LMFDB - Embedded newform 4620.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4620,2,Mod(1,4620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4620.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.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:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4620.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +3.41421 q^{13} -1.00000 q^{15} -4.65685 q^{17} +1.00000 q^{19} +1.00000 q^{21} +7.82843 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.41421 q^{29} -2.24264 q^{31} -1.00000 q^{33} -1.00000 q^{35} -7.41421 q^{37} -3.41421 q^{39} +9.89949 q^{41} +5.58579 q^{43} +1.00000 q^{45} +6.24264 q^{47} +1.00000 q^{49} +4.65685 q^{51} +0.656854 q^{53} +1.00000 q^{55} -1.00000 q^{57} -4.41421 q^{59} +5.48528 q^{61} -1.00000 q^{63} +3.41421 q^{65} +11.6569 q^{67} -7.82843 q^{69} -15.5563 q^{71} -2.82843 q^{73} -1.00000 q^{75} -1.00000 q^{77} -12.7279 q^{79} +1.00000 q^{81} +3.34315 q^{83} -4.65685 q^{85} +4.41421 q^{87} +9.24264 q^{89} -3.41421 q^{91} +2.24264 q^{93} +1.00000 q^{95} -14.8995 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 4 q^{13} - 2 q^{15} + 2 q^{17} + 2 q^{19} + 2 q^{21} + 10 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} + 4 q^{31} - 2 q^{33} - 2 q^{35} - 12 q^{37} - 4 q^{39} + 14 q^{43} + 2 q^{45} + 4 q^{47} + 2 q^{49} - 2 q^{51} - 10 q^{53} + 2 q^{55} - 2 q^{57} - 6 q^{59} - 6 q^{61} - 2 q^{63} + 4 q^{65} + 12 q^{67} - 10 q^{69} - 2 q^{75} - 2 q^{77} + 2 q^{81} + 18 q^{83} + 2 q^{85} + 6 q^{87} + 10 q^{89} - 4 q^{91} - 4 q^{93} + 2 q^{95} - 10 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 4 * q^13 - 2 * q^15 + 2 * q^17 + 2 * q^19 + 2 * q^21 + 10 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 + 4 * q^31 - 2 * q^33 - 2 * q^35 - 12 * q^37 - 4 * q^39 + 14 * q^43 + 2 * q^45 + 4 * q^47 + 2 * q^49 - 2 * q^51 - 10 * q^53 + 2 * q^55 - 2 * q^57 - 6 * q^59 - 6 * q^61 - 2 * q^63 + 4 * q^65 + 12 * q^67 - 10 * q^69 - 2 * q^75 - 2 * q^77 + 2 * q^81 + 18 * q^83 + 2 * q^85 + 6 * q^87 + 10 * q^89 - 4 * q^91 - 4 * q^93 + 2 * q^95 - 10 * q^97 + 2 * 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
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	3.41421	0.946932	0.473466	−	0.880812i	$-0.343003\pi$
$13$	3.41421	0.946932	0.473466	+	0.880812i	$0.343003\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−4.65685	−1.12945	−0.564727	−	0.825278i	$-0.691018\pi$
$17$	−4.65685	−1.12945	−0.564727	+	0.825278i	$0.691018\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	7.82843	1.63234	0.816170	−	0.577812i	$-0.196093\pi$
$23$	7.82843	1.63234	0.816170	+	0.577812i	$0.196093\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.41421	−0.819699	−0.409849	−	0.912153i	$-0.634419\pi$
$29$	−4.41421	−0.819699	−0.409849	+	0.912153i	$0.634419\pi$
$30$	0	0
$31$	−2.24264	−0.402790	−0.201395	−	0.979510i	$-0.564548\pi$
$31$	−2.24264	−0.402790	−0.201395	+	0.979510i	$0.564548\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−7.41421	−1.21889	−0.609445	−	0.792829i	$-0.708608\pi$
$37$	−7.41421	−1.21889	−0.609445	+	0.792829i	$0.708608\pi$
$38$	0	0
$39$	−3.41421	−0.546712
$40$	0	0
$41$	9.89949	1.54604	0.773021	−	0.634381i	$-0.218745\pi$
$41$	9.89949	1.54604	0.773021	+	0.634381i	$0.218745\pi$
$42$	0	0
$43$	5.58579	0.851824	0.425912	−	0.904764i	$-0.359953\pi$
$43$	5.58579	0.851824	0.425912	+	0.904764i	$0.359953\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	6.24264	0.910583	0.455291	−	0.890343i	$-0.349535\pi$
$47$	6.24264	0.910583	0.455291	+	0.890343i	$0.349535\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.65685	0.652090
$52$	0	0
$53$	0.656854	0.0902259	0.0451129	−	0.998982i	$-0.485635\pi$
$53$	0.656854	0.0902259	0.0451129	+	0.998982i	$0.485635\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−4.41421	−0.574682	−0.287341	−	0.957828i	$-0.592771\pi$
$59$	−4.41421	−0.574682	−0.287341	+	0.957828i	$0.592771\pi$
$60$	0	0
$61$	5.48528	0.702318	0.351159	−	0.936316i	$-0.385788\pi$
$61$	5.48528	0.702318	0.351159	+	0.936316i	$0.385788\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	3.41421	0.423481
$66$	0	0
$67$	11.6569	1.42411	0.712056	−	0.702123i	$-0.247764\pi$
$67$	11.6569	1.42411	0.712056	+	0.702123i	$0.247764\pi$
$68$	0	0
$69$	−7.82843	−0.942432
$70$	0	0
$71$	−15.5563	−1.84620	−0.923099	−	0.384561i	$-0.874353\pi$
$71$	−15.5563	−1.84620	−0.923099	+	0.384561i	$0.874353\pi$
$72$	0	0
$73$	−2.82843	−0.331042	−0.165521	−	0.986206i	$-0.552931\pi$
