LMFDB - Embedded newform 4005.2.a.n.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.10407557.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 2x^{3} + 18x^{2} + 7x - 2 $ x^6 - x^5 - 8x^4 + 2x^3 + 18x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.63450$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30610 q^{2} +3.31809 q^{4} +1.00000 q^{5} +4.21574 q^{7} +3.03965 q^{8} +O(q^{10})$	$q+2.30610 q^{2} +3.31809 q^{4} +1.00000 q^{5} +4.21574 q^{7} +3.03965 q^{8} +2.30610 q^{10} -1.82808 q^{13} +9.72192 q^{14} +0.373553 q^{16} +5.67809 q^{17} +2.00000 q^{19} +3.31809 q^{20} +5.92164 q^{23} +1.00000 q^{25} -4.21574 q^{26} +13.9882 q^{28} -3.41262 q^{29} -6.43148 q^{31} -5.21785 q^{32} +13.0942 q^{34} +4.21574 q^{35} -3.67809 q^{37} +4.61220 q^{38} +3.03965 q^{40} -7.65029 q^{41} -0.509013 q^{43} +13.6559 q^{46} -2.23973 q^{47} +10.7725 q^{49} +2.30610 q^{50} -6.06575 q^{52} +8.74864 q^{53} +12.8144 q^{56} -7.86985 q^{58} +2.09355 q^{59} +13.6006 q^{61} -14.8316 q^{62} -12.7800 q^{64} -1.82808 q^{65} -10.7229 q^{67} +18.8404 q^{68} +9.72192 q^{70} +10.9644 q^{71} -8.16189 q^{73} -8.48205 q^{74} +6.63618 q^{76} -1.36206 q^{79} +0.373553 q^{80} -17.6423 q^{82} -2.50985 q^{83} +5.67809 q^{85} -1.17383 q^{86} +1.00000 q^{89} -7.70673 q^{91} +19.6485 q^{92} -5.16504 q^{94} +2.00000 q^{95} -16.8022 q^{97} +24.8424 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) $ 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8	$ 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8} + 4 q^{10} - 3 q^{13} + 5 q^{14} + 12 q^{16} + 13 q^{17} + 12 q^{19} + 8 q^{20} + 19 q^{23} + 6 q^{25} - q^{26} - 6 q^{28} + 10 q^{31} + 17 q^{32} - 2 q^{34} + q^{35} - q^{37} + 8 q^{38} + 9 q^{40} + 4 q^{41} - 7 q^{43} - 6 q^{46} + 15 q^{47} - q^{49} + 4 q^{50} + q^{52} + 27 q^{53} + 14 q^{56} - 6 q^{58} + 4 q^{59} + 8 q^{61} - 2 q^{62} - q^{64} - 3 q^{65} - 11 q^{67} + 47 q^{68} + 5 q^{70} + 16 q^{71} + q^{73} + 10 q^{74} + 16 q^{76} + 12 q^{80} + q^{82} + 17 q^{83} + 13 q^{85} + 20 q^{86} + 6 q^{89} - 18 q^{91} + 36 q^{92} + 17 q^{94} + 12 q^{95} - 29 q^{97} + 23 q^{98}+O(q^{100}) $ 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8 + 4 * q^10 - 3 * q^13 + 5 * q^14 + 12 * q^16 + 13 * q^17 + 12 * q^19 + 8 * q^20 + 19 * q^23 + 6 * q^25 - q^26 - 6 * q^28 + 10 * q^31 + 17 * q^32 - 2 * q^34 + q^35 - q^37 + 8 * q^38 + 9 * q^40 + 4 * q^41 - 7 * q^43 - 6 * q^46 + 15 * q^47 - q^49 + 4 * q^50 + q^52 + 27 * q^53 + 14 * q^56 - 6 * q^58 + 4 * q^59 + 8 * q^61 - 2 * q^62 - q^64 - 3 * q^65 - 11 * q^67 + 47 * q^68 + 5 * q^70 + 16 * q^71 + q^73 + 10 * q^74 + 16 * q^76 + 12 * q^80 + q^82 + 17 * q^83 + 13 * q^85 + 20 * q^86 + 6 * q^89 - 18 * q^91 + 36 * q^92 + 17 * q^94 + 12 * q^95 - 29 * q^97 + 23 * q^98

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 $

$2$	2.30610	1.63066	0.815329	−	0.578998i	$-0.196556\pi$
$2$	2.30610	1.63066	0.815329	+	0.578998i	$0.196556\pi$
$3$	0	0
$3$	0	0
$4$	3.31809	1.65905
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.21574	1.59340	0.796700	−	0.604375i	$-0.206577\pi$
$7$	4.21574	1.59340	0.796700	+	0.604375i	$0.206577\pi$
$8$	3.03965	1.07468
$9$	0	0
$10$	2.30610	0.729253
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−1.82808	−0.507019	−0.253510	−	0.967333i	$-0.581585\pi$
$13$	−1.82808	−0.507019	−0.253510	+	0.967333i	$0.581585\pi$
$14$	9.72192	2.59829
$15$	0	0
$16$	0.373553	0.0933882
$17$	5.67809	1.37714	0.688570	−	0.725170i	$-0.258239\pi$
$17$	5.67809	1.37714	0.688570	+	0.725170i	$0.258239\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	3.31809	0.741948
$21$	0	0
$22$	0	0
$23$	5.92164	1.23475	0.617373	−	0.786670i	$-0.288197\pi$
$23$	5.92164	1.23475	0.617373	+	0.786670i	$0.288197\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.21574	−0.826775
$27$	0	0
$28$	13.9882	2.64353
$29$	−3.41262	−0.633708	−0.316854	−	0.948474i	$-0.602627\pi$
$29$	−3.41262	−0.633708	−0.316854	+	0.948474i	$0.602627\pi$
$30$	0	0
$31$	−6.43148	−1.15513	−0.577564	−	0.816345i	$-0.695997\pi$
$31$	−6.43148	−1.15513	−0.577564	+	0.816345i	$0.695997\pi$
$32$	−5.21785	−0.922395
$33$	0	0
$34$	13.0942	2.24564
$35$	4.21574	0.712590
$36$	0	0
$37$	−3.67809	−0.604675	−0.302337	−	0.953201i	$-0.597767\pi$
$37$	−3.67809	−0.604675	−0.302337	+	0.953201i	$0.597767\pi$
$38$	4.61220	0.748197
$39$	0	0
$40$	3.03965	0.480611
$41$	−7.65029	−1.19477	−0.597387	−	0.801953i	$-0.703794\pi$
$41$	−7.65029	−1.19477	−0.597387	+	0.801953i	$0.703794\pi$
$42$	0	0
$43$	−0.509013	−0.0776238	−0.0388119	−	0.999247i	$-0.512357\pi$
$43$	−0.509013	−0.0776238	−0.0388119	+	0.999247i	$0.512357\pi$
$44$	0	0
$45$	0	0
$46$	13.6559	2.01345
$47$	−2.23973	−0.326698	−0.163349	−	0.986568i	$-0.552230\pi$
$47$	−2.23973	−0.326698	−0.163349	+	0.986568i	$0.552230\pi$
$48$	0	0
$49$	10.7725	1.53893
$50$	2.30610	0.326132
$51$	0	0
$52$	−6.06575	−0.841168
$53$	8.74864	1.20172	0.600859	−	0.799355i	$-0.294825\pi$
$53$	8.74864	1.20172	0.600859	+	0.799355i	$0.294825\pi$
$54$	0	0
$55$	0	0
$56$	12.8144	1.71239
$57$	0	0
$58$	−7.86985	−1.03336
