * q^15 - 8 * q^17 + 5 * q^19 + q^21 - 12 * q^23 + 4 * q^25 + 4 * q^27 - 5 * q^29 - q^31 - 5 * q^33 - q^35 + 14 * q^37 - 5 * q^39 - 17 * q^41 - 9 * q^43 - 4 * q^45 - 7 * q^47 - 11 * q^49 - 8 * q^51 - 3 * q^53 + 5 * q^55 + 5 * q^57 - 23 * q^59 - 27 * q^61 + q^63 + 5 * q^65 - 4 * q^67 - 12 * q^69 - 20 * q^71 - 2 * q^73 + 4 * q^75 - 23 * q^77 - 17 * q^79 + 4 * q^81 + 10 * q^83 + 8 * q^85 - 5 * q^87 - 18 * q^89 - 5 * q^91 - q^93 - 5 * q^95 - 11 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.184077	−0.0695747	−0.0347874	−	0.999395i	$-0.511075\pi$
$7$	−0.184077	−0.0695747	−0.0347874	+	0.999395i	$0.511075\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.61657	1.09044	0.545219	−	0.838294i	$-0.316446\pi$
$11$	3.61657	1.09044	0.545219	+	0.838294i	$0.316446\pi$
$12$	0	0
$13$	−1.46638	−0.406701	−0.203351	−	0.979106i	$-0.565183\pi$
$13$	−1.46638	−0.406701	−0.203351	+	0.979106i	$0.565183\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−5.78204	−1.40235	−0.701175	−	0.712989i	$-0.747341\pi$
$17$	−5.78204	−1.40235	−0.701175	+	0.712989i	$0.747341\pi$
$18$	0	0
$19$	−2.96612	−0.680474	−0.340237	−	0.940340i	$-0.610507\pi$
$19$	−2.96612	−0.680474	−0.340237	+	0.940340i	$0.610507\pi$
$20$	0	0
$21$	−0.184077	−0.0401690
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.815923	−0.151513	−0.0757565	−	0.997126i	$-0.524137\pi$
$29$	−0.815923	−0.151513	−0.0757565	+	0.997126i	$0.524137\pi$
$30$	0	0
$31$	−4.89888	−0.879865	−0.439932	−	0.898031i	$-0.644998\pi$
$31$	−4.89888	−0.879865	−0.439932	+	0.898031i	$0.644998\pi$
$32$	0	0
$33$	3.61657	0.629565
$34$	0	0
$35$	0.184077	0.0311148
$36$	0	0
$37$	2.63185	0.432673	0.216336	−	0.976319i	$-0.430589\pi$
$37$	2.63185	0.432673	0.216336	+	0.976319i	$0.430589\pi$
$38$	0	0
$39$	−1.46638	−0.234809
$40$	0	0
$41$	−0.0338845	−0.00529187	−0.00264593	−	0.999996i	$-0.500842\pi$
$41$	−0.0338845	−0.00529187	−0.00264593	+	0.999996i	$0.500842\pi$
$42$	0	0
$43$	6.39861	0.975779	0.487890	−	0.872905i	$-0.337767\pi$
$43$	6.39861	0.975779	0.487890	+	0.872905i	$0.337767\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−8.66230	−1.26353	−0.631763	−	0.775161i	$-0.717669\pi$
$47$	−8.66230	−1.26353	−0.631763	+	0.775161i	$0.717669\pi$
$48$	0	0
$49$	−6.96612	−0.995159
$50$	0	0
$51$	−5.78204	−0.809647
$52$	0	0
$53$	3.89888	0.535552	0.267776	−	0.963481i	$-0.413711\pi$
$53$	3.89888	0.535552	0.267776	+	0.963481i	$0.413711\pi$
$54$	0	0
$55$	−3.61657	−0.487659
$56$	0	0
$57$	−2.96612	−0.392872
$58$	0	0
$59$	−7.26993	−0.946464	−0.473232	−	0.880938i	$-0.656913\pi$
$59$	−7.26993	−0.946464	−0.473232	+	0.880938i	$0.656913\pi$
$60$	0	0
$61$	−10.9661	−1.40407	−0.702034	−	0.712144i	$-0.747724\pi$
$61$	−10.9661	−1.40407	−0.702034	+	0.712144i	$0.747724\pi$
$62$	0	0
$63$	−0.184077	−0.0231916
$64$	0	0
$65$	1.46638	0.181882
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−1.86842	−0.221741	−0.110870	−	0.993835i	$-0.535364\pi$
$71$	−1.86842	−0.221741	−0.110870	+	0.993835i	$0.535364\pi$
$72$	0	0
$73$	5.45111	0.638004	0.319002	−	0.947754i	$-0.396652\pi$
$73$	5.45111	0.638004	0.319002	+	0.947754i	$0.396652\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−0.665730	−0.0758669
$78$	0	0
$79$	−4.46638	−0.502507	−0.251254	−	0.967921i	$-0.580843\pi$
$79$	−4.46638	−0.502507	−0.251254	+	0.967921i	$0.580843\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.92934	−1.08989	−0.544943	−	0.838473i	$-0.683449\pi$
$83$	−9.92934	−1.08989	−0.544943	+	0.838473i	$0.683449\pi$
$84$	0	0
$85$	5.78204	0.627150
$86$	0	0
$87$	−0.815923	−0.0874761
$88$	0	0
$89$	16.7943	1.78020	0.890098	−	0.455770i	$-0.150636\pi$
$89$	16.7943	1.78020	0.890098	+	0.455770i	$0.150636\pi$
$90$	0	0
$91$	0.269928	0.0282961
$92$	0	0
$93$	−4.89888	−0.507990
$94$	0	0
$95$	2.96612	0.304317
$96$	0	0
$97$	8.37711	0.850566	0.425283	−	0.905060i	$-0.360175\pi$
$97$	8.37711	0.850566	0.425283	+	0.905060i	$0.360175\pi$
$98$	0	0
$99$	3.61657	0.363479
$100$	0	0
$101$	−11.1659	−1.11105	−0.555525	−	0.831500i	$-0.687483\pi$
$101$	−11.1659	−1.11105	−0.555525	+	0.831500i	$0.687483\pi$
$102$	0	0
$103$	8.76677	0.863815	0.431908	−	0.901918i	$-0.357841\pi$
$103$	8.76677	0.863815	0.431908	+	0.901918i	$0.357841\pi$
$104$	0	0
$105$	0.184077	0.0179641
$106$	0	0
$107$	2.14730	0.207587	0.103794	−	0.994599i	$-0.466902\pi$
$107$	2.14730	0.207587	0.103794	+	0.994599i	$0.466902\pi$
$108$	0	0
$109$	−15.9293	−1.52575	−0.762877	−	0.646544i	$-0.776214\pi$
$109$	−15.9293	−1.52575	−0.762877	+	0.646544i	$0.776214\pi$