$73$	−2.82843	−0.331042	−0.165521	+	0.986206i	$0.552931\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−12.7279	−1.43200	−0.716002	−	0.698099i	$-0.754030\pi$
$79$	−12.7279	−1.43200	−0.716002	+	0.698099i	$0.754030\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.34315	0.366958	0.183479	−	0.983024i	$-0.441264\pi$
$83$	3.34315	0.366958	0.183479	+	0.983024i	$0.441264\pi$
$84$	0	0
$85$	−4.65685	−0.505107
$86$	0	0
$87$	4.41421	0.473253
$88$	0	0
$89$	9.24264	0.979718	0.489859	−	0.871802i	$-0.337048\pi$
$89$	9.24264	0.979718	0.489859	+	0.871802i	$0.337048\pi$
$90$	0	0
$91$	−3.41421	−0.357907
$92$	0	0
$93$	2.24264	0.232551
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−14.8995	−1.51281	−0.756407	−	0.654101i	$-0.773047\pi$
$97$	−14.8995	−1.51281	−0.756407	+	0.654101i	$0.773047\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−4.34315	−0.432159	−0.216080	−	0.976376i	$-0.569327\pi$
$101$	−4.34315	−0.432159	−0.216080	+	0.976376i	$0.569327\pi$
$102$	0	0
$103$	13.2426	1.30484	0.652418	−	0.757859i	$-0.273755\pi$
$103$	13.2426	1.30484	0.652418	+	0.757859i	$0.273755\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	7.41421	0.716759	0.358380	−	0.933576i	$-0.383329\pi$
$107$	7.41421	0.716759	0.358380	+	0.933576i	$0.383329\pi$
$108$	0	0
$109$	−11.8995	−1.13976	−0.569882	−	0.821726i	$-0.693011\pi$
$109$	−11.8995	−1.13976	−0.569882	+	0.821726i	$0.693011\pi$
$110$	0	0
$111$	7.41421	0.703726
$112$	0	0
$113$	6.65685	0.626224	0.313112	−	0.949716i	$-0.398628\pi$
$113$	6.65685	0.626224	0.313112	+	0.949716i	$0.398628\pi$
$114$	0	0
$115$	7.82843	0.730005
$116$	0	0
$117$	3.41421	0.315644
$118$	0	0
$119$	4.65685	0.426893
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−9.89949	−0.892607
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	13.7279	1.21816	0.609078	−	0.793110i	$-0.291540\pi$
$127$	13.7279	1.21816	0.609078	+	0.793110i	$0.291540\pi$
$128$	0	0
$129$	−5.58579	−0.491801
$130$	0	0
$131$	13.0711	1.14202	0.571012	−	0.820942i	$-0.306551\pi$
$131$	13.0711	1.14202	0.571012	+	0.820942i	$0.306551\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−3.31371	−0.283109	−0.141555	−	0.989930i	$-0.545210\pi$
$137$	−3.31371	−0.283109	−0.141555	+	0.989930i	$0.545210\pi$
$138$	0	0
$139$	15.6569	1.32800	0.663999	−	0.747734i	$-0.268858\pi$
$139$	15.6569	1.32800	0.663999	+	0.747734i	$0.268858\pi$
$140$	0	0
$141$	−6.24264	−0.525725
$142$	0	0
$143$	3.41421	0.285511
$144$	0	0
$145$	−4.41421	−0.366580
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	22.4853	1.84207	0.921033	−	0.389485i	$-0.127347\pi$
$149$	22.4853	1.84207	0.921033	+	0.389485i	$0.127347\pi$
$150$	0	0
$151$	18.9706	1.54380	0.771901	−	0.635742i	$-0.219306\pi$
$151$	18.9706	1.54380	0.771901	+	0.635742i	$0.219306\pi$
$152$	0	0
$153$	−4.65685	−0.376484
$154$	0	0
$155$	−2.24264	−0.180133
$156$	0	0
$157$	−0.414214	−0.0330578	−0.0165289	−	0.999863i	$-0.505262\pi$
$157$	−0.414214	−0.0330578	−0.0165289	+	0.999863i	$0.505262\pi$
$158$	0	0
$159$	−0.656854	−0.0520919
$160$	0	0
$161$	−7.82843	−0.616966
$162$	0	0
$163$	6.24264	0.488961	0.244481	−	0.969654i	$-0.421383\pi$
$163$	6.24264	0.488961	0.244481	+	0.969654i	$0.421383\pi$
$164$	0	0
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	−1.34315	−0.103319
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−8.97056	−0.682019	−0.341010	−	0.940060i	$-0.610769\pi$
$173$	−8.97056	−0.682019	−0.341010	+	0.940060i	$0.610769\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	4.41421	0.331793
$178$	0	0
$179$	−8.24264	−0.616084	−0.308042	−	0.951373i	$-0.599674\pi$
$179$	−8.24264	−0.616084	−0.308042	+	0.951373i	$0.599674\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−5.48528	−0.405484
$184$	0	0
$185$	−7.41421	−0.545104
$186$	0	0
$187$	−4.65685	−0.340543
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−26.6274	−1.92669	−0.963346	−	0.268261i	$-0.913551\pi$
$191$	−26.6274	−1.92669	−0.963346	+	0.268261i	$0.913551\pi$
$192$	0	0
$193$	−8.82843	−0.635484	−0.317742	−	0.948177i	$-0.602925\pi$
$193$	−8.82843	−0.635484	−0.317742	+	0.948177i	$0.602925\pi$
$194$	0	0
$195$	−3.41421	−0.244497
$196$	0	0
$197$	−1.17157	−0.0834711	−0.0417356	−	0.999129i	$-0.513289\pi$
$197$	−1.17157	−0.0834711	−0.0417356	+	0.999129i	$0.513289\pi$
$198$	0	0
$199$	21.6569	1.53521	0.767607	−	0.640921i	$-0.221447\pi$
$199$	21.6569	1.53521	0.767607	+	0.640921i	$0.221447\pi$
$200$	0	0
$201$	−11.6569	−0.822211
$202$	0	0
$203$	4.41421	0.309817
$204$	0	0
$205$	9.89949	0.691411
$206$	0	0
$207$	7.82843	0.544113
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−7.17157	−0.493711	−0.246856	−	0.969052i	$-0.579397\pi$