$59$	2.09355	0.272557	0.136279	−	0.990671i	$-0.456486\pi$
$59$	2.09355	0.272557	0.136279	+	0.990671i	$0.456486\pi$
$60$	0	0
$61$	13.6006	1.74137	0.870687	−	0.491838i	$-0.163675\pi$
$61$	13.6006	1.74137	0.870687	+	0.491838i	$0.163675\pi$
$62$	−14.8316	−1.88362
$63$	0	0
$64$	−12.7800	−1.59750
$65$	−1.82808	−0.226746
$66$	0	0
$67$	−10.7229	−1.31001	−0.655005	−	0.755624i	$-0.727334\pi$
$67$	−10.7229	−1.31001	−0.655005	+	0.755624i	$0.727334\pi$
$68$	18.8404	2.28474
$69$	0	0
$70$	9.72192	1.16199
$71$	10.9644	1.30123	0.650616	−	0.759407i	$-0.274511\pi$
$71$	10.9644	1.30123	0.650616	+	0.759407i	$0.274511\pi$
$72$	0	0
$73$	−8.16189	−0.955277	−0.477639	−	0.878556i	$-0.658507\pi$
$73$	−8.16189	−0.955277	−0.477639	+	0.878556i	$0.658507\pi$
$74$	−8.48205	−0.986018
$75$	0	0
$76$	6.63618	0.761223
$77$	0	0
$78$	0	0
$79$	−1.36206	−0.153244	−0.0766220	−	0.997060i	$-0.524413\pi$
$79$	−1.36206	−0.153244	−0.0766220	+	0.997060i	$0.524413\pi$
$80$	0.373553	0.0417645
$81$	0	0
$82$	−17.6423	−1.94827
$83$	−2.50985	−0.275492	−0.137746	−	0.990468i	$-0.543986\pi$
$83$	−2.50985	−0.275492	−0.137746	+	0.990468i	$0.543986\pi$
$84$	0	0
$85$	5.67809	0.615876
$86$	−1.17383	−0.126578
$87$	0	0
$88$	0	0
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−7.70673	−0.807885
$92$	19.6485	2.04850
$93$	0	0
$94$	−5.16504	−0.532733
$95$	2.00000	0.205196
$96$	0	0
$97$	−16.8022	−1.70600	−0.853002	−	0.521907i	$-0.825221\pi$
$97$	−16.8022	−1.70600	−0.853002	+	0.521907i	$0.825221\pi$
$98$	24.8424	2.50946
$99$	0	0
$100$	3.31809	0.331809
$101$	4.76013	0.473650	0.236825	−	0.971552i	$-0.423893\pi$
$101$	4.76013	0.473650	0.236825	+	0.971552i	$0.423893\pi$
$102$	0	0
$103$	12.9416	1.27517	0.637585	−	0.770380i	$-0.279934\pi$
$103$	12.9416	1.27517	0.637585	+	0.770380i	$0.279934\pi$
$104$	−5.55674	−0.544883
$105$	0	0
$106$	20.1752	1.95959
$107$	−12.8536	−1.24260	−0.621300	−	0.783572i	$-0.713395\pi$
$107$	−12.8536	−1.24260	−0.621300	+	0.783572i	$0.713395\pi$
$108$	0	0
$109$	17.4894	1.67518	0.837591	−	0.546298i	$-0.183963\pi$
$109$	17.4894	1.67518	0.837591	+	0.546298i	$0.183963\pi$
$110$	0	0
$111$	0	0
$112$	1.57480	0.148805
$113$	−7.33314	−0.689844	−0.344922	−	0.938631i	$-0.612095\pi$
$113$	−7.33314	−0.689844	−0.344922	+	0.938631i	$0.612095\pi$
$114$	0	0
$115$	5.92164	0.552195
$116$	−11.3234	−1.05135
$117$	0	0
$118$	4.82794	0.444448
$119$	23.9374	2.19434
$120$	0	0
$121$	−11.0000	−1.00000
$122$	31.3642	2.83959
$123$	0	0
$124$	−21.3403	−1.91641
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.9578	−0.972348	−0.486174	−	0.873862i	$-0.661608\pi$
$127$	−10.9578	−0.972348	−0.486174	+	0.873862i	$0.661608\pi$
$128$	−19.0362	−1.68258
$129$	0	0
$130$	−4.21574	−0.369745
$131$	15.2526	1.33263	0.666313	−	0.745672i	$-0.267871\pi$
$131$	15.2526	1.33263	0.666313	+	0.745672i	$0.267871\pi$
$132$	0	0
$133$	8.43148	0.731102
$134$	−24.7281	−2.13618
$135$	0	0
$136$	17.2594	1.47998
$137$	16.2924	1.39195	0.695975	−	0.718066i	$-0.254973\pi$
$137$	16.2924	1.39195	0.695975	+	0.718066i	$0.254973\pi$
$138$	0	0
$139$	−10.0933	−0.856100	−0.428050	−	0.903755i	$-0.640799\pi$
$139$	−10.0933	−0.856100	−0.428050	+	0.903755i	$0.640799\pi$
$140$	13.9882	1.18222
$141$	0	0
$142$	25.2849	2.12187
$143$	0	0
$144$	0	0
$145$	−3.41262	−0.283403
$146$	−18.8221	−1.55773
$147$	0	0
$148$	−12.2043	−1.00318
$149$	−0.217495	−0.0178179	−0.00890894	−	0.999960i	$-0.502836\pi$
$149$	−0.217495	−0.0178179	−0.00890894	+	0.999960i	$0.502836\pi$
$150$	0	0
$151$	−7.86058	−0.639685	−0.319842	−	0.947471i	$-0.603630\pi$
$151$	−7.86058	−0.639685	−0.319842	+	0.947471i	$0.603630\pi$
$152$	6.07930	0.493097
$153$	0	0
$154$	0	0
$155$	−6.43148	−0.516589
$156$	0	0
$157$	−11.0581	−0.882532	−0.441266	−	0.897376i	$-0.645470\pi$
$157$	−11.0581	−0.882532	−0.441266	+	0.897376i	$0.645470\pi$
$158$	−3.14105	−0.249888
$159$	0	0
$160$	−5.21785	−0.412508
$161$	24.9641	1.96745
$162$	0	0
$163$	2.46427	0.193016	0.0965082	−	0.995332i	$-0.469233\pi$
$163$	2.46427	0.193016	0.0965082	+	0.995332i	$0.469233\pi$
$164$	−25.3844	−1.98219
$165$	0	0
$166$	−5.78796	−0.449233
$167$	13.3547	1.03342	0.516710	−	0.856160i	$-0.327157\pi$
$167$	13.3547	1.03342	0.516710	+	0.856160i	$0.327157\pi$
$168$	0	0
$169$	−9.65811	−0.742932
$170$	13.0942	1.00428
$171$	0	0
$172$	−1.68895	−0.128781
$173$	−0.947233	−0.0720168	−0.0360084	−	0.999351i	$-0.511464\pi$
$173$	−0.947233	−0.0720168	−0.0360084	+	0.999351i	$0.511464\pi$
$174$	0	0
$175$	4.21574	0.318680
$176$	0	0
$177$	0	0
$178$	2.30610	0.172849
$179$	1.11376	0.0832466	0.0416233	−	0.999133i	$-0.486747\pi$
$179$	1.11376	0.0832466	0.0416233	+	0.999133i	$0.486747\pi$
$180$	0	0
$181$	−10.7853	−0.801664	−0.400832	−	0.916152i	$-0.631279\pi$
$181$	−10.7853	−0.801664	−0.400832	+	0.916152i	$0.631279\pi$
$182$	−17.7725	−1.31738
$183$	0	0
$184$	17.9997	1.32696
$185$	−3.67809	−0.270419
$186$	0	0
$187$	0	0
$188$	−7.43163	−0.542007
$189$	0	0
$190$	4.61220	0.334604
$191$	−7.00261	−0.506691	−0.253346	−	0.967376i	$-0.581531\pi$
$191$	−7.00261	−0.506691	−0.253346	+	0.967376i	$0.581531\pi$
$192$	0	0