$110$	0	0
$111$	2.63185	0.249804
$112$	0	0
$113$	12.4845	1.17444	0.587220	−	0.809428i	$-0.300222\pi$
$113$	12.4845	1.17444	0.587220	+	0.809428i	$0.300222\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	−1.46638	−0.135567
$118$	0	0
$119$	1.06434	0.0975681
$120$	0	0
$121$	2.07961	0.189056
$122$	0	0
$123$	−0.0338845	−0.00305526
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.58269	0.584119	0.292060	−	0.956400i	$-0.405659\pi$
$127$	6.58269	0.584119	0.292060	+	0.956400i	$0.405659\pi$
$128$	0	0
$129$	6.39861	0.563366
$130$	0	0
$131$	−21.7085	−1.89668	−0.948339	−	0.317258i	$-0.897238\pi$
$131$	−21.7085	−1.89668	−0.948339	+	0.317258i	$0.897238\pi$
$132$	0	0
$133$	0.545995	0.0473438
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−2.05197	−0.175311	−0.0876556	−	0.996151i	$-0.527938\pi$
$137$	−2.05197	−0.175311	−0.0876556	+	0.996151i	$0.527938\pi$
$138$	0	0
$139$	−13.7854	−1.16926	−0.584630	−	0.811300i	$-0.698760\pi$
$139$	−13.7854	−1.16926	−0.584630	+	0.811300i	$0.698760\pi$
$140$	0	0
$141$	−8.66230	−0.729498
$142$	0	0
$143$	−5.30328	−0.443483
$144$	0	0
$145$	0.815923	0.0677587
$146$	0	0
$147$	−6.96612	−0.574556
$148$	0	0
$149$	−12.2670	−1.00495	−0.502477	−	0.864590i	$-0.667578\pi$
$149$	−12.2670	−1.00495	−0.502477	+	0.864590i	$0.667578\pi$
$150$	0	0
$151$	11.4939	0.935364	0.467682	−	0.883897i	$-0.345089\pi$
$151$	11.4939	0.935364	0.467682	+	0.883897i	$0.345089\pi$
$152$	0	0
$153$	−5.78204	−0.467450
$154$	0	0
$155$	4.89888	0.393487
$156$	0	0
$157$	−4.81592	−0.384352	−0.192176	−	0.981360i	$-0.561555\pi$
$157$	−4.81592	−0.384352	−0.192176	+	0.981360i	$0.561555\pi$
$158$	0	0
$159$	3.89888	0.309201
$160$	0	0
$161$	0.552232	0.0435220
$162$	0	0
$163$	9.38677	0.735228	0.367614	−	0.929978i	$-0.380175\pi$
$163$	9.38677	0.735228	0.367614	+	0.929978i	$0.380175\pi$
$164$	0	0
$165$	−3.61657	−0.281550
$166$	0	0
$167$	−9.65046	−0.746775	−0.373387	−	0.927675i	$-0.621804\pi$
$167$	−9.65046	−0.746775	−0.373387	+	0.927675i	$0.621804\pi$
$168$	0	0
$169$	−10.8497	−0.834594
$170$	0	0
$171$	−2.96612	−0.226825
$172$	0	0
$173$	8.02756	0.610324	0.305162	−	0.952300i	$-0.401289\pi$
$173$	8.02756	0.610324	0.305162	+	0.952300i	$0.401289\pi$
$174$	0	0
$175$	−0.184077	−0.0139149
$176$	0	0
$177$	−7.26993	−0.546441
$178$	0	0
$179$	−20.1253	−1.50423	−0.752116	−	0.659030i	$-0.770967\pi$
$179$	−20.1253	−1.50423	−0.752116	+	0.659030i	$0.770967\pi$
$180$	0	0
$181$	0.681450	0.0506518	0.0253259	−	0.999679i	$-0.491938\pi$
$181$	0.681450	0.0506518	0.0253259	+	0.999679i	$0.491938\pi$
$182$	0	0
$183$	−10.9661	−0.810639
$184$	0	0
$185$	−2.63185	−0.193497
$186$	0	0
$187$	−20.9112	−1.52918
$188$	0	0
$189$	−0.184077	−0.0133897
$190$	0	0
$191$	−7.12018	−0.515198	−0.257599	−	0.966252i	$-0.582931\pi$
$191$	−7.12018	−0.515198	−0.257599	+	0.966252i	$0.582931\pi$
$192$	0	0
$193$	−7.84304	−0.564554	−0.282277	−	0.959333i	$-0.591090\pi$
$193$	−7.84304	−0.564554	−0.282277	+	0.959333i	$0.591090\pi$
$194$	0	0
$195$	1.46638	0.105010
$196$	0	0
$197$	−9.48210	−0.675572	−0.337786	−	0.941223i	$-0.609678\pi$
$197$	−9.48210	−0.675572	−0.337786	+	0.941223i	$0.609678\pi$
$198$	0	0
$199$	19.4172	1.37645	0.688225	−	0.725497i	$-0.258390\pi$
$199$	19.4172	1.37645	0.688225	+	0.725497i	$0.258390\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	0.150193	0.0105415
$204$	0	0
$205$	0.0338845	0.00236660
$206$	0	0
$207$	−3.00000	−0.208514
$208$	0	0
$209$	−10.7272	−0.742014
$210$	0	0
$211$	22.0580	1.51854	0.759268	−	0.650778i	$-0.225557\pi$
$211$	22.0580	1.51854	0.759268	+	0.650778i	$0.225557\pi$
$212$	0	0
$213$	−1.86842	−0.128022
$214$	0	0
$215$	−6.39861	−0.436382
$216$	0	0
$217$	0.901773	0.0612163
$218$	0	0
$219$	5.45111	0.368352
$220$	0	0
$221$	8.47868	0.570337
$222$	0	0
$223$	−9.02756	−0.604530	−0.302265	−	0.953224i	$-0.597743\pi$
$223$	−9.02756	−0.604530	−0.302265	+	0.953224i	$0.597743\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−19.5517	−1.29769	−0.648846	−	0.760920i	$-0.724748\pi$
$227$	−19.5517	−1.29769	−0.648846	+	0.760920i	$0.724748\pi$
$228$	0	0
$229$	18.5430	1.22536	0.612679	−	0.790332i	$-0.290092\pi$
$229$	18.5430	1.22536	0.612679	+	0.790332i	$0.290092\pi$
$230$	0	0
$231$	−0.665730	−0.0438018
$232$	0	0
$233$	3.60139	0.235935	0.117967	−	0.993017i	$-0.462362\pi$
$233$	3.60139	0.235935	0.117967	+	0.993017i	$0.462362\pi$
$234$	0	0
$235$	8.66230	0.565066
$236$	0	0
$237$	−4.46638	−0.290123
$238$	0	0
$239$	−5.77019	−0.373243	−0.186621	−	0.982432i	$-0.559754\pi$