$211$	−7.17157	−0.493711	−0.246856	+	0.969052i	$0.579397\pi$
$212$	0	0
$213$	15.5563	1.06590
$214$	0	0
$215$	5.58579	0.380947
$216$	0	0
$217$	2.24264	0.152240
$218$	0	0
$219$	2.82843	0.191127
$220$	0	0
$221$	−15.8995	−1.06952
$222$	0	0
$223$	18.5563	1.24263	0.621313	−	0.783563i	$-0.286600\pi$
$223$	18.5563	1.24263	0.621313	+	0.783563i	$0.286600\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−1.48528	−0.0985816	−0.0492908	−	0.998784i	$-0.515696\pi$
$227$	−1.48528	−0.0985816	−0.0492908	+	0.998784i	$0.515696\pi$
$228$	0	0
$229$	3.41421	0.225618	0.112809	−	0.993617i	$-0.464015\pi$
$229$	3.41421	0.225618	0.112809	+	0.993617i	$0.464015\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	15.7574	1.03230	0.516149	−	0.856499i	$-0.327365\pi$
$233$	15.7574	1.03230	0.516149	+	0.856499i	$0.327365\pi$
$234$	0	0
$235$	6.24264	0.407225
$236$	0	0
$237$	12.7279	0.826767
$238$	0	0
$239$	7.72792	0.499878	0.249939	−	0.968262i	$-0.419589\pi$
$239$	7.72792	0.499878	0.249939	+	0.968262i	$0.419589\pi$
$240$	0	0
$241$	−27.7990	−1.79069	−0.895345	−	0.445373i	$-0.853071\pi$
$241$	−27.7990	−1.79069	−0.895345	+	0.445373i	$0.853071\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	3.41421	0.217241
$248$	0	0
$249$	−3.34315	−0.211863
$250$	0	0
$251$	−9.65685	−0.609535	−0.304768	−	0.952427i	$-0.598579\pi$
$251$	−9.65685	−0.609535	−0.304768	+	0.952427i	$0.598579\pi$
$252$	0	0
$253$	7.82843	0.492169
$254$	0	0
$255$	4.65685	0.291624
$256$	0	0
$257$	6.58579	0.410810	0.205405	−	0.978677i	$-0.434149\pi$
$257$	6.58579	0.410810	0.205405	+	0.978677i	$0.434149\pi$
$258$	0	0
$259$	7.41421	0.460697
$260$	0	0
$261$	−4.41421	−0.273233
$262$	0	0
$263$	15.7990	0.974207	0.487104	−	0.873344i	$-0.338053\pi$
$263$	15.7990	0.974207	0.487104	+	0.873344i	$0.338053\pi$
$264$	0	0
$265$	0.656854	0.0403502
$266$	0	0
$267$	−9.24264	−0.565640
$268$	0	0
$269$	8.41421	0.513024	0.256512	−	0.966541i	$-0.417427\pi$
$269$	8.41421	0.513024	0.256512	+	0.966541i	$0.417427\pi$
$270$	0	0
$271$	−10.3137	−0.626513	−0.313257	−	0.949669i	$-0.601420\pi$
$271$	−10.3137	−0.626513	−0.313257	+	0.949669i	$0.601420\pi$
$272$	0	0
$273$	3.41421	0.206638
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	21.1716	1.27208	0.636038	−	0.771658i	$-0.280572\pi$
$277$	21.1716	1.27208	0.636038	+	0.771658i	$0.280572\pi$
$278$	0	0
$279$	−2.24264	−0.134263
$280$	0	0
$281$	22.1421	1.32089	0.660445	−	0.750875i	$-0.270368\pi$
$281$	22.1421	1.32089	0.660445	+	0.750875i	$0.270368\pi$
$282$	0	0
$283$	14.9289	0.887433	0.443716	−	0.896167i	$-0.353660\pi$
$283$	14.9289	0.887433	0.443716	+	0.896167i	$0.353660\pi$
$284$	0	0
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	−9.89949	−0.584349
$288$	0	0
$289$	4.68629	0.275664
$290$	0	0
$291$	14.8995	0.873424
$292$	0	0
$293$	17.3431	1.01320	0.506599	−	0.862182i	$-0.330903\pi$
$293$	17.3431	1.01320	0.506599	+	0.862182i	$0.330903\pi$
$294$	0	0
$295$	−4.41421	−0.257005
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	26.7279	1.54572
$300$	0	0
$301$	−5.58579	−0.321959
$302$	0	0
$303$	4.34315	0.249507
$304$	0	0
$305$	5.48528	0.314086
$306$	0	0
$307$	2.48528	0.141843	0.0709213	−	0.997482i	$-0.477406\pi$
$307$	2.48528	0.141843	0.0709213	+	0.997482i	$0.477406\pi$
$308$	0	0
$309$	−13.2426	−0.753348
$310$	0	0
$311$	33.4558	1.89711	0.948553	−	0.316617i	$-0.102547\pi$
$311$	33.4558	1.89711	0.948553	+	0.316617i	$0.102547\pi$
$312$	0	0
$313$	−9.72792	−0.549855	−0.274927	−	0.961465i	$-0.588654\pi$
$313$	−9.72792	−0.549855	−0.274927	+	0.961465i	$0.588654\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	10.8284	0.608185	0.304093	−	0.952643i	$-0.401647\pi$
$317$	10.8284	0.608185	0.304093	+	0.952643i	$0.401647\pi$
$318$	0	0
$319$	−4.41421	−0.247149
$320$	0	0
$321$	−7.41421	−0.413821
$322$	0	0
$323$	−4.65685	−0.259114
$324$	0	0
$325$	3.41421	0.189386
$326$	0	0
$327$	11.8995	0.658044
$328$	0	0
$329$	−6.24264	−0.344168
$330$	0	0
$331$	−3.14214	−0.172707	−0.0863537	−	0.996265i	$-0.527522\pi$
$331$	−3.14214	−0.172707	−0.0863537	+	0.996265i	$0.527522\pi$
$332$	0	0
$333$	−7.41421	−0.406296
$334$	0	0
$335$	11.6569	0.636882
$336$	0	0
$337$	−20.4142	−1.11203	−0.556017	−	0.831171i	$-0.687671\pi$
$337$	−20.4142	−1.11203	−0.556017	+	0.831171i	$0.687671\pi$
$338$	0	0
$339$	−6.65685	−0.361551
$340$	0	0
$341$	−2.24264	−0.121446
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−7.82843	−0.421468
$346$	0	0
$347$	9.17157	0.492356	0.246178	−	0.969225i	$-0.420825\pi$
$347$	9.17157	0.492356	0.246178	+	0.969225i	$0.420825\pi$
$348$	0	0