$193$	8.36484	0.602114	0.301057	−	0.953606i	$-0.402661\pi$
$193$	8.36484	0.602114	0.301057	+	0.953606i	$0.402661\pi$
$194$	−38.7475	−2.78191
$195$	0	0
$196$	35.7441	2.55315
$197$	5.59673	0.398750	0.199375	−	0.979923i	$-0.436109\pi$
$197$	5.59673	0.398750	0.199375	+	0.979923i	$0.436109\pi$
$198$	0	0
$199$	−4.58170	−0.324788	−0.162394	−	0.986726i	$-0.551922\pi$
$199$	−4.58170	−0.324788	−0.162394	+	0.986726i	$0.551922\pi$
$200$	3.03965	0.214936
$201$	0	0
$202$	10.9773	0.772362
$203$	−14.3867	−1.00975
$204$	0	0
$205$	−7.65029	−0.534319
$206$	29.8445	2.07937
$207$	0	0
$208$	−0.682886	−0.0473496
$209$	0	0
$210$	0	0
$211$	−7.75946	−0.534183	−0.267092	−	0.963671i	$-0.586063\pi$
$211$	−7.75946	−0.534183	−0.267092	+	0.963671i	$0.586063\pi$
$212$	29.0288	1.99371
$213$	0	0
$214$	−29.6416	−2.02626
$215$	−0.509013	−0.0347144
$216$	0	0
$217$	−27.1135	−1.84058
$218$	40.3323	2.73165
$219$	0	0
$220$	0	0
$221$	−10.3800	−0.698236
$222$	0	0
$223$	−22.8585	−1.53072	−0.765360	−	0.643603i	$-0.777439\pi$
$223$	−22.8585	−1.53072	−0.765360	+	0.643603i	$0.777439\pi$
$224$	−21.9971	−1.46974
$225$	0	0
$226$	−16.9109	−1.12490
$227$	−4.42467	−0.293675	−0.146838	−	0.989161i	$-0.546910\pi$
$227$	−4.42467	−0.293675	−0.146838	+	0.989161i	$0.546910\pi$
$228$	0	0
$229$	25.0674	1.65650	0.828250	−	0.560359i	$-0.189337\pi$
$229$	25.0674	1.65650	0.828250	+	0.560359i	$0.189337\pi$
$230$	13.6559	0.900442
$231$	0	0
$232$	−10.3732	−0.681033
$233$	12.3075	0.806294	0.403147	−	0.915135i	$-0.367916\pi$
$233$	12.3075	0.806294	0.403147	+	0.915135i	$0.367916\pi$
$234$	0	0
$235$	−2.23973	−0.146104
$236$	6.94660	0.452185
$237$	0	0
$238$	55.2019	3.57821
$239$	−24.2881	−1.57107	−0.785535	−	0.618818i	$-0.787612\pi$
$239$	−24.2881	−1.57107	−0.785535	+	0.618818i	$0.787612\pi$
$240$	0	0
$241$	−26.8729	−1.73104	−0.865519	−	0.500876i	$-0.833011\pi$
$241$	−26.8729	−1.73104	−0.865519	+	0.500876i	$0.833011\pi$
$242$	−25.3671	−1.63066
$243$	0	0
$244$	45.1279	2.88902
$245$	10.7725	0.688228
$246$	0	0
$247$	−3.65617	−0.232636
$248$	−19.5495	−1.24139
$249$	0	0
$250$	2.30610	0.145851
$251$	4.64031	0.292894	0.146447	−	0.989219i	$-0.453216\pi$
$251$	4.64031	0.292894	0.146447	+	0.989219i	$0.453216\pi$
$252$	0	0
$253$	0	0
$254$	−25.2698	−1.58557
$255$	0	0
$256$	−18.3394	−1.14621
$257$	−14.6572	−0.914290	−0.457145	−	0.889392i	$-0.651128\pi$
$257$	−14.6572	−0.914290	−0.457145	+	0.889392i	$0.651128\pi$
$258$	0	0
$259$	−15.5059	−0.963489
$260$	−6.06575	−0.376182
$261$	0	0
$262$	35.1740	2.17306
$263$	−12.6859	−0.782248	−0.391124	−	0.920338i	$-0.627914\pi$
$263$	−12.6859	−0.782248	−0.391124	+	0.920338i	$0.627914\pi$
$264$	0	0
$265$	8.74864	0.537424
$266$	19.4438	1.19218
$267$	0	0
$268$	−35.5796	−2.17337
$269$	27.7915	1.69448	0.847239	−	0.531212i	$-0.178263\pi$
$269$	27.7915	1.69448	0.847239	+	0.531212i	$0.178263\pi$
$270$	0	0
$271$	15.6387	0.949984	0.474992	−	0.879990i	$-0.342451\pi$
$271$	15.6387	0.949984	0.474992	+	0.879990i	$0.342451\pi$
$272$	2.12107	0.128609
$273$	0	0
$274$	37.5718	2.26979
$275$	0	0
$276$	0	0
$277$	−15.3885	−0.924603	−0.462301	−	0.886723i	$-0.652976\pi$
$277$	−15.3885	−0.924603	−0.462301	+	0.886723i	$0.652976\pi$
$278$	−23.2761	−1.39601
$279$	0	0
$280$	12.8144	0.765806
$281$	20.4076	1.21742	0.608708	−	0.793395i	$-0.291688\pi$
$281$	20.4076	1.21742	0.608708	+	0.793395i	$0.291688\pi$
$282$	0	0
$283$	26.9349	1.60112	0.800558	−	0.599256i	$-0.204537\pi$
$283$	26.9349	1.60112	0.800558	+	0.599256i	$0.204537\pi$
$284$	36.3808	2.15880
$285$	0	0
$286$	0	0
$287$	−32.2516	−1.90375
$288$	0	0
$289$	15.2407	0.896514
$290$	−7.86985	−0.462133
$291$	0	0
$292$	−27.0819	−1.58485
$293$	−10.1280	−0.591684	−0.295842	−	0.955237i	$-0.595600\pi$
$293$	−10.1280	−0.591684	−0.295842	+	0.955237i	$0.595600\pi$
$294$	0	0
$295$	2.09355	0.121891
$296$	−11.1801	−0.649831
$297$	0	0
$298$	−0.501565	−0.0290549
$299$	−10.8252	−0.626040
$300$	0	0
$301$	−2.14587	−0.123686
$302$	−18.1273	−1.04311
$303$	0	0
$304$	0.747106	0.0428494
$305$	13.6006	0.778766
$306$	0	0
$307$	14.7609	0.842449	0.421225	−	0.906956i	$-0.361600\pi$
$307$	14.7609	0.842449	0.421225	+	0.906956i	$0.361600\pi$
$308$	0	0
$309$	0	0
$310$	−14.8316	−0.842380
$311$	−13.6939	−0.776509	−0.388255	−	0.921552i	$-0.626922\pi$
$311$	−13.6939	−0.776509	−0.388255	+	0.921552i	$0.626922\pi$
$312$	0	0
$313$	28.1646	1.59196	0.795979	−	0.605325i	$-0.206957\pi$
$313$	28.1646	1.59196	0.795979	+	0.605325i	$0.206957\pi$
$314$	−25.5011	−1.43911
$315$	0	0
$316$	−4.51945	−0.254239
$317$	17.4734	0.981403	0.490702	−	0.871328i	$-0.336740\pi$
$317$	17.4734	0.981403	0.490702	+	0.871328i	$0.336740\pi$
$318$	0	0
$319$	0	0
$320$	−12.7800	−0.714423
$321$	0	0
$322$	57.5697	3.20823
$323$	11.3562	0.631875
$324$	0	0
$325$	−1.82808	−0.101404
$326$	5.68285	0.314744
$327$	0	0
$328$	−23.2542	−1.28400
$329$	−9.44212	−0.520561
$330$	0	0
$331$	−10.9354	−0.601066	−0.300533	−	0.953771i	$-0.597164\pi$
$331$	−10.9354	−0.601066	−0.300533	+	0.953771i	$0.597164\pi$
$332$	−8.32791	−0.457053
$333$	0	0
$334$	30.7973	1.68515
$335$	−10.7229	−0.585854
$336$	0	0