$239$	−5.77019	−0.373243	−0.186621	+	0.982432i	$0.559754\pi$
$240$	0	0
$241$	−28.0212	−1.80501	−0.902504	−	0.430683i	$-0.858273\pi$
$241$	−28.0212	−1.80501	−0.902504	+	0.430683i	$0.858273\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.96612	0.445049
$246$	0	0
$247$	4.34946	0.276749
$248$	0	0
$249$	−9.92934	−0.629246
$250$	0	0
$251$	18.9751	1.19770	0.598848	−	0.800863i	$-0.295625\pi$
$251$	18.9751	1.19770	0.598848	+	0.800863i	$0.295625\pi$
$252$	0	0
$253$	−10.8497	−0.682116
$254$	0	0
$255$	5.78204	0.362085
$256$	0	0
$257$	−23.2946	−1.45308	−0.726539	−	0.687126i	$-0.758872\pi$
$257$	−23.2946	−1.45308	−0.726539	+	0.687126i	$0.758872\pi$
$258$	0	0
$259$	−0.484463	−0.0301031
$260$	0	0
$261$	−0.815923	−0.0505043
$262$	0	0
$263$	−16.8311	−1.03785	−0.518925	−	0.854820i	$-0.673668\pi$
$263$	−16.8311	−1.03785	−0.518925	+	0.854820i	$0.673668\pi$
$264$	0	0
$265$	−3.89888	−0.239506
$266$	0	0
$267$	16.7943	1.02780
$268$	0	0
$269$	−22.6719	−1.38233	−0.691165	−	0.722697i	$-0.742902\pi$
$269$	−22.6719	−1.38233	−0.691165	+	0.722697i	$0.742902\pi$
$270$	0	0
$271$	−2.59462	−0.157612	−0.0788059	−	0.996890i	$-0.525111\pi$
$271$	−2.59462	−0.157612	−0.0788059	+	0.996890i	$0.525111\pi$
$272$	0	0
$273$	0.269928	0.0163368
$274$	0	0
$275$	3.61657	0.218088
$276$	0	0
$277$	−1.34901	−0.0810540	−0.0405270	−	0.999178i	$-0.512904\pi$
$277$	−1.34901	−0.0810540	−0.0405270	+	0.999178i	$0.512904\pi$
$278$	0	0
$279$	−4.89888	−0.293288
$280$	0	0
$281$	13.3347	0.795483	0.397741	−	0.917498i	$-0.369794\pi$
$281$	13.3347	0.795483	0.397741	+	0.917498i	$0.369794\pi$
$282$	0	0
$283$	19.7791	1.17575	0.587874	−	0.808952i	$-0.299965\pi$
$283$	19.7791	1.17575	0.587874	+	0.808952i	$0.299965\pi$
$284$	0	0
$285$	2.96612	0.175698
$286$	0	0
$287$	0.00623737	0.000368180 0
$288$	0	0
$289$	16.4320	0.966586
$290$	0	0
$291$	8.37711	0.491075
$292$	0	0
$293$	−12.3285	−0.720238	−0.360119	−	0.932906i	$-0.617264\pi$
$293$	−12.3285	−0.720238	−0.360119	+	0.932906i	$0.617264\pi$
$294$	0	0
$295$	7.26993	0.423272
$296$	0	0
$297$	3.61657	0.209855
$298$	0	0
$299$	4.39915	0.254409
$300$	0	0
$301$	−1.17784	−0.0678896
$302$	0	0
$303$	−11.1659	−0.641465
$304$	0	0
$305$	10.9661	0.627918
$306$	0	0
$307$	13.8712	0.791673	0.395837	−	0.918321i	$-0.370455\pi$
$307$	13.8712	0.791673	0.395837	+	0.918321i	$0.370455\pi$
$308$	0	0
$309$	8.76677	0.498724
$310$	0	0
$311$	−14.2269	−0.806734	−0.403367	−	0.915038i	$-0.632160\pi$
$311$	−14.2269	−0.806734	−0.403367	+	0.915038i	$0.632160\pi$
$312$	0	0
$313$	−6.03335	−0.341025	−0.170513	−	0.985355i	$-0.554542\pi$
$313$	−6.03335	−0.341025	−0.170513	+	0.985355i	$0.554542\pi$
$314$	0	0
$315$	0.184077	0.0103716
$316$	0	0
$317$	−5.39915	−0.303246	−0.151623	−	0.988438i	$-0.548450\pi$
$317$	−5.39915	−0.303246	−0.151623	+	0.988438i	$0.548450\pi$
$318$	0	0
$319$	−2.95085	−0.165216
$320$	0	0
$321$	2.14730	0.119851
$322$	0	0
$323$	17.1502	0.954262
$324$	0	0
$325$	−1.46638	−0.0813402
$326$	0	0
$327$	−15.9293	−0.880894
$328$	0	0
$329$	1.59453	0.0879095
$330$	0	0
$331$	10.8408	0.595863	0.297931	−	0.954587i	$-0.403703\pi$
$331$	10.8408	0.595863	0.297931	+	0.954587i	$0.403703\pi$
$332$	0	0
$333$	2.63185	0.144224
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	17.9441	0.977476	0.488738	−	0.872431i	$-0.337457\pi$
$337$	17.9441	0.977476	0.488738	+	0.872431i	$0.337457\pi$
$338$	0	0
$339$	12.4845	0.678063
$340$	0	0
$341$	−17.7172	−0.959438
$342$	0	0
$343$	2.57085	0.138813
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	15.4663	0.830274	0.415137	−	0.909759i	$-0.363734\pi$
$347$	15.4663	0.830274	0.415137	+	0.909759i	$0.363734\pi$
$348$	0	0
$349$	−27.3589	−1.46449	−0.732245	−	0.681041i	$-0.761528\pi$
$349$	−27.3589	−1.46449	−0.732245	+	0.681041i	$0.761528\pi$
$350$	0	0
$351$	−1.46638	−0.0782697
$352$	0	0
$353$	9.65046	0.513642	0.256821	−	0.966459i	$-0.417325\pi$
$353$	9.65046	0.513642	0.256821	+	0.966459i	$0.417325\pi$
$354$	0	0
$355$	1.86842	0.0991655
$356$	0	0
$357$	1.06434	0.0563310
$358$	0	0
$359$	23.9230	1.26261	0.631304	−	0.775535i	$-0.282520\pi$
$359$	23.9230	1.26261	0.631304	+	0.775535i	$0.282520\pi$
$360$	0	0
$361$	−10.2022	−0.536956
$362$	0	0
$363$	2.07961	0.109151
$364$	0	0
$365$	−5.45111	−0.285324
$366$	0	0
$367$	−13.4325	−0.701171	−0.350585	−	0.936531i	$-0.614017\pi$
$367$	−13.4325	−0.701171	−0.350585	+	0.936531i	$0.614017\pi$
$368$	0	0
$369$	−0.0338845	−0.00176396
$370$	0	0
$371$	−0.717696	−0.0372609
$372$	0	0
$373$	32.7108	1.69370	0.846852	−	0.531829i	$-0.178495\pi$