$349$	−7.82843	−0.419046	−0.209523	−	0.977804i	$-0.567191\pi$
$349$	−7.82843	−0.419046	−0.209523	+	0.977804i	$0.567191\pi$
$350$	0	0
$351$	−3.41421	−0.182237
$352$	0	0
$353$	32.8284	1.74728	0.873640	−	0.486572i	$-0.161753\pi$
$353$	32.8284	1.74728	0.873640	+	0.486572i	$0.161753\pi$
$354$	0	0
$355$	−15.5563	−0.825645
$356$	0	0
$357$	−4.65685	−0.246467
$358$	0	0
$359$	26.2132	1.38348	0.691740	−	0.722147i	$-0.256844\pi$
$359$	26.2132	1.38348	0.691740	+	0.722147i	$0.256844\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−2.82843	−0.148047
$366$	0	0
$367$	−5.24264	−0.273664	−0.136832	−	0.990594i	$-0.543692\pi$
$367$	−5.24264	−0.273664	−0.136832	+	0.990594i	$0.543692\pi$
$368$	0	0
$369$	9.89949	0.515347
$370$	0	0
$371$	−0.656854	−0.0341022
$372$	0	0
$373$	17.3848	0.900150	0.450075	−	0.892991i	$-0.351397\pi$
$373$	17.3848	0.900150	0.450075	+	0.892991i	$0.351397\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−15.0711	−0.776199
$378$	0	0
$379$	3.14214	0.161401	0.0807003	−	0.996738i	$-0.474284\pi$
$379$	3.14214	0.161401	0.0807003	+	0.996738i	$0.474284\pi$
$380$	0	0
$381$	−13.7279	−0.703303
$382$	0	0
$383$	5.41421	0.276653	0.138327	−	0.990387i	$-0.455828\pi$
$383$	5.41421	0.276653	0.138327	+	0.990387i	$0.455828\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	5.58579	0.283941
$388$	0	0
$389$	−22.2426	−1.12775	−0.563873	−	0.825861i	$-0.690689\pi$
$389$	−22.2426	−1.12775	−0.563873	+	0.825861i	$0.690689\pi$
$390$	0	0
$391$	−36.4558	−1.84365
$392$	0	0
$393$	−13.0711	−0.659348
$394$	0	0
$395$	−12.7279	−0.640411
$396$	0	0
$397$	15.7990	0.792929	0.396464	−	0.918050i	$-0.370237\pi$
$397$	15.7990	0.792929	0.396464	+	0.918050i	$0.370237\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	37.2132	1.85834	0.929169	−	0.369654i	$-0.120524\pi$
$401$	37.2132	1.85834	0.929169	+	0.369654i	$0.120524\pi$
$402$	0	0
$403$	−7.65685	−0.381415
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−7.41421	−0.367509
$408$	0	0
$409$	4.48528	0.221783	0.110891	−	0.993833i	$-0.464629\pi$
$409$	4.48528	0.221783	0.110891	+	0.993833i	$0.464629\pi$
$410$	0	0
$411$	3.31371	0.163453
$412$	0	0
$413$	4.41421	0.217209
$414$	0	0
$415$	3.34315	0.164109
$416$	0	0
$417$	−15.6569	−0.766719
$418$	0	0
$419$	−4.21320	−0.205828	−0.102914	−	0.994690i	$-0.532817\pi$
$419$	−4.21320	−0.205828	−0.102914	+	0.994690i	$0.532817\pi$
$420$	0	0
$421$	1.97056	0.0960394	0.0480197	−	0.998846i	$-0.484709\pi$
$421$	1.97056	0.0960394	0.0480197	+	0.998846i	$0.484709\pi$
$422$	0	0
$423$	6.24264	0.303528
$424$	0	0
$425$	−4.65685	−0.225891
$426$	0	0
$427$	−5.48528	−0.265451
$428$	0	0
$429$	−3.41421	−0.164840
$430$	0	0
$431$	−37.4558	−1.80418	−0.902092	−	0.431543i	$-0.857969\pi$
$431$	−37.4558	−1.80418	−0.902092	+	0.431543i	$0.857969\pi$
$432$	0	0
$433$	−27.4558	−1.31944	−0.659722	−	0.751510i	$-0.729326\pi$
$433$	−27.4558	−1.31944	−0.659722	+	0.751510i	$0.729326\pi$
$434$	0	0
$435$	4.41421	0.211645
$436$	0	0
$437$	7.82843	0.374484
$438$	0	0
$439$	0.514719	0.0245662	0.0122831	−	0.999925i	$-0.496090\pi$
$439$	0.514719	0.0245662	0.0122831	+	0.999925i	$0.496090\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	2.68629	0.127630	0.0638148	−	0.997962i	$-0.479673\pi$
$443$	2.68629	0.127630	0.0638148	+	0.997962i	$0.479673\pi$
$444$	0	0
$445$	9.24264	0.438143
$446$	0	0
$447$	−22.4853	−1.06352
$448$	0	0
$449$	−13.2132	−0.623570	−0.311785	−	0.950153i	$-0.600927\pi$
$449$	−13.2132	−0.623570	−0.311785	+	0.950153i	$0.600927\pi$
$450$	0	0
$451$	9.89949	0.466149
$452$	0	0
$453$	−18.9706	−0.891315
$454$	0	0
$455$	−3.41421	−0.160061
$456$	0	0
$457$	13.9289	0.651568	0.325784	−	0.945444i	$-0.394372\pi$
$457$	13.9289	0.651568	0.325784	+	0.945444i	$0.394372\pi$
$458$	0	0
$459$	4.65685	0.217363
$460$	0	0
$461$	4.14214	0.192918	0.0964592	−	0.995337i	$-0.469248\pi$
$461$	4.14214	0.192918	0.0964592	+	0.995337i	$0.469248\pi$
$462$	0	0
$463$	−12.1421	−0.564293	−0.282146	−	0.959371i	$-0.591046\pi$
$463$	−12.1421	−0.564293	−0.282146	+	0.959371i	$0.591046\pi$
$464$	0	0
$465$	2.24264	0.104000
$466$	0	0
$467$	−0.201010	−0.00930164	−0.00465082	−	0.999989i	$-0.501480\pi$
$467$	−0.201010	−0.00930164	−0.00465082	+	0.999989i	$0.501480\pi$
$468$	0	0
$469$	−11.6569	−0.538264
$470$	0	0
$471$	0.414214	0.0190860
$472$	0	0
$473$	5.58579	0.256835
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0.656854	0.0300753
$478$	0	0
$479$	10.4437	0.477183	0.238591	−	0.971120i	$-0.423314\pi$
$479$	10.4437	0.477183	0.238591	+	0.971120i	$0.423314\pi$
$480$	0	0
$481$	−25.3137	−1.15421
$482$	0	0
$483$	7.82843	0.356206
$484$	0	0