$337$	−18.5580	−1.01092	−0.505460	−	0.862850i	$-0.668677\pi$
$337$	−18.5580	−1.01092	−0.505460	+	0.862850i	$0.668677\pi$
$338$	−22.2726	−1.21147
$339$	0	0
$340$	18.8404	1.02177
$341$	0	0
$342$	0	0
$343$	15.9038	0.858724
$344$	−1.54722	−0.0834206
$345$	0	0
$346$	−2.18441	−0.117435
$347$	−18.4571	−0.990831	−0.495416	−	0.868656i	$-0.664984\pi$
$347$	−18.4571	−0.990831	−0.495416	+	0.868656i	$0.664984\pi$
$348$	0	0
$349$	−6.60908	−0.353776	−0.176888	−	0.984231i	$-0.556603\pi$
$349$	−6.60908	−0.353776	−0.176888	+	0.984231i	$0.556603\pi$
$350$	9.72192	0.519658
$351$	0	0
$352$	0	0
$353$	−28.2346	−1.50277	−0.751387	−	0.659861i	$-0.770615\pi$
$353$	−28.2346	−1.50277	−0.751387	+	0.659861i	$0.770615\pi$
$354$	0	0
$355$	10.9644	0.581929
$356$	3.31809	0.175859
$357$	0	0
$358$	2.56845	0.135747
$359$	17.8602	0.942623	0.471311	−	0.881967i	$-0.343781\pi$
$359$	17.8602	0.942623	0.471311	+	0.881967i	$0.343781\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−24.8719	−1.30724
$363$	0	0
$364$	−25.5716	−1.34032
$365$	−8.16189	−0.427213
$366$	0	0
$367$	−9.14697	−0.477468	−0.238734	−	0.971085i	$-0.576732\pi$
$367$	−9.14697	−0.477468	−0.238734	+	0.971085i	$0.576732\pi$
$368$	2.21204	0.115311
$369$	0	0
$370$	−8.48205	−0.440961
$371$	36.8820	1.91482
$372$	0	0
$373$	18.3913	0.952265	0.476133	−	0.879373i	$-0.342038\pi$
$373$	18.3913	0.952265	0.476133	+	0.879373i	$0.342038\pi$
$374$	0	0
$375$	0	0
$376$	−6.80800	−0.351096
$377$	6.23856	0.321302
$378$	0	0
$379$	20.3030	1.04289	0.521447	−	0.853284i	$-0.325393\pi$
$379$	20.3030	1.04289	0.521447	+	0.853284i	$0.325393\pi$
$380$	6.63618	0.340429
$381$	0	0
$382$	−16.1487	−0.826241
$383$	29.8354	1.52452	0.762258	−	0.647274i	$-0.224091\pi$
$383$	29.8354	1.52452	0.762258	+	0.647274i	$0.224091\pi$
$384$	0	0
$385$	0	0
$386$	19.2901	0.981843
$387$	0	0
$388$	−55.7513	−2.83034
$389$	10.8845	0.551869	0.275934	−	0.961177i	$-0.411013\pi$
$389$	10.8845	0.551869	0.275934	+	0.961177i	$0.411013\pi$
$390$	0	0
$391$	33.6236	1.70042
$392$	32.7446	1.65385
$393$	0	0
$394$	12.9066	0.650225
$395$	−1.36206	−0.0685328
$396$	0	0
$397$	−29.0231	−1.45663	−0.728314	−	0.685244i	$-0.759696\pi$
$397$	−29.0231	−1.45663	−0.728314	+	0.685244i	$0.759696\pi$
$398$	−10.5659	−0.529619
$399$	0	0
$400$	0.373553	0.0186776
$401$	21.3579	1.06656	0.533280	−	0.845939i	$-0.320959\pi$
$401$	21.3579	1.06656	0.533280	+	0.845939i	$0.320959\pi$
$402$	0	0
$403$	11.7573	0.585672
$404$	15.7945	0.785808
$405$	0	0
$406$	−33.1772	−1.64656
$407$	0	0
$408$	0	0
$409$	−18.8424	−0.931694	−0.465847	−	0.884865i	$-0.654250\pi$
$409$	−18.8424	−0.931694	−0.465847	+	0.884865i	$0.654250\pi$
$410$	−17.6423	−0.871292
$411$	0	0
$412$	42.9413	2.11557
$413$	8.82588	0.434293
$414$	0	0
$415$	−2.50985	−0.123204
$416$	9.53867	0.467672
$417$	0	0
$418$	0	0
$419$	−3.86297	−0.188718	−0.0943591	−	0.995538i	$-0.530080\pi$
$419$	−3.86297	−0.188718	−0.0943591	+	0.995538i	$0.530080\pi$
$420$	0	0
$421$	−29.6982	−1.44740	−0.723700	−	0.690114i	$-0.757560\pi$
$421$	−29.6982	−1.44740	−0.723700	+	0.690114i	$0.757560\pi$
$422$	−17.8941	−0.871070
$423$	0	0
$424$	26.5928	1.29146
$425$	5.67809	0.275428
$426$	0	0
$427$	57.3365	2.77471
$428$	−42.6493	−2.06153
$429$	0	0
$430$	−1.17383	−0.0566073
$431$	−1.41028	−0.0679306	−0.0339653	−	0.999423i	$-0.510814\pi$
$431$	−1.41028	−0.0679306	−0.0339653	+	0.999423i	$0.510814\pi$
$432$	0	0
$433$	−4.10313	−0.197184	−0.0985919	−	0.995128i	$-0.531434\pi$
$433$	−4.10313	−0.197184	−0.0985919	+	0.995128i	$0.531434\pi$
$434$	−62.5264	−3.00136
$435$	0	0
$436$	58.0315	2.77920
$437$	11.8433	0.566541
$438$	0	0
$439$	−8.74877	−0.417556	−0.208778	−	0.977963i	$-0.566949\pi$
$439$	−8.74877	−0.417556	−0.208778	+	0.977963i	$0.566949\pi$
$440$	0	0
$441$	0	0
$442$	−23.9374	−1.13858
$443$	−17.5652	−0.834549	−0.417274	−	0.908781i	$-0.637015\pi$
$443$	−17.5652	−0.834549	−0.417274	+	0.908781i	$0.637015\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−52.7140	−2.49608
$447$	0	0
$448$	−53.8772	−2.54546
$449$	25.8180	1.21843	0.609214	−	0.793005i	$-0.291485\pi$
$449$	25.8180	1.21843	0.609214	+	0.793005i	$0.291485\pi$
$450$	0	0
$451$	0	0
$452$	−24.3320	−1.14448
$453$	0	0
$454$	−10.2037	−0.478884
$455$	−7.70673	−0.361297
$456$	0	0
$457$	17.9406	0.839226	0.419613	−	0.907703i	$-0.362166\pi$
$457$	17.9406	0.839226	0.419613	+	0.907703i	$0.362166\pi$
$458$	57.8079	2.70118
$459$	0	0
$460$	19.6485	0.916118
$461$	1.02582	0.0477774	0.0238887	−	0.999715i	$-0.492395\pi$
$461$	1.02582	0.0477774	0.0238887	+	0.999715i	$0.492395\pi$
$462$	0	0
$463$	−40.6127	−1.88743	−0.943717	−	0.330755i	$-0.892697\pi$
$463$	−40.6127	−1.88743	−0.943717	+	0.330755i	$0.892697\pi$
$464$	−1.27479	−0.0591809
$465$	0	0
$466$	28.3824	1.31479
$467$	−20.6865	−0.957259	−0.478629	−	0.878017i	$-0.658866\pi$
$467$	−20.6865	−0.957259	−0.478629	+	0.878017i	$0.658866\pi$
$468$	0	0
$469$	−45.2050	−2.08737
$470$	−5.16504	−0.238245
$471$	0	0
$472$	6.36367	0.292912
$473$	0	0
$474$	0	0
$475$	2.00000	0.0917663
$476$	79.4264	3.64050
$477$	0	0
$478$	−56.0109	−2.56188