$373$	32.7108	1.69370	0.846852	+	0.531829i	$0.178495\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	1.19645	0.0616205
$378$	0	0
$379$	35.1133	1.80365	0.901825	−	0.432101i	$-0.142228\pi$
$379$	35.1133	1.80365	0.901825	+	0.432101i	$0.142228\pi$
$380$	0	0
$381$	6.58269	0.337241
$382$	0	0
$383$	7.22981	0.369426	0.184713	−	0.982793i	$-0.440864\pi$
$383$	7.22981	0.369426	0.184713	+	0.982793i	$0.440864\pi$
$384$	0	0
$385$	0.665730	0.0339287
$386$	0	0
$387$	6.39861	0.325260
$388$	0	0
$389$	11.9170	0.604214	0.302107	−	0.953274i	$-0.402310\pi$
$389$	11.9170	0.604214	0.302107	+	0.953274i	$0.402310\pi$
$390$	0	0
$391$	17.3461	0.877231
$392$	0	0
$393$	−21.7085	−1.09505
$394$	0	0
$395$	4.46638	0.224728
$396$	0	0
$397$	16.1219	0.809136	0.404568	−	0.914508i	$-0.367422\pi$
$397$	16.1219	0.809136	0.404568	+	0.914508i	$0.367422\pi$
$398$	0	0
$399$	0.545995	0.0273339
$400$	0	0
$401$	−11.2636	−0.562478	−0.281239	−	0.959638i	$-0.590745\pi$
$401$	−11.2636	−0.562478	−0.281239	+	0.959638i	$0.590745\pi$
$402$	0	0
$403$	7.18363	0.357842
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	9.51826	0.471803
$408$	0	0
$409$	22.5335	1.11421	0.557106	−	0.830442i	$-0.311912\pi$
$409$	22.5335	1.11421	0.557106	+	0.830442i	$0.311912\pi$
$410$	0	0
$411$	−2.05197	−0.101216
$412$	0	0
$413$	1.33823	0.0658500
$414$	0	0
$415$	9.92934	0.487412
$416$	0	0
$417$	−13.7854	−0.675073
$418$	0	0
$419$	−32.0884	−1.56762	−0.783810	−	0.621000i	$-0.786727\pi$
$419$	−32.0884	−1.56762	−0.783810	+	0.621000i	$0.786727\pi$
$420$	0	0
$421$	20.9451	1.02080	0.510400	−	0.859937i	$-0.329497\pi$
$421$	20.9451	1.02080	0.510400	+	0.859937i	$0.329497\pi$
$422$	0	0
$423$	−8.66230	−0.421176
$424$	0	0
$425$	−5.78204	−0.280470
$426$	0	0
$427$	2.01861	0.0976876
$428$	0	0
$429$	−5.30328	−0.256045
$430$	0	0
$431$	7.37764	0.355368	0.177684	−	0.984088i	$-0.443139\pi$
$431$	7.37764	0.355368	0.177684	+	0.984088i	$0.443139\pi$
$432$	0	0
$433$	24.8186	1.19270	0.596352	−	0.802723i	$-0.296616\pi$
$433$	24.8186	1.19270	0.596352	+	0.802723i	$0.296616\pi$
$434$	0	0
$435$	0.815923	0.0391205
$436$	0	0
$437$	8.89835	0.425666
$438$	0	0
$439$	−7.10999	−0.339341	−0.169671	−	0.985501i	$-0.554270\pi$
$439$	−7.10999	−0.339341	−0.169671	+	0.985501i	$0.554270\pi$
$440$	0	0
$441$	−6.96612	−0.331720
$442$	0	0
$443$	−24.8310	−1.17976	−0.589879	−	0.807492i	$-0.700825\pi$
$443$	−24.8310	−1.17976	−0.589879	+	0.807492i	$0.700825\pi$
$444$	0	0
$445$	−16.7943	−0.796128
$446$	0	0
$447$	−12.2670	−0.580211
$448$	0	0
$449$	−2.11639	−0.0998787	−0.0499393	−	0.998752i	$-0.515903\pi$
$449$	−2.11639	−0.0998787	−0.0499393	+	0.998752i	$0.515903\pi$
$450$	0	0
$451$	−0.122546	−0.00577046
$452$	0	0
$453$	11.4939	0.540033
$454$	0	0
$455$	−0.269928	−0.0126544
$456$	0	0
$457$	−11.5449	−0.540049	−0.270025	−	0.962853i	$-0.587032\pi$
$457$	−11.5449	−0.540049	−0.270025	+	0.962853i	$0.587032\pi$
$458$	0	0
$459$	−5.78204	−0.269882
$460$	0	0
$461$	−26.2049	−1.22048	−0.610241	−	0.792216i	$-0.708928\pi$
$461$	−26.2049	−1.22048	−0.610241	+	0.792216i	$0.708928\pi$
$462$	0	0
$463$	−29.1624	−1.35529	−0.677646	−	0.735388i	$-0.737000\pi$
$463$	−29.1624	−1.35529	−0.677646	+	0.735388i	$0.737000\pi$
$464$	0	0
$465$	4.89888	0.227180
$466$	0	0
$467$	−8.98130	−0.415605	−0.207803	−	0.978171i	$-0.566631\pi$
$467$	−8.98130	−0.415605	−0.207803	+	0.978171i	$0.566631\pi$
$468$	0	0
$469$	0.184077	0.00849990
$470$	0	0
$471$	−4.81592	−0.221906
$472$	0	0
$473$	23.1411	1.06403
$474$	0	0
$475$	−2.96612	−0.136095
$476$	0	0
$477$	3.89888	0.178517
$478$	0	0
$479$	3.55459	0.162414	0.0812068	−	0.996697i	$-0.474123\pi$
$479$	3.55459	0.162414	0.0812068	+	0.996697i	$0.474123\pi$
$480$	0	0
$481$	−3.85929	−0.175968
$482$	0	0
$483$	0.552232	0.0251274
$484$	0	0
$485$	−8.37711	−0.380385
$486$	0	0
$487$	−1.49973	−0.0679594	−0.0339797	−	0.999423i	$-0.510818\pi$
$487$	−1.49973	−0.0679594	−0.0339797	+	0.999423i	$0.510818\pi$
$488$	0	0
$489$	9.38677	0.424484
$490$	0	0
$491$	16.0423	0.723979	0.361989	−	0.932182i	$-0.382098\pi$
$491$	16.0423	0.723979	0.361989	+	0.932182i	$0.382098\pi$
$492$	0	0
$493$	4.71770	0.212474
$494$	0	0
$495$	−3.61657	−0.162553
$496$	0	0
$497$	0.343934	0.0154276
$498$	0	0
$499$	6.49078	0.290567	0.145284	−	0.989390i	$-0.453591\pi$
$499$	6.49078	0.290567	0.145284	+	0.989390i	$0.453591\pi$
$500$	0	0
$501$	−9.65046	−0.431151
$502$	0	0
$503$	22.0389	0.982665	0.491332	−	0.870972i	$-0.336510\pi$
$503$	22.0389	0.982665	0.491332	+	0.870972i	$0.336510\pi$