$485$	−14.8995	−0.676551
$486$	0	0
$487$	−5.51472	−0.249896	−0.124948	−	0.992163i	$-0.539876\pi$
$487$	−5.51472	−0.249896	−0.124948	+	0.992163i	$0.539876\pi$
$488$	0	0
$489$	−6.24264	−0.282302
$490$	0	0
$491$	34.6985	1.56592	0.782960	−	0.622072i	$-0.213709\pi$
$491$	34.6985	1.56592	0.782960	+	0.622072i	$0.213709\pi$
$492$	0	0
$493$	20.5563	0.925811
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	15.5563	0.697798
$498$	0	0
$499$	23.2843	1.04235	0.521174	−	0.853451i	$-0.325494\pi$
$499$	23.2843	1.04235	0.521174	+	0.853451i	$0.325494\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	0	0
$503$	−6.31371	−0.281514	−0.140757	−	0.990044i	$-0.544954\pi$
$503$	−6.31371	−0.281514	−0.140757	+	0.990044i	$0.544954\pi$
$504$	0	0
$505$	−4.34315	−0.193267
$506$	0	0
$507$	1.34315	0.0596512
$508$	0	0
$509$	9.58579	0.424883	0.212441	−	0.977174i	$-0.431859\pi$
$509$	9.58579	0.424883	0.212441	+	0.977174i	$0.431859\pi$
$510$	0	0
$511$	2.82843	0.125122
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	13.2426	0.583540
$516$	0	0
$517$	6.24264	0.274551
$518$	0	0
$519$	8.97056	0.393764
$520$	0	0
$521$	17.7279	0.776674	0.388337	−	0.921517i	$-0.373050\pi$
$521$	17.7279	0.776674	0.388337	+	0.921517i	$0.373050\pi$
$522$	0	0
$523$	−27.1127	−1.18556	−0.592778	−	0.805366i	$-0.701969\pi$
$523$	−27.1127	−1.18556	−0.592778	+	0.805366i	$0.701969\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	10.4437	0.454933
$528$	0	0
$529$	38.2843	1.66453
$530$	0	0
$531$	−4.41421	−0.191561
$532$	0	0
$533$	33.7990	1.46400
$534$	0	0
$535$	7.41421	0.320544
$536$	0	0
$537$	8.24264	0.355696
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−15.8995	−0.683573	−0.341786	−	0.939778i	$-0.611032\pi$
$541$	−15.8995	−0.683573	−0.341786	+	0.939778i	$0.611032\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	−11.8995	−0.509718
$546$	0	0
$547$	−34.2132	−1.46285	−0.731425	−	0.681921i	$-0.761145\pi$
$547$	−34.2132	−1.46285	−0.731425	+	0.681921i	$0.761145\pi$
$548$	0	0
$549$	5.48528	0.234106
$550$	0	0
$551$	−4.41421	−0.188052
$552$	0	0
$553$	12.7279	0.541246
$554$	0	0
$555$	7.41421	0.314716
$556$	0	0
$557$	20.1421	0.853450	0.426725	−	0.904382i	$-0.359667\pi$
$557$	20.1421	0.853450	0.426725	+	0.904382i	$0.359667\pi$
$558$	0	0
$559$	19.0711	0.806620
$560$	0	0
$561$	4.65685	0.196613
$562$	0	0
$563$	21.7990	0.918718	0.459359	−	0.888251i	$-0.348079\pi$
$563$	21.7990	0.918718	0.459359	+	0.888251i	$0.348079\pi$
$564$	0	0
$565$	6.65685	0.280056
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−11.3848	−0.477275	−0.238637	−	0.971109i	$-0.576701\pi$
$569$	−11.3848	−0.477275	−0.238637	+	0.971109i	$0.576701\pi$
$570$	0	0
$571$	−24.2426	−1.01452	−0.507261	−	0.861792i	$-0.669342\pi$
$571$	−24.2426	−1.01452	−0.507261	+	0.861792i	$0.669342\pi$
$572$	0	0
$573$	26.6274	1.11238
$574$	0	0
$575$	7.82843	0.326468
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	8.82843	0.366897
$580$	0	0
$581$	−3.34315	−0.138697
$582$	0	0
$583$	0.656854	0.0272041
$584$	0	0
$585$	3.41421	0.141160
$586$	0	0
$587$	38.8701	1.60434	0.802169	−	0.597096i	$-0.203679\pi$
$587$	38.8701	1.60434	0.802169	+	0.597096i	$0.203679\pi$
$588$	0	0
$589$	−2.24264	−0.0924064
$590$	0	0
$591$	1.17157	0.0481921
$592$	0	0
$593$	24.4853	1.00549	0.502745	−	0.864435i	$-0.332323\pi$
$593$	24.4853	1.00549	0.502745	+	0.864435i	$0.332323\pi$
$594$	0	0
$595$	4.65685	0.190912
$596$	0	0
$597$	−21.6569	−0.886356
$598$	0	0
$599$	−12.1421	−0.496114	−0.248057	−	0.968745i	$-0.579792\pi$
$599$	−12.1421	−0.496114	−0.248057	+	0.968745i	$0.579792\pi$
$600$	0	0
$601$	27.2843	1.11295	0.556474	−	0.830865i	$-0.312154\pi$
$601$	27.2843	1.11295	0.556474	+	0.830865i	$0.312154\pi$
$602$	0	0
$603$	11.6569	0.474704
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−6.48528	−0.263229	−0.131615	−	0.991301i	$-0.542016\pi$
$607$	−6.48528	−0.263229	−0.131615	+	0.991301i	$0.542016\pi$
$608$	0	0
$609$	−4.41421	−0.178873
$610$	0	0
$611$	21.3137	0.862260
$612$	0	0
$613$	−30.9706	−1.25089	−0.625445	−	0.780269i	$-0.715082\pi$
$613$	−30.9706	−1.25089	−0.625445	+	0.780269i	$0.715082\pi$
$614$	0	0
$615$	−9.89949	−0.399186
$616$	0	0
$617$	−38.1421	−1.53554	−0.767772	−	0.640723i	$-0.778635\pi$
$617$	−38.1421	−1.53554	−0.767772	+	0.640723i	$0.778635\pi$
$618$	0	0
$619$	−22.2843	−0.895680	−0.447840	−	0.894114i	$-0.647807\pi$
$619$	−22.2843	−0.895680	−0.447840	+	0.894114i	$0.647807\pi$
$620$	0	0
$621$	−7.82843	−0.314144
$622$	0	0
$623$	−9.24264	−0.370299
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.00000	−0.0399362
$628$	0	0
$629$	34.5269	1.37668