$479$	−26.6183	−1.21622	−0.608111	−	0.793852i	$-0.708072\pi$
$479$	−26.6183	−1.21622	−0.608111	+	0.793852i	$0.708072\pi$
$480$	0	0
$481$	6.72386	0.306582
$482$	−61.9716	−2.82273
$483$	0	0
$484$	−36.4990	−1.65905
$485$	−16.8022	−0.762949
$486$	0	0
$487$	6.81405	0.308774	0.154387	−	0.988010i	$-0.450660\pi$
$487$	6.81405	0.308774	0.154387	+	0.988010i	$0.450660\pi$
$488$	41.3410	1.87142
$489$	0	0
$490$	24.8424	1.12227
$491$	−18.2232	−0.822402	−0.411201	−	0.911545i	$-0.634891\pi$
$491$	−18.2232	−0.822402	−0.411201	+	0.911545i	$0.634891\pi$
$492$	0	0
$493$	−19.3772	−0.872705
$494$	−8.43148	−0.379350
$495$	0	0
$496$	−2.40250	−0.107875
$497$	46.2230	2.07338
$498$	0	0
$499$	−10.1980	−0.456526	−0.228263	−	0.973600i	$-0.573305\pi$
$499$	−10.1980	−0.456526	−0.228263	+	0.973600i	$0.573305\pi$
$500$	3.31809	0.148390
$501$	0	0
$502$	10.7010	0.477610
$503$	6.64765	0.296404	0.148202	−	0.988957i	$-0.452651\pi$
$503$	6.64765	0.296404	0.148202	+	0.988957i	$0.452651\pi$
$504$	0	0
$505$	4.76013	0.211823
$506$	0	0
$507$	0	0
$508$	−36.3590	−1.61317
$509$	−23.1634	−1.02670	−0.513350	−	0.858179i	$-0.671596\pi$
$509$	−23.1634	−1.02670	−0.513350	+	0.858179i	$0.671596\pi$
$510$	0	0
$511$	−34.4084	−1.52214
$512$	−4.22008	−0.186503
$513$	0	0
$514$	−33.8009	−1.49090
$515$	12.9416	0.570273
$516$	0	0
$517$	0	0
$518$	−35.7581	−1.57112
$519$	0	0
$520$	−5.55674	−0.243679
$521$	−28.9773	−1.26952	−0.634759	−	0.772710i	$-0.718901\pi$
$521$	−28.9773	−1.26952	−0.634759	+	0.772710i	$0.718901\pi$
$522$	0	0
$523$	−0.997875	−0.0436340	−0.0218170	−	0.999762i	$-0.506945\pi$
$523$	−0.997875	−0.0436340	−0.0218170	+	0.999762i	$0.506945\pi$
$524$	50.6096	2.21089
$525$	0	0
$526$	−29.2550	−1.27558
$527$	−36.5186	−1.59077
$528$	0	0
$529$	12.0658	0.524599
$530$	20.1752	0.876356
$531$	0	0
$532$	27.9764	1.21293
$533$	13.9854	0.605774
$534$	0	0
$535$	−12.8536	−0.555708
$536$	−32.5939	−1.40784
$537$	0	0
$538$	64.0900	2.76311
$539$	0	0
$540$	0	0
$541$	−4.47145	−0.192243	−0.0961213	−	0.995370i	$-0.530644\pi$
$541$	−4.47145	−0.192243	−0.0961213	+	0.995370i	$0.530644\pi$
$542$	36.0644	1.54910
$543$	0	0
$544$	−29.6275	−1.27027
$545$	17.4894	0.749164
$546$	0	0
$547$	0.832925	0.0356133	0.0178067	−	0.999841i	$-0.494332\pi$
$547$	0.832925	0.0356133	0.0178067	+	0.999841i	$0.494332\pi$
$548$	54.0595	2.30931
$549$	0	0
$550$	0	0
$551$	−6.82525	−0.290765
$552$	0	0
$553$	−5.74210	−0.244179
$554$	−35.4873	−1.50771
$555$	0	0
$556$	−33.4904	−1.42031
$557$	11.2279	0.475741	0.237871	−	0.971297i	$-0.423551\pi$
$557$	11.2279	0.475741	0.237871	+	0.971297i	$0.423551\pi$
$558$	0	0
$559$	0.930518	0.0393567
$560$	1.57480	0.0665475
$561$	0	0
$562$	47.0620	1.98519
$563$	6.04448	0.254744	0.127372	−	0.991855i	$-0.459346\pi$
$563$	6.04448	0.254744	0.127372	+	0.991855i	$0.459346\pi$
$564$	0	0
$565$	−7.33314	−0.308507
$566$	62.1146	2.61087
$567$	0	0
$568$	33.3279	1.39841
$569$	−19.3252	−0.810155	−0.405078	−	0.914282i	$-0.632755\pi$
$569$	−19.3252	−0.810155	−0.405078	+	0.914282i	$0.632755\pi$
$570$	0	0
$571$	−15.4816	−0.647883	−0.323941	−	0.946077i	$-0.605008\pi$
$571$	−15.4816	−0.647883	−0.323941	+	0.946077i	$0.605008\pi$
$572$	0	0
$573$	0	0
$574$	−74.3755	−3.10437
$575$	5.92164	0.246949
$576$	0	0
$577$	−19.9579	−0.830858	−0.415429	−	0.909626i	$-0.636369\pi$
$577$	−19.9579	−0.830858	−0.415429	+	0.909626i	$0.636369\pi$
$578$	35.1466	1.46191
$579$	0	0
$580$	−11.3234	−0.470179
$581$	−10.5809	−0.438968
$582$	0	0
$583$	0	0
$584$	−24.8093	−1.02662
$585$	0	0
$586$	−23.3562	−0.964835
$587$	26.5531	1.09596	0.547982	−	0.836490i	$-0.315396\pi$
$587$	26.5531	1.09596	0.547982	+	0.836490i	$0.315396\pi$
$588$	0	0
$589$	−12.8630	−0.530009
$590$	4.82794	0.198763
$591$	0	0
$592$	−1.37396	−0.0564695
$593$	−28.6235	−1.17542	−0.587712	−	0.809070i	$-0.699971\pi$
$593$	−28.6235	−1.17542	−0.587712	+	0.809070i	$0.699971\pi$
$594$	0	0
$595$	23.9374	0.981337
$596$	−0.721669	−0.0295607
$597$	0	0
$598$	−24.9641	−1.02086
$599$	35.7399	1.46029	0.730146	−	0.683291i	$-0.239452\pi$
$599$	35.7399	1.46029	0.730146	+	0.683291i	$0.239452\pi$
$600$	0	0
$601$	−23.4986	−0.958528	−0.479264	−	0.877671i	$-0.659096\pi$
$601$	−23.4986	−0.958528	−0.479264	+	0.877671i	$0.659096\pi$
$602$	−4.94858	−0.201689
$603$	0	0
$604$	−26.0821	−1.06127
$605$	−11.0000	−0.447214
$606$	0	0
$607$	8.73808	0.354668	0.177334	−	0.984151i	$-0.443253\pi$
$607$	8.73808	0.354668	0.177334	+	0.984151i	$0.443253\pi$
$608$	−10.4357	−0.423224
$609$	0	0
$610$	31.3642	1.26990
$611$	4.09441	0.165642
$612$	0	0
$613$	−6.75039	−0.272646	−0.136323	−	0.990664i	$-0.543528\pi$
$613$	−6.75039	−0.272646	−0.136323	+	0.990664i	$0.543528\pi$
$614$	34.0401	1.37375
$615$	0	0
$616$	0	0
$617$	−38.2284	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$617$	−38.2284	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$618$	0	0
$619$	−11.3887	−0.457749	−0.228875	−	0.973456i	$-0.573505\pi$
$619$	−11.3887	−0.457749	−0.228875	+	0.973456i	$0.573505\pi$
$620$	−21.3403	−0.857045
$621$	0	0
$622$	−31.5795	−1.26622
$623$	4.21574	0.168900
$624$	0	0
$625$	1.00000	0.0400000
$626$	64.9504	2.59594