$504$	0	0
$505$	11.1659	0.496877
$506$	0	0
$507$	−10.8497	−0.481853
$508$	0	0
$509$	8.23026	0.364800	0.182400	−	0.983224i	$-0.441613\pi$
$509$	8.23026	0.364800	0.182400	+	0.983224i	$0.441613\pi$
$510$	0	0
$511$	−1.00343	−0.0443890
$512$	0	0
$513$	−2.96612	−0.130957
$514$	0	0
$515$	−8.76677	−0.386310
$516$	0	0
$517$	−31.3279	−1.37780
$518$	0	0
$519$	8.02756	0.352371
$520$	0	0
$521$	2.85649	0.125145	0.0625726	−	0.998040i	$-0.480069\pi$
$521$	2.85649	0.125145	0.0625726	+	0.998040i	$0.480069\pi$
$522$	0	0
$523$	10.2852	0.449740	0.224870	−	0.974389i	$-0.427804\pi$
$523$	10.2852	0.449740	0.224870	+	0.974389i	$0.427804\pi$
$524$	0	0
$525$	−0.184077	−0.00803380
$526$	0	0
$527$	28.3255	1.23388
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−7.26993	−0.315488
$532$	0	0
$533$	0.0496876	0.00215221
$534$	0	0
$535$	−2.14730	−0.0928358
$536$	0	0
$537$	−20.1253	−0.868469
$538$	0	0
$539$	−25.1935	−1.08516
$540$	0	0
$541$	31.3765	1.34898	0.674490	−	0.738284i	$-0.264363\pi$
$541$	31.3765	1.34898	0.674490	+	0.738284i	$0.264363\pi$
$542$	0	0
$543$	0.681450	0.0292438
$544$	0	0
$545$	15.9293	0.682338
$546$	0	0
$547$	−22.1506	−0.947093	−0.473547	−	0.880769i	$-0.657026\pi$
$547$	−22.1506	−0.947093	−0.473547	+	0.880769i	$0.657026\pi$
$548$	0	0
$549$	−10.9661	−0.468022
$550$	0	0
$551$	2.42012	0.103101
$552$	0	0
$553$	0.822160	0.0349618
$554$	0	0
$555$	−2.63185	−0.111716
$556$	0	0
$557$	−33.9012	−1.43644	−0.718219	−	0.695817i	$-0.755042\pi$
$557$	−33.9012	−1.43644	−0.718219	+	0.695817i	$0.755042\pi$
$558$	0	0
$559$	−9.38281	−0.396851
$560$	0	0
$561$	−20.9112	−0.882870
$562$	0	0
$563$	39.1495	1.64995	0.824977	−	0.565166i	$-0.191188\pi$
$563$	39.1495	1.64995	0.824977	+	0.565166i	$0.191188\pi$
$564$	0	0
$565$	−12.4845	−0.525225
$566$	0	0
$567$	−0.184077	−0.00773052
$568$	0	0
$569$	−13.2823	−0.556823	−0.278412	−	0.960462i	$-0.589808\pi$
$569$	−13.2823	−0.556823	−0.278412	+	0.960462i	$0.589808\pi$
$570$	0	0
$571$	1.11982	0.0468629	0.0234315	−	0.999725i	$-0.492541\pi$
$571$	1.11982	0.0468629	0.0234315	+	0.999725i	$0.492541\pi$
$572$	0	0
$573$	−7.12018	−0.297450
$574$	0	0
$575$	−3.00000	−0.125109
$576$	0	0
$577$	−17.7825	−0.740295	−0.370147	−	0.928973i	$-0.620693\pi$
$577$	−17.7825	−0.740295	−0.370147	+	0.928973i	$0.620693\pi$
$578$	0	0
$579$	−7.84304	−0.325945
$580$	0	0
$581$	1.82777	0.0758285
$582$	0	0
$583$	14.1006	0.583987
$584$	0	0
$585$	1.46638	0.0606274
$586$	0	0
$587$	32.1801	1.32822	0.664108	−	0.747637i	$-0.268812\pi$
$587$	32.1801	1.32822	0.664108	+	0.747637i	$0.268812\pi$
$588$	0	0
$589$	14.5306	0.598725
$590$	0	0
$591$	−9.48210	−0.390042
$592$	0	0
$593$	16.4658	0.676171	0.338086	−	0.941115i	$-0.390221\pi$
$593$	16.4658	0.676171	0.338086	+	0.941115i	$0.390221\pi$
$594$	0	0
$595$	−1.06434	−0.0436338
$596$	0	0
$597$	19.4172	0.794694
$598$	0	0
$599$	1.63808	0.0669302	0.0334651	−	0.999440i	$-0.489346\pi$
$599$	1.63808	0.0669302	0.0334651	+	0.999440i	$0.489346\pi$
$600$	0	0
$601$	−1.15126	−0.0469608	−0.0234804	−	0.999724i	$-0.507475\pi$
$601$	−1.15126	−0.0469608	−0.0234804	+	0.999724i	$0.507475\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−2.07961	−0.0845483
$606$	0	0
$607$	0.724285	0.0293978	0.0146989	−	0.999892i	$-0.495321\pi$
$607$	0.724285	0.0293978	0.0146989	+	0.999892i	$0.495321\pi$
$608$	0	0
$609$	0.150193	0.00608612
$610$	0	0
$611$	12.7022	0.513878
$612$	0	0
$613$	−18.8744	−0.762330	−0.381165	−	0.924507i	$-0.624477\pi$
$613$	−18.8744	−0.762330	−0.381165	+	0.924507i	$0.624477\pi$
$614$	0	0
$615$	0.0338845	0.00136635
$616$	0	0
$617$	−41.1501	−1.65664	−0.828321	−	0.560255i	$-0.810703\pi$
$617$	−41.1501	−1.65664	−0.828321	+	0.560255i	$0.810703\pi$
$618$	0	0
$619$	27.2416	1.09493	0.547467	−	0.836827i	$-0.315592\pi$
$619$	27.2416	1.09493	0.547467	+	0.836827i	$0.315592\pi$
$620$	0	0
$621$	−3.00000	−0.120386
$622$	0	0
$623$	−3.09146	−0.123857
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−10.7272	−0.428402
$628$	0	0
$629$	−15.2174	−0.606759
$630$	0	0
$631$	−36.8773	−1.46806	−0.734031	−	0.679116i	$-0.762363\pi$
$631$	−36.8773	−1.46806	−0.734031	+	0.679116i	$0.762363\pi$
$632$	0	0
$633$	22.0580	0.876728
$634$	0	0
$635$	−6.58269	−0.261226
$636$	0	0
$637$	10.2150	0.404732
$638$	0	0
$639$	−1.86842	−0.0739136
$640$	0	0
$641$	44.9344	1.77480	0.887401	−	0.460998i	$-0.152508\pi$
$641$	44.9344	1.77480	0.887401	+	0.460998i	$0.152508\pi$
$642$	0	0
$643$	36.4634	1.43798	0.718988	−	0.695022i	$-0.244606\pi$