$630$	0	0
$631$	−37.8284	−1.50593	−0.752963	−	0.658063i	$-0.771376\pi$
$631$	−37.8284	−1.50593	−0.752963	+	0.658063i	$0.771376\pi$
$632$	0	0
$633$	7.17157	0.285044
$634$	0	0
$635$	13.7279	0.544776
$636$	0	0
$637$	3.41421	0.135276
$638$	0	0
$639$	−15.5563	−0.615400
$640$	0	0
$641$	−26.9706	−1.06527	−0.532637	−	0.846344i	$-0.678799\pi$
$641$	−26.9706	−1.06527	−0.532637	+	0.846344i	$0.678799\pi$
$642$	0	0
$643$	−7.10051	−0.280017	−0.140008	−	0.990150i	$-0.544713\pi$
$643$	−7.10051	−0.280017	−0.140008	+	0.990150i	$0.544713\pi$
$644$	0	0
$645$	−5.58579	−0.219940
$646$	0	0
$647$	23.7990	0.935635	0.467817	−	0.883825i	$-0.345040\pi$
$647$	23.7990	0.935635	0.467817	+	0.883825i	$0.345040\pi$
$648$	0	0
$649$	−4.41421	−0.173273
$650$	0	0
$651$	−2.24264	−0.0878960
$652$	0	0
$653$	−36.6569	−1.43449	−0.717247	−	0.696819i	$-0.754598\pi$
$653$	−36.6569	−1.43449	−0.717247	+	0.696819i	$0.754598\pi$
$654$	0	0
$655$	13.0711	0.510729
$656$	0	0
$657$	−2.82843	−0.110347
$658$	0	0
$659$	−24.6985	−0.962116	−0.481058	−	0.876689i	$-0.659747\pi$
$659$	−24.6985	−0.962116	−0.481058	+	0.876689i	$0.659747\pi$
$660$	0	0
$661$	−21.5563	−0.838445	−0.419222	−	0.907884i	$-0.637697\pi$
$661$	−21.5563	−0.838445	−0.419222	+	0.907884i	$0.637697\pi$
$662$	0	0
$663$	15.8995	0.617485
$664$	0	0
$665$	−1.00000	−0.0387783
$666$	0	0
$667$	−34.5563	−1.33803
$668$	0	0
$669$	−18.5563	−0.717430
$670$	0	0
$671$	5.48528	0.211757
$672$	0	0
$673$	4.07107	0.156928	0.0784641	−	0.996917i	$-0.474998\pi$
$673$	4.07107	0.156928	0.0784641	+	0.996917i	$0.474998\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−11.9706	−0.460066	−0.230033	−	0.973183i	$-0.573883\pi$
$677$	−11.9706	−0.460066	−0.230033	+	0.973183i	$0.573883\pi$
$678$	0	0
$679$	14.8995	0.571790
$680$	0	0
$681$	1.48528	0.0569161
$682$	0	0
$683$	37.9411	1.45178	0.725888	−	0.687812i	$-0.241429\pi$
$683$	37.9411	1.45178	0.725888	+	0.687812i	$0.241429\pi$
$684$	0	0
$685$	−3.31371	−0.126610
$686$	0	0
$687$	−3.41421	−0.130260
$688$	0	0
$689$	2.24264	0.0854378
$690$	0	0
$691$	14.2843	0.543399	0.271700	−	0.962382i	$-0.412414\pi$
$691$	14.2843	0.543399	0.271700	+	0.962382i	$0.412414\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	15.6569	0.593898
$696$	0	0
$697$	−46.1005	−1.74618
$698$	0	0
$699$	−15.7574	−0.595998
$700$	0	0
$701$	26.7574	1.01061	0.505306	−	0.862940i	$-0.331380\pi$
$701$	26.7574	1.01061	0.505306	+	0.862940i	$0.331380\pi$
$702$	0	0
$703$	−7.41421	−0.279632
$704$	0	0
$705$	−6.24264	−0.235111
$706$	0	0
$707$	4.34315	0.163341
$708$	0	0
$709$	−47.6274	−1.78869	−0.894343	−	0.447383i	$-0.852356\pi$
$709$	−47.6274	−1.78869	−0.894343	+	0.447383i	$0.852356\pi$
$710$	0	0
$711$	−12.7279	−0.477334
$712$	0	0
$713$	−17.5563	−0.657490
$714$	0	0
$715$	3.41421	0.127684
$716$	0	0
$717$	−7.72792	−0.288605
$718$	0	0
$719$	−19.7279	−0.735727	−0.367864	−	0.929880i	$-0.619911\pi$
$719$	−19.7279	−0.735727	−0.367864	+	0.929880i	$0.619911\pi$
$720$	0	0
$721$	−13.2426	−0.493182
$722$	0	0
$723$	27.7990	1.03386
$724$	0	0
$725$	−4.41421	−0.163940
$726$	0	0
$727$	28.2721	1.04855	0.524277	−	0.851548i	$-0.324336\pi$
$727$	28.2721	1.04855	0.524277	+	0.851548i	$0.324336\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−26.0122	−0.962096
$732$	0	0
$733$	36.1421	1.33494	0.667470	−	0.744637i	$-0.267377\pi$
$733$	36.1421	1.33494	0.667470	+	0.744637i	$0.267377\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	11.6569	0.429386
$738$	0	0
$739$	3.85786	0.141914	0.0709569	−	0.997479i	$-0.477395\pi$
$739$	3.85786	0.141914	0.0709569	+	0.997479i	$0.477395\pi$
$740$	0	0
$741$	−3.41421	−0.125424
$742$	0	0
$743$	4.14214	0.151960	0.0759801	−	0.997109i	$-0.475791\pi$
$743$	4.14214	0.151960	0.0759801	+	0.997109i	$0.475791\pi$
$744$	0	0
$745$	22.4853	0.823797
$746$	0	0
$747$	3.34315	0.122319
$748$	0	0
$749$	−7.41421	−0.270909
$750$	0	0
$751$	−5.20101	−0.189788	−0.0948938	−	0.995487i	$-0.530251\pi$
$751$	−5.20101	−0.189788	−0.0948938	+	0.995487i	$0.530251\pi$
$752$	0	0
$753$	9.65685	0.351915
$754$	0	0
$755$	18.9706	0.690409
$756$	0	0
$757$	−28.8284	−1.04779	−0.523894	−	0.851784i	$-0.675521\pi$
$757$	−28.8284	−1.04779	−0.523894	+	0.851784i	$0.675521\pi$
$758$	0	0
$759$	−7.82843	−0.284154
$760$	0	0
$761$	4.34315	0.157439	0.0787195	−	0.996897i	$-0.474917\pi$
$761$	4.34315	0.157439	0.0787195	+	0.996897i	$0.474917\pi$
$762$	0	0
$763$	11.8995	0.430791
$764$	0	0
$765$	−4.65685	−0.168369
$766$	0	0
$767$	−15.0711	−0.544185
$768$	0	0
$769$	−23.1421	−0.834527	−0.417263	−	0.908786i	$-0.637011\pi$
$769$	−23.1421	−0.834527	−0.417263	+	0.908786i	$0.637011\pi$