$627$	0	0
$628$	−36.6918	−1.46416
$629$	−20.8845	−0.832722
$630$	0	0
$631$	−6.72173	−0.267588	−0.133794	−	0.991009i	$-0.542716\pi$
$631$	−6.72173	−0.267588	−0.133794	+	0.991009i	$0.542716\pi$
$632$	−4.14019	−0.164688
$633$	0	0
$634$	40.2954	1.60033
$635$	−10.9578	−0.434847
$636$	0	0
$637$	−19.6930	−0.780265
$638$	0	0
$639$	0	0
$640$	−19.0362	−0.752473
$641$	−2.06300	−0.0814836	−0.0407418	−	0.999170i	$-0.512972\pi$
$641$	−2.06300	−0.0814836	−0.0407418	+	0.999170i	$0.512972\pi$
$642$	0	0
$643$	39.3610	1.55224	0.776122	−	0.630582i	$-0.217184\pi$
$643$	39.3610	1.55224	0.776122	+	0.630582i	$0.217184\pi$
$644$	82.8332	3.26408
$645$	0	0
$646$	26.1885	1.03037
$647$	28.2764	1.11166	0.555829	−	0.831297i	$-0.312401\pi$
$647$	28.2764	1.11166	0.555829	+	0.831297i	$0.312401\pi$
$648$	0	0
$649$	0	0
$650$	−4.21574	−0.165355
$651$	0	0
$652$	8.17667	0.320223
$653$	9.32513	0.364920	0.182460	−	0.983213i	$-0.441594\pi$
$653$	9.32513	0.364920	0.182460	+	0.983213i	$0.441594\pi$
$654$	0	0
$655$	15.2526	0.595969
$656$	−2.85779	−0.111578
$657$	0	0
$658$	−21.7745	−0.848857
$659$	−20.7945	−0.810037	−0.405019	−	0.914308i	$-0.632735\pi$
$659$	−20.7945	−0.810037	−0.405019	+	0.914308i	$0.632735\pi$
$660$	0	0
$661$	25.5971	0.995610	0.497805	−	0.867289i	$-0.334140\pi$
$661$	25.5971	0.995610	0.497805	+	0.867289i	$0.334140\pi$
$662$	−25.2182	−0.980133
$663$	0	0
$664$	−7.62906	−0.296065
$665$	8.43148	0.326959
$666$	0	0
$667$	−20.2083	−0.782469
$668$	44.3122	1.71449
$669$	0	0
$670$	−24.7281	−0.955328
$671$	0	0
$672$	0	0
$673$	−37.8618	−1.45946	−0.729732	−	0.683733i	$-0.760355\pi$
$673$	−37.8618	−1.45946	−0.729732	+	0.683733i	$0.760355\pi$
$674$	−42.7966	−1.64847
$675$	0	0
$676$	−32.0465	−1.23256
$677$	−39.6250	−1.52291	−0.761456	−	0.648217i	$-0.775515\pi$
$677$	−39.6250	−1.52291	−0.761456	+	0.648217i	$0.775515\pi$
$678$	0	0
$679$	−70.8337	−2.71835
$680$	17.2594	0.661869
$681$	0	0
$682$	0	0
$683$	37.7302	1.44371	0.721854	−	0.692046i	$-0.243290\pi$
$683$	37.7302	1.44371	0.721854	+	0.692046i	$0.243290\pi$
$684$	0	0
$685$	16.2924	0.622499
$686$	36.6757	1.40029
$687$	0	0
$688$	−0.190143	−0.00724914
$689$	−15.9932	−0.609294
$690$	0	0
$691$	−0.755781	−0.0287513	−0.0143756	−	0.999897i	$-0.504576\pi$
$691$	−0.755781	−0.0287513	−0.0143756	+	0.999897i	$0.504576\pi$
$692$	−3.14301	−0.119479
$693$	0	0
$694$	−42.5640	−1.61571
$695$	−10.0933	−0.382859
$696$	0	0
$697$	−43.4391	−1.64537
$698$	−15.2412	−0.576888
$699$	0	0
$700$	13.9882	0.528705
$701$	9.44244	0.356636	0.178318	−	0.983973i	$-0.442934\pi$
$701$	9.44244	0.356636	0.178318	+	0.983973i	$0.442934\pi$
$702$	0	0
$703$	−7.35618	−0.277444
$704$	0	0
$705$	0	0
$706$	−65.1117	−2.45051
$707$	20.0675	0.754715
$708$	0	0
$709$	−9.63575	−0.361878	−0.180939	−	0.983494i	$-0.557914\pi$
$709$	−9.63575	−0.361878	−0.180939	+	0.983494i	$0.557914\pi$
$710$	25.2849	0.948927
$711$	0	0
$712$	3.03965	0.113916
$713$	−38.0849	−1.42629
$714$	0	0
$715$	0	0
$716$	3.69557	0.138110
$717$	0	0
$718$	41.1873	1.53710
$719$	−20.7572	−0.774114	−0.387057	−	0.922056i	$-0.626508\pi$
$719$	−20.7572	−0.774114	−0.387057	+	0.922056i	$0.626508\pi$
$720$	0	0
$721$	54.5583	2.03186
$722$	−34.5915	−1.28736
$723$	0	0
$724$	−35.7866	−1.33000
$725$	−3.41262	−0.126742
$726$	0	0
$727$	43.4484	1.61141	0.805707	−	0.592315i	$-0.201786\pi$
$727$	43.4484	1.61141	0.805707	+	0.592315i	$0.201786\pi$
$728$	−23.4258	−0.868217
$729$	0	0
$730$	−18.8221	−0.696638
$731$	−2.89022	−0.106899
$732$	0	0
$733$	39.6649	1.46506	0.732528	−	0.680737i	$-0.238340\pi$
$733$	39.6649	1.46506	0.732528	+	0.680737i	$0.238340\pi$
$734$	−21.0938	−0.778587
$735$	0	0
$736$	−30.8982	−1.13892
$737$	0	0
$738$	0	0
$739$	29.7759	1.09532	0.547661	−	0.836700i	$-0.315518\pi$
$739$	29.7759	1.09532	0.547661	+	0.836700i	$0.315518\pi$
$740$	−12.2043	−0.448637
$741$	0	0
$742$	85.0535	3.12241
$743$	−46.7483	−1.71503	−0.857514	−	0.514461i	$-0.827992\pi$
$743$	−46.7483	−1.71503	−0.857514	+	0.514461i	$0.827992\pi$
$744$	0	0
$745$	−0.217495	−0.00796840
$746$	42.4122	1.55282
$747$	0	0
$748$	0	0
$749$	−54.1873	−1.97996
$750$	0	0
$751$	−15.1922	−0.554372	−0.277186	−	0.960816i	$-0.589402\pi$
$751$	−15.1922	−0.554372	−0.277186	+	0.960816i	$0.589402\pi$
$752$	−0.836657	−0.0305097
$753$	0	0
$754$	14.3867	0.523934
$755$	−7.86058	−0.286076
$756$	0	0
$757$	16.8706	0.613172	0.306586	−	0.951843i	$-0.400813\pi$
$757$	16.8706	0.613172	0.306586	+	0.951843i	$0.400813\pi$
$758$	46.8207	1.70060
$759$	0	0
$760$	6.07930	0.220520
$761$	47.3147	1.71516	0.857579	−	0.514352i	$-0.171968\pi$
$761$	47.3147	1.71516	0.857579	+	0.514352i	$0.171968\pi$
$762$	0	0
$763$	73.7309	2.66924
$764$	−23.2353	−0.840624
$765$	0	0
$766$	68.8033	2.48596
$767$	−3.82719	−0.138192
$768$	0	0
$769$	34.0268	1.22704	0.613518	−	0.789681i	$-0.289754\pi$
$769$	34.0268	1.22704	0.613518	+	0.789681i	$0.289754\pi$
$770$	0	0
$771$	0	0
$772$	27.7553	0.998935
$773$	41.3934	1.48882	0.744408	−	0.667725i	$-0.232732\pi$
$773$	41.3934	1.48882	0.744408	+	0.667725i	$0.232732\pi$
$774$	0	0
$775$	−6.43148	−0.231026
$776$	−51.0728	−1.83341