$643$	36.4634	1.43798	0.718988	+	0.695022i	$0.244606\pi$
$644$	0	0
$645$	−6.39861	−0.251945
$646$	0	0
$647$	−7.33815	−0.288492	−0.144246	−	0.989542i	$-0.546076\pi$
$647$	−7.33815	−0.288492	−0.144246	+	0.989542i	$0.546076\pi$
$648$	0	0
$649$	−26.2922	−1.03206
$650$	0	0
$651$	0.901773	0.0353433
$652$	0	0
$653$	19.4387	0.760696	0.380348	−	0.924843i	$-0.375804\pi$
$653$	19.4387	0.760696	0.380348	+	0.924843i	$0.375804\pi$
$654$	0	0
$655$	21.7085	0.848221
$656$	0	0
$657$	5.45111	0.212668
$658$	0	0
$659$	15.8773	0.618491	0.309246	−	0.950982i	$-0.399923\pi$
$659$	15.8773	0.618491	0.309246	+	0.950982i	$0.399923\pi$
$660$	0	0
$661$	−10.9882	−0.427390	−0.213695	−	0.976900i	$-0.568550\pi$
$661$	−10.9882	−0.427390	−0.213695	+	0.976900i	$0.568550\pi$
$662$	0	0
$663$	8.47868	0.329284
$664$	0	0
$665$	−0.545995	−0.0211728
$666$	0	0
$667$	2.44777	0.0947779
$668$	0	0
$669$	−9.02756	−0.349026
$670$	0	0
$671$	−39.6598	−1.53105
$672$	0	0
$673$	33.6368	1.29660	0.648302	−	0.761383i	$-0.275479\pi$
$673$	33.6368	1.29660	0.648302	+	0.761383i	$0.275479\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	50.9716	1.95900	0.979498	−	0.201454i	$-0.0645666\pi$
$677$	50.9716	1.95900	0.979498	+	0.201454i	$0.0645666\pi$
$678$	0	0
$679$	−1.54204	−0.0591779
$680$	0	0
$681$	−19.5517	−0.749223
$682$	0	0
$683$	36.9808	1.41503	0.707515	−	0.706698i	$-0.249816\pi$
$683$	36.9808	1.41503	0.707515	+	0.706698i	$0.249816\pi$
$684$	0	0
$685$	2.05197	0.0784016
$686$	0	0
$687$	18.5430	0.707460
$688$	0	0
$689$	−5.71725	−0.217810
$690$	0	0
$691$	25.0153	0.951626	0.475813	−	0.879547i	$-0.342154\pi$
$691$	25.0153	0.951626	0.475813	+	0.879547i	$0.342154\pi$
$692$	0	0
$693$	−0.665730	−0.0252890
$694$	0	0
$695$	13.7854	0.522909
$696$	0	0
$697$	0.195921	0.00742105
$698$	0	0
$699$	3.60139	0.136217
$700$	0	0
$701$	−31.0422	−1.17245	−0.586224	−	0.810149i	$-0.699386\pi$
$701$	−31.0422	−1.17245	−0.586224	+	0.810149i	$0.699386\pi$
$702$	0	0
$703$	−7.80636	−0.294422
$704$	0	0
$705$	8.66230	0.326241
$706$	0	0
$707$	2.05539	0.0773010
$708$	0	0
$709$	−29.0175	−1.08978	−0.544888	−	0.838509i	$-0.683428\pi$
$709$	−29.0175	−1.08978	−0.544888	+	0.838509i	$0.683428\pi$
$710$	0	0
$711$	−4.46638	−0.167502
$712$	0	0
$713$	14.6966	0.550393
$714$	0	0
$715$	5.30328	0.198331
$716$	0	0
$717$	−5.77019	−0.215492
$718$	0	0
$719$	5.85658	0.218413	0.109207	−	0.994019i	$-0.465169\pi$
$719$	5.85658	0.218413	0.109207	+	0.994019i	$0.465169\pi$
$720$	0	0
$721$	−1.61376	−0.0600997
$722$	0	0
$723$	−28.0212	−1.04212
$724$	0	0
$725$	−0.815923	−0.0303026
$726$	0	0
$727$	5.92381	0.219702	0.109851	−	0.993948i	$-0.464963\pi$
$727$	5.92381	0.219702	0.109851	+	0.993948i	$0.464963\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−36.9970	−1.36838
$732$	0	0
$733$	7.86789	0.290607	0.145304	−	0.989387i	$-0.453584\pi$
$733$	7.86789	0.290607	0.145304	+	0.989387i	$0.453584\pi$
$734$	0	0
$735$	6.96612	0.256949
$736$	0	0
$737$	−3.61657	−0.133218
$738$	0	0
$739$	12.9538	0.476514	0.238257	−	0.971202i	$-0.423424\pi$
$739$	12.9538	0.476514	0.238257	+	0.971202i	$0.423424\pi$
$740$	0	0
$741$	4.34946	0.159781
$742$	0	0
$743$	17.1873	0.630542	0.315271	−	0.949002i	$-0.397905\pi$
$743$	17.1873	0.630542	0.315271	+	0.949002i	$0.397905\pi$
$744$	0	0
$745$	12.2670	0.449429
$746$	0	0
$747$	−9.92934	−0.363295
$748$	0	0
$749$	−0.395269	−0.0144428
$750$	0	0
$751$	−16.2293	−0.592215	−0.296107	−	0.955155i	$-0.595689\pi$
$751$	−16.2293	−0.592215	−0.296107	+	0.955155i	$0.595689\pi$
$752$	0	0
$753$	18.9751	0.691490
$754$	0	0
$755$	−11.4939	−0.418308
$756$	0	0
$757$	44.8864	1.63142	0.815712	−	0.578458i	$-0.196345\pi$
$757$	44.8864	1.63142	0.815712	+	0.578458i	$0.196345\pi$
$758$	0	0
$759$	−10.8497	−0.393820
$760$	0	0
$761$	7.42907	0.269304	0.134652	−	0.990893i	$-0.457008\pi$
$761$	7.42907	0.269304	0.134652	+	0.990893i	$0.457008\pi$
$762$	0	0
$763$	2.93223	0.106154
$764$	0	0
$765$	5.78204	0.209050
$766$	0	0
$767$	10.6605	0.384928
$768$	0	0
$769$	−3.94986	−0.142436	−0.0712178	−	0.997461i	$-0.522689\pi$
$769$	−3.94986	−0.142436	−0.0712178	+	0.997461i	$0.522689\pi$
$770$	0	0
$771$	−23.2946	−0.838934
$772$	0	0
$773$	−29.1438	−1.04823	−0.524115	−	0.851648i	$-0.675604\pi$
$773$	−29.1438	−1.04823	−0.524115	+	0.851648i	$0.675604\pi$
$774$	0	0
$775$	−4.89888	−0.175973
$776$	0	0
$777$	−0.484463	−0.0173800
$778$	0	0
$779$	0.100505	0.00360098
$780$	0	0
$781$	−6.75728	−0.241795
$782$	0	0
$783$	−0.815923	−0.0291587
$784$	0	0
$785$	4.81592	0.171888
$786$	0	0