$770$	0	0
$771$	−6.58579	−0.237181
$772$	0	0
$773$	−43.4558	−1.56300	−0.781499	−	0.623906i	$-0.785545\pi$
$773$	−43.4558	−1.56300	−0.781499	+	0.623906i	$0.785545\pi$
$774$	0	0
$775$	−2.24264	−0.0805580
$776$	0	0
$777$	−7.41421	−0.265983
$778$	0	0
$779$	9.89949	0.354686
$780$	0	0
$781$	−15.5563	−0.556650
$782$	0	0
$783$	4.41421	0.157751
$784$	0	0
$785$	−0.414214	−0.0147839
$786$	0	0
$787$	20.2843	0.723056	0.361528	−	0.932361i	$-0.382255\pi$
$787$	20.2843	0.723056	0.361528	+	0.932361i	$0.382255\pi$
$788$	0	0
$789$	−15.7990	−0.562459
$790$	0	0
$791$	−6.65685	−0.236690
$792$	0	0
$793$	18.7279	0.665048
$794$	0	0
$795$	−0.656854	−0.0232962
$796$	0	0
$797$	−21.7990	−0.772160	−0.386080	−	0.922465i	$-0.626171\pi$
$797$	−21.7990	−0.772160	−0.386080	+	0.922465i	$0.626171\pi$
$798$	0	0
$799$	−29.0711	−1.02846
$800$	0	0
$801$	9.24264	0.326573
$802$	0	0
$803$	−2.82843	−0.0998130
$804$	0	0
$805$	−7.82843	−0.275916
$806$	0	0
$807$	−8.41421	−0.296194
$808$	0	0
$809$	−49.7990	−1.75084	−0.875420	−	0.483364i	$-0.839415\pi$
$809$	−49.7990	−1.75084	−0.875420	+	0.483364i	$0.839415\pi$
$810$	0	0
$811$	−36.9706	−1.29821	−0.649106	−	0.760698i	$-0.724857\pi$
$811$	−36.9706	−1.29821	−0.649106	+	0.760698i	$0.724857\pi$
$812$	0	0
$813$	10.3137	0.361718
$814$	0	0
$815$	6.24264	0.218670
$816$	0	0
$817$	5.58579	0.195422
$818$	0	0
$819$	−3.41421	−0.119302
$820$	0	0
$821$	−45.8701	−1.60088	−0.800438	−	0.599416i	$-0.795400\pi$
$821$	−45.8701	−1.60088	−0.800438	+	0.599416i	$0.795400\pi$
$822$	0	0
$823$	−38.2426	−1.33305	−0.666527	−	0.745481i	$-0.732220\pi$
$823$	−38.2426	−1.33305	−0.666527	+	0.745481i	$0.732220\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	−14.8284	−0.515635	−0.257817	−	0.966194i	$-0.583003\pi$
$827$	−14.8284	−0.515635	−0.257817	+	0.966194i	$0.583003\pi$
$828$	0	0
$829$	−13.5147	−0.469386	−0.234693	−	0.972070i	$-0.575408\pi$
$829$	−13.5147	−0.469386	−0.234693	+	0.972070i	$0.575408\pi$
$830$	0	0
$831$	−21.1716	−0.734434
$832$	0	0
$833$	−4.65685	−0.161350
$834$	0	0
$835$	18.0000	0.622916
$836$	0	0
$837$	2.24264	0.0775170
$838$	0	0
$839$	−35.1005	−1.21180	−0.605902	−	0.795539i	$-0.707188\pi$
$839$	−35.1005	−1.21180	−0.605902	+	0.795539i	$0.707188\pi$
$840$	0	0
$841$	−9.51472	−0.328094
$842$	0	0
$843$	−22.1421	−0.762616
$844$	0	0
$845$	−1.34315	−0.0462056
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−14.9289	−0.512360
$850$	0	0
$851$	−58.0416	−1.98964
$852$	0	0
$853$	35.3553	1.21054	0.605272	−	0.796019i	$-0.293064\pi$
$853$	35.3553	1.21054	0.605272	+	0.796019i	$0.293064\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	24.6274	0.841257	0.420628	−	0.907233i	$-0.361810\pi$
$857$	24.6274	0.841257	0.420628	+	0.907233i	$0.361810\pi$
$858$	0	0
$859$	18.2426	0.622431	0.311215	−	0.950339i	$-0.399264\pi$
$859$	18.2426	0.622431	0.311215	+	0.950339i	$0.399264\pi$
$860$	0	0
$861$	9.89949	0.337374
$862$	0	0
$863$	33.3431	1.13501	0.567507	−	0.823369i	$-0.307908\pi$
$863$	33.3431	1.13501	0.567507	+	0.823369i	$0.307908\pi$
$864$	0	0
$865$	−8.97056	−0.305008
$866$	0	0
$867$	−4.68629	−0.159155
$868$	0	0
$869$	−12.7279	−0.431765
$870$	0	0
$871$	39.7990	1.34854
$872$	0	0
$873$	−14.8995	−0.504272
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−41.2426	−1.39267	−0.696333	−	0.717719i	$-0.745186\pi$
$877$	−41.2426	−1.39267	−0.696333	+	0.717719i	$0.745186\pi$
$878$	0	0
$879$	−17.3431	−0.584970
$880$	0	0
$881$	6.41421	0.216100	0.108050	−	0.994145i	$-0.465539\pi$
$881$	6.41421	0.216100	0.108050	+	0.994145i	$0.465539\pi$
$882$	0	0
$883$	−29.4558	−0.991268	−0.495634	−	0.868531i	$-0.665064\pi$
$883$	−29.4558	−0.991268	−0.495634	+	0.868531i	$0.665064\pi$
$884$	0	0
$885$	4.41421	0.148382
$886$	0	0
$887$	−23.8284	−0.800080	−0.400040	−	0.916498i	$-0.631004\pi$
$887$	−23.8284	−0.800080	−0.400040	+	0.916498i	$0.631004\pi$
$888$	0	0
$889$	−13.7279	−0.460420
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	6.24264	0.208902
$894$	0	0
$895$	−8.24264	−0.275521
$896$	0	0
$897$	−26.7279	−0.892419
$898$	0	0
$899$	9.89949	0.330167
$900$	0	0
$901$	−3.05887	−0.101906
$902$	0	0
$903$	5.58579	0.185883
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	−52.3848	−1.73941	−0.869704	−	0.493574i	$-0.835690\pi$
$907$	−52.3848	−1.73941	−0.869704	+	0.493574i	$0.835690\pi$
$908$	0	0
$909$	−4.34315	−0.144053
$910$	0	0
$911$	−9.85786	−0.326606	−0.163303	−	0.986576i	$-0.552215\pi$
$911$	−9.85786	−0.326606	−0.163303	+	0.986576i	$0.552215\pi$
$912$	0	0
$913$	3.34315	0.110642
$914$	0	0
$915$	−5.48528	−0.181338
$916$	0	0