$777$	0	0
$778$	25.1008	0.899909
$779$	−15.3006	−0.548200
$780$	0	0
$781$	0	0
$782$	77.5393	2.77280
$783$	0	0
$784$	4.02409	0.143717
$785$	−11.0581	−0.394680
$786$	0	0
$787$	30.5226	1.08801	0.544007	−	0.839081i	$-0.316906\pi$
$787$	30.5226	1.08801	0.544007	+	0.839081i	$0.316906\pi$
$788$	18.5705	0.661545
$789$	0	0
$790$	−3.14105	−0.111754
$791$	−30.9146	−1.09920
$792$	0	0
$793$	−24.8630	−0.882910
$794$	−66.9302	−2.37526
$795$	0	0
$796$	−15.2025	−0.538839
$797$	−43.4758	−1.53999	−0.769997	−	0.638048i	$-0.779742\pi$
$797$	−43.4758	−1.53999	−0.769997	+	0.638048i	$0.779742\pi$
$798$	0	0
$799$	−12.7174	−0.449909
$800$	−5.21785	−0.184479
$801$	0	0
$802$	49.2533	1.73920
$803$	0	0
$804$	0	0
$805$	24.9641	0.879869
$806$	27.1135	0.955031
$807$	0	0
$808$	14.4691	0.509022
$809$	10.5965	0.372553	0.186277	−	0.982497i	$-0.440358\pi$
$809$	10.5965	0.372553	0.186277	+	0.982497i	$0.440358\pi$
$810$	0	0
$811$	53.9732	1.89525	0.947627	−	0.319380i	$-0.103474\pi$
$811$	53.9732	1.89525	0.947627	+	0.319380i	$0.103474\pi$
$812$	−47.7365	−1.67522
$813$	0	0
$814$	0	0
$815$	2.46427	0.0863196
$816$	0	0
$817$	−1.01803	−0.0356162
$818$	−43.4523	−1.51928
$819$	0	0
$820$	−25.3844	−0.886461
$821$	−2.45153	−0.0855591	−0.0427796	−	0.999085i	$-0.513621\pi$
$821$	−2.45153	−0.0855591	−0.0427796	+	0.999085i	$0.513621\pi$
$822$	0	0
$823$	−38.4975	−1.34194	−0.670970	−	0.741484i	$-0.734122\pi$
$823$	−38.4975	−1.34194	−0.670970	+	0.741484i	$0.734122\pi$
$824$	39.3378	1.37040
$825$	0	0
$826$	20.3533	0.708184
$827$	−56.9039	−1.97874	−0.989370	−	0.145417i	$-0.953548\pi$
$827$	−56.9039	−1.97874	−0.989370	+	0.145417i	$0.953548\pi$
$828$	0	0
$829$	16.6863	0.579540	0.289770	−	0.957096i	$-0.406421\pi$
$829$	16.6863	0.579540	0.289770	+	0.957096i	$0.406421\pi$
$830$	−5.78796	−0.200903
$831$	0	0
$832$	23.3629	0.809963
$833$	61.1671	2.11932
$834$	0	0
$835$	13.3547	0.462159
$836$	0	0
$837$	0	0
$838$	−8.90838	−0.307735
$839$	−45.9495	−1.58635	−0.793177	−	0.608991i	$-0.791575\pi$
$839$	−45.9495	−1.58635	−0.793177	+	0.608991i	$0.791575\pi$
$840$	0	0
$841$	−17.3540	−0.598414
$842$	−68.4869	−2.36022
$843$	0	0
$844$	−25.7466	−0.886234
$845$	−9.65811	−0.332249
$846$	0	0
$847$	−46.3732	−1.59340
$848$	3.26808	0.112226
$849$	0	0
$850$	13.0942	0.449129
$851$	−21.7803	−0.746620
$852$	0	0
$853$	24.6220	0.843041	0.421520	−	0.906819i	$-0.361497\pi$
$853$	24.6220	0.843041	0.421520	+	0.906819i	$0.361497\pi$
$854$	132.224	4.52460
$855$	0	0
$856$	−39.0704	−1.33540
$857$	28.7093	0.980690	0.490345	−	0.871528i	$-0.336871\pi$
$857$	28.7093	0.980690	0.490345	+	0.871528i	$0.336871\pi$
$858$	0	0
$859$	54.5033	1.85963	0.929814	−	0.368029i	$-0.119967\pi$
$859$	54.5033	1.85963	0.929814	+	0.368029i	$0.119967\pi$
$860$	−1.68895	−0.0575928
$861$	0	0
$862$	−3.25224	−0.110772
$863$	15.7275	0.535370	0.267685	−	0.963507i	$-0.413741\pi$
$863$	15.7275	0.535370	0.267685	+	0.963507i	$0.413741\pi$
$864$	0	0
$865$	−0.947233	−0.0322069
$866$	−9.46222	−0.321539
$867$	0	0
$868$	−89.9650	−3.05361
$869$	0	0
$870$	0	0
$871$	19.6024	0.664200
$872$	53.1617	1.80028
$873$	0	0
$874$	27.3118	0.923834
$875$	4.21574	0.142518
$876$	0	0
$877$	5.32597	0.179845	0.0899227	−	0.995949i	$-0.471338\pi$
$877$	5.32597	0.179845	0.0899227	+	0.995949i	$0.471338\pi$
$878$	−20.1755	−0.680891
$879$	0	0
$880$	0	0
$881$	−8.57090	−0.288761	−0.144381	−	0.989522i	$-0.546119\pi$
$881$	−8.57090	−0.288761	−0.144381	+	0.989522i	$0.546119\pi$
$882$	0	0
$883$	20.8260	0.700852	0.350426	−	0.936590i	$-0.386037\pi$
$883$	20.8260	0.700852	0.350426	+	0.936590i	$0.386037\pi$
$884$	−34.4419	−1.15841
$885$	0	0
$886$	−40.5071	−1.36086
$887$	57.5249	1.93150	0.965749	−	0.259480i	$-0.0835511\pi$
$887$	57.5249	1.93150	0.965749	+	0.259480i	$0.0835511\pi$
$888$	0	0
$889$	−46.1953	−1.54934
$890$	2.30610	0.0773006
$891$	0	0
$892$	−75.8467	−2.53953
$893$	−4.47946	−0.149899
$894$	0	0
$895$	1.11376	0.0372290
$896$	−80.2518	−2.68102
$897$	0	0
$898$	59.5390	1.98684
$899$	21.9482	0.732015
$900$	0	0
$901$	49.6756	1.65493
$902$	0	0
$903$	0	0
$904$	−22.2902	−0.741361
$905$	−10.7853	−0.358515
$906$	0	0
$907$	−31.8857	−1.05875	−0.529374	−	0.848389i	$-0.677573\pi$
$907$	−31.8857	−1.05875	−0.529374	+	0.848389i	$0.677573\pi$
$908$	−14.6815	−0.487221
$909$	0	0
$910$	−17.7725	−0.589152
$911$	−33.7969	−1.11974	−0.559870	−	0.828580i	$-0.689149\pi$
$911$	−33.7969	−1.11974	−0.559870	+	0.828580i	$0.689149\pi$
$912$	0	0
$913$	0	0
$914$	41.3728	1.36849
$915$	0	0
$916$	83.1759	2.74821
$917$	64.3011	2.12341
$918$	0	0
$919$	17.8701	0.589482	0.294741	−	0.955577i	$-0.404767\pi$
$919$	17.8701	0.589482	0.294741	+	0.955577i	$0.404767\pi$
$920$	17.9997	0.593433
$921$	0	0
$922$	2.36565	0.0779086
$923$	−20.0438	−0.659750
$924$	0	0
$925$	−3.67809	−0.120935
$926$	−93.6570	−3.07776
$927$	0	0
$928$	17.8066	0.584529
$929$	32.5742	1.06873	0.534363	−	0.845255i	$-0.320552\pi$
$929$	32.5742	1.06873	0.534363	+	0.845255i	$0.320552\pi$
$930$	0	0
$931$	21.5450	0.706108
$932$	40.8376	1.33768
$933$	0	0
$934$	−47.7052	−1.56096
$935$	0	0
$936$	0	0