$787$	−16.8073	−0.599117	−0.299558	−	0.954078i	$-0.596839\pi$
$787$	−16.8073	−0.599117	−0.299558	+	0.954078i	$0.596839\pi$
$788$	0	0
$789$	−16.8311	−0.599203
$790$	0	0
$791$	−2.29811	−0.0817113
$792$	0	0
$793$	16.0805	0.571036
$794$	0	0
$795$	−3.89888	−0.138279
$796$	0	0
$797$	−25.8092	−0.914207	−0.457104	−	0.889413i	$-0.651113\pi$
$797$	−25.8092	−0.914207	−0.457104	+	0.889413i	$0.651113\pi$
$798$	0	0
$799$	50.0858	1.77191
$800$	0	0
$801$	16.7943	0.593399
$802$	0	0
$803$	19.7144	0.695704
$804$	0	0
$805$	−0.552232	−0.0194636
$806$	0	0
$807$	−22.6719	−0.798088
$808$	0	0
$809$	−48.2675	−1.69699	−0.848497	−	0.529200i	$-0.822492\pi$
$809$	−48.2675	−1.69699	−0.848497	+	0.529200i	$0.822492\pi$
$810$	0	0
$811$	30.0009	1.05347	0.526737	−	0.850028i	$-0.323415\pi$
$811$	30.0009	1.05347	0.526737	+	0.850028i	$0.323415\pi$
$812$	0	0
$813$	−2.59462	−0.0909972
$814$	0	0
$815$	−9.38677	−0.328804
$816$	0	0
$817$	−18.9790	−0.663992
$818$	0	0
$819$	0.269928	0.00943204
$820$	0	0
$821$	25.5335	0.891127	0.445563	−	0.895250i	$-0.353003\pi$
$821$	25.5335	0.891127	0.445563	+	0.895250i	$0.353003\pi$
$822$	0	0
$823$	26.2833	0.916178	0.458089	−	0.888906i	$-0.348534\pi$
$823$	26.2833	0.916178	0.458089	+	0.888906i	$0.348534\pi$
$824$	0	0
$825$	3.61657	0.125913
$826$	0	0
$827$	−23.5954	−0.820494	−0.410247	−	0.911975i	$-0.634557\pi$
$827$	−23.5954	−0.820494	−0.410247	+	0.911975i	$0.634557\pi$
$828$	0	0
$829$	47.8366	1.66143	0.830716	−	0.556696i	$-0.187931\pi$
$829$	47.8366	1.66143	0.830716	+	0.556696i	$0.187931\pi$
$830$	0	0
$831$	−1.34901	−0.0467966
$832$	0	0
$833$	40.2783	1.39556
$834$	0	0
$835$	9.65046	0.333968
$836$	0	0
$837$	−4.89888	−0.169330
$838$	0	0
$839$	−3.99439	−0.137902	−0.0689509	−	0.997620i	$-0.521965\pi$
$839$	−3.99439	−0.137902	−0.0689509	+	0.997620i	$0.521965\pi$
$840$	0	0
$841$	−28.3343	−0.977044
$842$	0	0
$843$	13.3347	0.459272
$844$	0	0
$845$	10.8497	0.373242
$846$	0	0
$847$	−0.382810	−0.0131535
$848$	0	0
$849$	19.7791	0.678818
$850$	0	0
$851$	−7.89554	−0.270655
$852$	0	0
$853$	−45.4589	−1.55648	−0.778242	−	0.627965i	$-0.783888\pi$
$853$	−45.4589	−1.55648	−0.778242	+	0.627965i	$0.783888\pi$
$854$	0	0
$855$	2.96612	0.101439
$856$	0	0
$857$	−7.50018	−0.256201	−0.128101	−	0.991761i	$-0.540888\pi$
$857$	−7.50018	−0.256201	−0.128101	+	0.991761i	$0.540888\pi$
$858$	0	0
$859$	52.0618	1.77633	0.888163	−	0.459528i	$-0.151982\pi$
$859$	52.0618	1.77633	0.888163	+	0.459528i	$0.151982\pi$
$860$	0	0
$861$	0.00623737	0.000212569 0
$862$	0	0
$863$	−6.25564	−0.212944	−0.106472	−	0.994316i	$-0.533956\pi$
$863$	−6.25564	−0.212944	−0.106472	+	0.994316i	$0.533956\pi$
$864$	0	0
$865$	−8.02756	−0.272945
$866$	0	0
$867$	16.4320	0.558059
$868$	0	0
$869$	−16.1530	−0.547953
$870$	0	0
$871$	1.46638	0.0496865
$872$	0	0
$873$	8.37711	0.283522
$874$	0	0
$875$	0.184077	0.00622295
$876$	0	0
$877$	12.0897	0.408241	0.204121	−	0.978946i	$-0.434567\pi$
$877$	12.0897	0.408241	0.204121	+	0.978946i	$0.434567\pi$
$878$	0	0
$879$	−12.3285	−0.415829
$880$	0	0
$881$	3.23955	0.109143	0.0545716	−	0.998510i	$-0.482621\pi$
$881$	3.23955	0.109143	0.0545716	+	0.998510i	$0.482621\pi$
$882$	0	0
$883$	51.6893	1.73948	0.869741	−	0.493509i	$-0.164286\pi$
$883$	51.6893	1.73948	0.869741	+	0.493509i	$0.164286\pi$
$884$	0	0
$885$	7.26993	0.244376
$886$	0	0
$887$	−9.72006	−0.326368	−0.163184	−	0.986596i	$-0.552176\pi$
$887$	−9.72006	−0.326368	−0.163184	+	0.986596i	$0.552176\pi$
$888$	0	0
$889$	−1.21172	−0.0406399
$890$	0	0
$891$	3.61657	0.121160
$892$	0	0
$893$	25.6934	0.859797
$894$	0	0
$895$	20.1253	0.672713
$896$	0	0
$897$	4.39915	0.146883
$898$	0	0
$899$	3.99711	0.133311
$900$	0	0
$901$	−22.5435	−0.751032
$902$	0	0
$903$	−1.17784	−0.0391961
$904$	0	0
$905$	−0.681450	−0.0226522
$906$	0	0
$907$	30.1601	1.00145	0.500725	−	0.865606i	$-0.333067\pi$
$907$	30.1601	1.00145	0.500725	+	0.865606i	$0.333067\pi$
$908$	0	0
$909$	−11.1659	−0.370350
$910$	0	0
$911$	22.3381	0.740096	0.370048	−	0.929013i	$-0.379341\pi$
$911$	22.3381	0.740096	0.370048	+	0.929013i	$0.379341\pi$
$912$	0	0
$913$	−35.9102	−1.18845
$914$	0	0
$915$	10.9661	0.362529
$916$	0	0
$917$	3.99604	0.131961
$918$	0	0
$919$	36.2484	1.19573	0.597863	−	0.801599i	$-0.296017\pi$
$919$	36.2484	1.19573	0.597863	+	0.801599i	$0.296017\pi$
$920$	0	0
$921$	13.8712	0.457073
$922$	0	0
$923$	2.73982	0.0901822
$924$	0	0
$925$	2.63185	0.0865345
$926$	0	0
$927$	8.76677	0.287938
$928$	0	0
$929$	−38.2596	−1.25526	−0.627629	−	0.778512i	$-0.715975\pi$