$917$	−13.0711	−0.431645
$918$	0	0
$919$	54.9706	1.81331	0.906656	−	0.421871i	$-0.138627\pi$
$919$	54.9706	1.81331	0.906656	+	0.421871i	$0.138627\pi$
$920$	0	0
$921$	−2.48528	−0.0818928
$922$	0	0
$923$	−53.1127	−1.74823
$924$	0	0
$925$	−7.41421	−0.243778
$926$	0	0
$927$	13.2426	0.434945
$928$	0	0
$929$	44.7696	1.46884	0.734421	−	0.678695i	$-0.237454\pi$
$929$	44.7696	1.46884	0.734421	+	0.678695i	$0.237454\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−33.4558	−1.09530
$934$	0	0
$935$	−4.65685	−0.152295
$936$	0	0
$937$	43.8406	1.43221	0.716105	−	0.697992i	$-0.245923\pi$
$937$	43.8406	1.43221	0.716105	+	0.697992i	$0.245923\pi$
$938$	0	0
$939$	9.72792	0.317459
$940$	0	0
$941$	−48.3848	−1.57730	−0.788649	−	0.614843i	$-0.789219\pi$
$941$	−48.3848	−1.57730	−0.788649	+	0.614843i	$0.789219\pi$
$942$	0	0
$943$	77.4975	2.52366
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	33.9706	1.10389	0.551947	−	0.833879i	$-0.313885\pi$
$947$	33.9706	1.10389	0.551947	+	0.833879i	$0.313885\pi$
$948$	0	0
$949$	−9.65685	−0.313475
$950$	0	0
$951$	−10.8284	−0.351136
$952$	0	0
$953$	−15.9411	−0.516384	−0.258192	−	0.966094i	$-0.583127\pi$
$953$	−15.9411	−0.516384	−0.258192	+	0.966094i	$0.583127\pi$
$954$	0	0
$955$	−26.6274	−0.861643
$956$	0	0
$957$	4.41421	0.142691
$958$	0	0
$959$	3.31371	0.107005
$960$	0	0
$961$	−25.9706	−0.837760
$962$	0	0
$963$	7.41421	0.238920
$964$	0	0
$965$	−8.82843	−0.284197
$966$	0	0
$967$	−58.4142	−1.87847	−0.939237	−	0.343269i	$-0.888466\pi$
$967$	−58.4142	−1.87847	−0.939237	+	0.343269i	$0.888466\pi$
$968$	0	0
$969$	4.65685	0.149600
$970$	0	0
$971$	19.0416	0.611075	0.305538	−	0.952180i	$-0.401164\pi$
$971$	19.0416	0.611075	0.305538	+	0.952180i	$0.401164\pi$
$972$	0	0
$973$	−15.6569	−0.501936
$974$	0	0
$975$	−3.41421	−0.109342
$976$	0	0
$977$	31.2843	1.00087	0.500436	−	0.865773i	$-0.333173\pi$
$977$	31.2843	1.00087	0.500436	+	0.865773i	$0.333173\pi$
$978$	0	0
$979$	9.24264	0.295396
$980$	0	0
$981$	−11.8995	−0.379922
$982$	0	0
$983$	5.51472	0.175892	0.0879461	−	0.996125i	$-0.471970\pi$
$983$	5.51472	0.175892	0.0879461	+	0.996125i	$0.471970\pi$
$984$	0	0
$985$	−1.17157	−0.0373294
$986$	0	0
$987$	6.24264	0.198705
$988$	0	0
$989$	43.7279	1.39047
$990$	0	0
$991$	14.9411	0.474620	0.237310	−	0.971434i	$-0.423734\pi$
$991$	14.9411	0.474620	0.237310	+	0.971434i	$0.423734\pi$
$992$	0	0
$993$	3.14214	0.0997127
$994$	0	0
$995$	21.6569	0.686568
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	0	0
$999$	7.41421	0.234575

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.a.r.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.a.r.1.2	✓	2	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:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +3.41421 q^{13} -1.00000 q^{15} -4.65685 q^{17} +1.00000 q^{19} +1.00000 q^{21} +7.82843 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.41421 q^{29} -2.24264 q^{31} -1.00000 q^{33} -1.00000 q^{35} -7.41421 q^{37} -3.41421 q^{39} +9.89949 q^{41} +5.58579 q^{43} +1.00000 q^{45} +6.24264 q^{47} +1.00000 q^{49} +4.65685 q^{51} +0.656854 q^{53} +1.00000 q^{55} -1.00000 q^{57} -4.41421 q^{59} +5.48528 q^{61} -1.00000 q^{63} +3.41421 q^{65} +11.6569 q^{67} -7.82843 q^{69} -15.5563 q^{71} -2.82843 q^{73} -1.00000 q^{75} -1.00000 q^{77} -12.7279 q^{79} +1.00000 q^{81} +3.34315 q^{83} -4.65685 q^{85} +4.41421 q^{87} +9.24264 q^{89} -3.41421 q^{91} +2.24264 q^{93} +1.00000 q^{95} -14.8995 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 4 q^{13} - 2 q^{15} + 2 q^{17} + 2 q^{19} + 2 q^{21} + 10 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} + 4 q^{31} - 2 q^{33} - 2 q^{35} - 12 q^{37} - 4 q^{39} + 14 q^{43} + 2 q^{45} + 4 q^{47} + 2 q^{49} - 2 q^{51} - 10 q^{53} + 2 q^{55} - 2 q^{57} - 6 q^{59} - 6 q^{61} - 2 q^{63} + 4 q^{65} + 12 q^{67} - 10 q^{69} - 2 q^{75} - 2 q^{77} + 2 q^{81} + 18 q^{83} + 2 q^{85} + 6 q^{87} + 10 q^{89} - 4 q^{91} - 4 q^{93} + 2 q^{95} - 10 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 4 * q^13 - 2 * q^15 + 2 * q^17 + 2 * q^19 + 2 * q^21 + 10 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 + 4 * q^31 - 2 * q^33 - 2 * q^35 - 12 * q^37 - 4 * q^39 + 14 * q^43 + 2 * q^45 + 4 * q^47 + 2 * q^49 - 2 * q^51 - 10 * q^53 + 2 * q^55 - 2 * q^57 - 6 * q^59 - 6 * q^61 - 2 * q^63 + 4 * q^65 + 12 * q^67 - 10 * q^69 - 2 * q^75 - 2 * q^77 + 2 * q^81 + 18 * q^83 + 2 * q^85 + 6 * q^87 + 10 * q^89 - 4 * q^91 - 4 * q^93 + 2 * q^95 - 10 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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