$937$	−14.0837	−0.460094	−0.230047	−	0.973180i	$-0.573888\pi$
$937$	−14.0837	−0.460094	−0.230047	+	0.973180i	$0.573888\pi$
$938$	−104.247	−3.40379
$939$	0	0
$940$	−7.43163	−0.242393
$941$	−51.4559	−1.67741	−0.838707	−	0.544583i	$-0.816688\pi$
$941$	−51.4559	−1.67741	−0.838707	+	0.544583i	$0.816688\pi$
$942$	0	0
$943$	−45.3022	−1.47524
$944$	0.782052	0.0254536
$945$	0	0
$946$	0	0
$947$	19.5647	0.635766	0.317883	−	0.948130i	$-0.397028\pi$
$947$	19.5647	0.635766	0.317883	+	0.948130i	$0.397028\pi$
$948$	0	0
$949$	14.9206	0.484344
$950$	4.61220	0.149639
$951$	0	0
$952$	72.7613	2.35821
$953$	34.5762	1.12003	0.560017	−	0.828481i	$-0.310795\pi$
$953$	34.5762	1.12003	0.560017	+	0.828481i	$0.310795\pi$
$954$	0	0
$955$	−7.00261	−0.226599
$956$	−80.5903	−2.60648
$957$	0	0
$958$	−61.3844	−1.98324
$959$	68.6844	2.21793
$960$	0	0
$961$	10.3640	0.334322
$962$	15.5059	0.499930
$963$	0	0
$964$	−89.1669	−2.87187
$965$	8.36484	0.269274
$966$	0	0
$967$	37.5519	1.20759	0.603793	−	0.797141i	$-0.293655\pi$
$967$	37.5519	1.20759	0.603793	+	0.797141i	$0.293655\pi$
$968$	−33.4362	−1.07468
$969$	0	0
$970$	−38.7475	−1.24411
$971$	47.3614	1.51990	0.759950	−	0.649982i	$-0.225223\pi$
$971$	47.3614	1.51990	0.759950	+	0.649982i	$0.225223\pi$
$972$	0	0
$973$	−42.5506	−1.36411
$974$	15.7139	0.503505
$975$	0	0
$976$	5.08053	0.162624
$977$	−2.06158	−0.0659559	−0.0329780	−	0.999456i	$-0.510499\pi$
$977$	−2.06158	−0.0659559	−0.0329780	+	0.999456i	$0.510499\pi$
$978$	0	0
$979$	0	0
$980$	35.7441	1.14180
$981$	0	0
$982$	−42.0245	−1.34106
$983$	−16.7254	−0.533459	−0.266729	−	0.963771i	$-0.585943\pi$
$983$	−16.7254	−0.533459	−0.266729	+	0.963771i	$0.585943\pi$
$984$	0	0
$985$	5.59673	0.178327
$986$	−44.6857	−1.42308
$987$	0	0
$988$	−12.1315	−0.385954
$989$	−3.01419	−0.0958457
$990$	0	0
$991$	49.0300	1.55749	0.778744	−	0.627341i	$-0.215857\pi$
$991$	49.0300	1.55749	0.778744	+	0.627341i	$0.215857\pi$
$992$	33.5585	1.06548
$993$	0	0
$994$	106.595	3.38098
$995$	−4.58170	−0.145250
$996$	0	0
$997$	42.2988	1.33962	0.669809	−	0.742534i	$-0.266376\pi$
$997$	42.2988	1.33962	0.669809	+	0.742534i	$0.266376\pi$
$998$	−23.5176	−0.744437
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.n.1.5		6
3.2	odd	2		1335.2.a.g.1.2	✓	6
15.14	odd	2		6675.2.a.u.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.g.1.2	✓	6	3.2	odd	2
4005.2.a.n.1.5		6	1.1	even	1	trivial
6675.2.a.u.1.5		6	15.14	odd	2

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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30610 q^{2} +3.31809 q^{4} +1.00000 q^{5} +4.21574 q^{7} +3.03965 q^{8} +O(q^{10})\)	\(q+2.30610 q^{2} +3.31809 q^{4} +1.00000 q^{5} +4.21574 q^{7} +3.03965 q^{8} +2.30610 q^{10} -1.82808 q^{13} +9.72192 q^{14} +0.373553 q^{16} +5.67809 q^{17} +2.00000 q^{19} +3.31809 q^{20} +5.92164 q^{23} +1.00000 q^{25} -4.21574 q^{26} +13.9882 q^{28} -3.41262 q^{29} -6.43148 q^{31} -5.21785 q^{32} +13.0942 q^{34} +4.21574 q^{35} -3.67809 q^{37} +4.61220 q^{38} +3.03965 q^{40} -7.65029 q^{41} -0.509013 q^{43} +13.6559 q^{46} -2.23973 q^{47} +10.7725 q^{49} +2.30610 q^{50} -6.06575 q^{52} +8.74864 q^{53} +12.8144 q^{56} -7.86985 q^{58} +2.09355 q^{59} +13.6006 q^{61} -14.8316 q^{62} -12.7800 q^{64} -1.82808 q^{65} -10.7229 q^{67} +18.8404 q^{68} +9.72192 q^{70} +10.9644 q^{71} -8.16189 q^{73} -8.48205 q^{74} +6.63618 q^{76} -1.36206 q^{79} +0.373553 q^{80} -17.6423 q^{82} -2.50985 q^{83} +5.67809 q^{85} -1.17383 q^{86} +1.00000 q^{89} -7.70673 q^{91} +19.6485 q^{92} -5.16504 q^{94} +2.00000 q^{95} -16.8022 q^{97} +24.8424 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) \) 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8	\( 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8} + 4 q^{10} - 3 q^{13} + 5 q^{14} + 12 q^{16} + 13 q^{17} + 12 q^{19} + 8 q^{20} + 19 q^{23} + 6 q^{25} - q^{26} - 6 q^{28} + 10 q^{31} + 17 q^{32} - 2 q^{34} + q^{35} - q^{37} + 8 q^{38} + 9 q^{40} + 4 q^{41} - 7 q^{43} - 6 q^{46} + 15 q^{47} - q^{49} + 4 q^{50} + q^{52} + 27 q^{53} + 14 q^{56} - 6 q^{58} + 4 q^{59} + 8 q^{61} - 2 q^{62} - q^{64} - 3 q^{65} - 11 q^{67} + 47 q^{68} + 5 q^{70} + 16 q^{71} + q^{73} + 10 q^{74} + 16 q^{76} + 12 q^{80} + q^{82} + 17 q^{83} + 13 q^{85} + 20 q^{86} + 6 q^{89} - 18 q^{91} + 36 q^{92} + 17 q^{94} + 12 q^{95} - 29 q^{97} + 23 q^{98}+O(q^{100}) \) 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8 + 4 * q^10 - 3 * q^13 + 5 * q^14 + 12 * q^16 + 13 * q^17 + 12 * q^19 + 8 * q^20 + 19 * q^23 + 6 * q^25 - q^26 - 6 * q^28 + 10 * q^31 + 17 * q^32 - 2 * q^34 + q^35 - q^37 + 8 * q^38 + 9 * q^40 + 4 * q^41 - 7 * q^43 - 6 * q^46 + 15 * q^47 - q^49 + 4 * q^50 + q^52 + 27 * q^53 + 14 * q^56 - 6 * q^58 + 4 * q^59 + 8 * q^61 - 2 * q^62 - q^64 - 3 * q^65 - 11 * q^67 + 47 * q^68 + 5 * q^70 + 16 * q^71 + q^73 + 10 * q^74 + 16 * q^76 + 12 * q^80 + q^82 + 17 * q^83 + 13 * q^85 + 20 * q^86 + 6 * q^89 - 18 * q^91 + 36 * q^92 + 17 * q^94 + 12 * q^95 - 29 * q^97 + 23 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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