$929$	−38.2596	−1.25526	−0.627629	+	0.778512i	$0.715975\pi$
$930$	0	0
$931$	20.6623	0.677180
$932$	0	0
$933$	−14.2269	−0.465768
$934$	0	0
$935$	20.9112	0.683868
$936$	0	0
$937$	−5.82442	−0.190276	−0.0951378	−	0.995464i	$-0.530329\pi$
$937$	−5.82442	−0.190276	−0.0951378	+	0.995464i	$0.530329\pi$
$938$	0	0
$939$	−6.03335	−0.196891
$940$	0	0
$941$	−52.7510	−1.71963	−0.859816	−	0.510604i	$-0.829422\pi$
$941$	−52.7510	−1.71963	−0.859816	+	0.510604i	$0.829422\pi$
$942$	0	0
$943$	0.101653	0.00331029
$944$	0	0
$945$	0.184077	0.00598804
$946$	0	0
$947$	24.7034	0.802753	0.401376	−	0.915913i	$-0.368532\pi$
$947$	24.7034	0.802753	0.401376	+	0.915913i	$0.368532\pi$
$948$	0	0
$949$	−7.99341	−0.259477
$950$	0	0
$951$	−5.39915	−0.175079
$952$	0	0
$953$	−1.88080	−0.0609250	−0.0304625	−	0.999536i	$-0.509698\pi$
$953$	−1.88080	−0.0609250	−0.0304625	+	0.999536i	$0.509698\pi$
$954$	0	0
$955$	7.12018	0.230404
$956$	0	0
$957$	−2.95085	−0.0953873
$958$	0	0
$959$	0.377721	0.0121972
$960$	0	0
$961$	−7.00098	−0.225838
$962$	0	0
$963$	2.14730	0.0691957
$964$	0	0
$965$	7.84304	0.252476
$966$	0	0
$967$	2.65054	0.0852357	0.0426178	−	0.999091i	$-0.486430\pi$
$967$	2.65054	0.0852357	0.0426178	+	0.999091i	$0.486430\pi$
$968$	0	0
$969$	17.1502	0.550944
$970$	0	0
$971$	25.3470	0.813424	0.406712	−	0.913556i	$-0.366675\pi$
$971$	25.3470	0.813424	0.406712	+	0.913556i	$0.366675\pi$
$972$	0	0
$973$	2.53758	0.0813510
$974$	0	0
$975$	−1.46638	−0.0469618
$976$	0	0
$977$	−37.2917	−1.19307	−0.596534	−	0.802588i	$-0.703456\pi$
$977$	−37.2917	−1.19307	−0.596534	+	0.802588i	$0.703456\pi$
$978$	0	0
$979$	60.7380	1.94119
$980$	0	0
$981$	−15.9293	−0.508585
$982$	0	0
$983$	21.3072	0.679593	0.339796	−	0.940499i	$-0.389642\pi$
$983$	21.3072	0.679593	0.339796	+	0.940499i	$0.389642\pi$
$984$	0	0
$985$	9.48210	0.302125
$986$	0	0
$987$	1.59453	0.0507546
$988$	0	0
$989$	−19.1958	−0.610392
$990$	0	0
$991$	−31.8958	−1.01320	−0.506602	−	0.862180i	$-0.669099\pi$
$991$	−31.8958	−1.01320	−0.506602	+	0.862180i	$0.669099\pi$
$992$	0	0
$993$	10.8408	0.344022
$994$	0	0
$995$	−19.4172	−0.615567
$996$	0	0
$997$	−29.8193	−0.944386	−0.472193	−	0.881495i	$-0.656537\pi$
$997$	−29.8193	−0.944386	−0.472193	+	0.881495i	$0.656537\pi$
$998$	0	0
$999$	2.63185	0.0832679

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.c.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.c.1.2	✓	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -0.184077 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -0.184077 q^{7} +1.00000 q^{9} +3.61657 q^{11} -1.46638 q^{13} -1.00000 q^{15} -5.78204 q^{17} -2.96612 q^{19} -0.184077 q^{21} -3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.815923 q^{29} -4.89888 q^{31} +3.61657 q^{33} +0.184077 q^{35} +2.63185 q^{37} -1.46638 q^{39} -0.0338845 q^{41} +6.39861 q^{43} -1.00000 q^{45} -8.66230 q^{47} -6.96612 q^{49} -5.78204 q^{51} +3.89888 q^{53} -3.61657 q^{55} -2.96612 q^{57} -7.26993 q^{59} -10.9661 q^{61} -0.184077 q^{63} +1.46638 q^{65} -1.00000 q^{67} -3.00000 q^{69} -1.86842 q^{71} +5.45111 q^{73} +1.00000 q^{75} -0.665730 q^{77} -4.46638 q^{79} +1.00000 q^{81} -9.92934 q^{83} +5.78204 q^{85} -0.815923 q^{87} +16.7943 q^{89} +0.269928 q^{91} -4.89888 q^{93} +2.96612 q^{95} +8.37711 q^{97} +3.61657 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} - 5 q^{11} - 5 q^{13} - 4 q^{15} - 8 q^{17} + 5 q^{19} + q^{21} - 12 q^{23} + 4 q^{25} + 4 q^{27} - 5 q^{29} - q^{31} - 5 q^{33} - q^{35} + 14 q^{37} - 5 q^{39} - 17 q^{41} - 9 q^{43} - 4 q^{45} - 7 q^{47} - 11 q^{49} - 8 q^{51} - 3 q^{53} + 5 q^{55} + 5 q^{57} - 23 q^{59} - 27 q^{61} + q^{63} + 5 q^{65} - 4 q^{67} - 12 q^{69} - 20 q^{71} - 2 q^{73} + 4 q^{75} - 23 q^{77} - 17 q^{79} + 4 q^{81} + 10 q^{83} + 8 q^{85} - 5 q^{87} - 18 q^{89} - 5 q^{91} - q^{93} - 5 q^{95} - 11 q^{97} - 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^5 + q^7 + 4 * q^9 - 5 * q^11 - 5 * q^13 - 4 * q^15 - 8 * q^17 + 5 * q^19 + q^21 - 12 * q^23 + 4 * q^25 + 4 * q^27 - 5 * q^29 - q^31 - 5 * q^33 - q^35 + 14 * q^37 - 5 * q^39 - 17 * q^41 - 9 * q^43 - 4 * q^45 - 7 * q^47 - 11 * q^49 - 8 * q^51 - 3 * q^53 + 5 * q^55 + 5 * q^57 - 23 * q^59 - 27 * q^61 + q^63 + 5 * q^65 - 4 * q^67 - 12 * q^69 - 20 * q^71 - 2 * q^73 + 4 * q^75 - 23 * q^77 - 17 * q^79 + 4 * q^81 + 10 * q^83 + 8 * q^85 - 5 * q^87 - 18 * q^89 - 5 * q^91 - q^93 - 5 * q^95 - 11 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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