LMFDB - Embedded newform 1035.2.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1035,2,Mod(1,1035)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1035, 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("1035.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1035 = 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1035.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:	$8.26451660920$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 345)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	1035.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.34292 q^{2} +3.48929 q^{4} -1.00000 q^{5} +4.48929 q^{7} +3.48929 q^{8} +O(q^{10})$	\(q+2.34292 q^{2} +3.48929 q^{4} -1.00000 q^{5} +4.48929 q^{7} +3.48929 q^{8} -2.34292 q^{10} +1.14637 q^{11} +0.853635 q^{13} +10.5181 q^{14} +1.19656 q^{16} -1.34292 q^{17} -3.83221 q^{19} -3.48929 q^{20} +2.68585 q^{22} -1.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +15.6644 q^{28} +8.02877 q^{29} +2.19656 q^{31} -4.17513 q^{32} -3.14637 q^{34} -4.48929 q^{35} -2.48929 q^{37} -8.97858 q^{38} -3.48929 q^{40} -11.3001 q^{41} +10.6858 q^{43} +4.00000 q^{44} -2.34292 q^{46} -1.53948 q^{47} +13.1537 q^{49} +2.34292 q^{50} +2.97858 q^{52} -4.02877 q^{53} -1.14637 q^{55} +15.6644 q^{56} +18.8108 q^{58} +15.0073 q^{59} -5.83221 q^{61} +5.14637 q^{62} -12.1751 q^{64} -0.853635 q^{65} -11.5682 q^{67} -4.68585 q^{68} -10.5181 q^{70} -4.32150 q^{71} -13.1035 q^{73} -5.83221 q^{74} -13.3717 q^{76} +5.14637 q^{77} +0.585462 q^{79} -1.19656 q^{80} -26.4752 q^{82} -5.63565 q^{83} +1.34292 q^{85} +25.0361 q^{86} +4.00000 q^{88} +0.100384 q^{89} +3.83221 q^{91} -3.48929 q^{92} -3.60688 q^{94} +3.83221 q^{95} +11.5640 q^{97} +30.8181 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + q^2 + 3 * q^4 - 3 * q^5 + 6 * q^7 + 3 * q^8	$ 3 q + q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{7} + 3 q^{8} - q^{10} + 2 q^{11} + 4 q^{13} + 6 q^{14} - q^{16} + 2 q^{17} + 2 q^{19} - 3 q^{20} - 4 q^{22} - 3 q^{23} + 3 q^{25} + 6 q^{26} + 20 q^{28} + 6 q^{29} + 2 q^{31} + 7 q^{32} - 8 q^{34} - 6 q^{35} - 12 q^{38} - 3 q^{40} + 2 q^{41} + 20 q^{43} + 12 q^{44} - q^{46} + 6 q^{47} + 5 q^{49} + q^{50} - 6 q^{52} + 6 q^{53} - 2 q^{55} + 20 q^{56} + 28 q^{58} + 12 q^{59} - 4 q^{61} + 14 q^{62} - 17 q^{64} - 4 q^{65} - 6 q^{67} - 2 q^{68} - 6 q^{70} + 8 q^{71} - 8 q^{73} - 4 q^{74} - 16 q^{76} + 14 q^{77} - 4 q^{79} + q^{80} - 24 q^{82} - 8 q^{83} - 2 q^{85} + 24 q^{86} + 12 q^{88} - 6 q^{89} - 2 q^{91} - 3 q^{92} - 20 q^{94} - 2 q^{95} + 14 q^{97} + 31 q^{98}+O(q^{100}) $ 3 * q + q^2 + 3 * q^4 - 3 * q^5 + 6 * q^7 + 3 * q^8 - q^10 + 2 * q^11 + 4 * q^13 + 6 * q^14 - q^16 + 2 * q^17 + 2 * q^19 - 3 * q^20 - 4 * q^22 - 3 * q^23 + 3 * q^25 + 6 * q^26 + 20 * q^28 + 6 * q^29 + 2 * q^31 + 7 * q^32 - 8 * q^34 - 6 * q^35 - 12 * q^38 - 3 * q^40 + 2 * q^41 + 20 * q^43 + 12 * q^44 - q^46 + 6 * q^47 + 5 * q^49 + q^50 - 6 * q^52 + 6 * q^53 - 2 * q^55 + 20 * q^56 + 28 * q^58 + 12 * q^59 - 4 * q^61 + 14 * q^62 - 17 * q^64 - 4 * q^65 - 6 * q^67 - 2 * q^68 - 6 * q^70 + 8 * q^71 - 8 * q^73 - 4 * q^74 - 16 * q^76 + 14 * q^77 - 4 * q^79 + q^80 - 24 * q^82 - 8 * q^83 - 2 * q^85 + 24 * q^86 + 12 * q^88 - 6 * q^89 - 2 * q^91 - 3 * q^92 - 20 * q^94 - 2 * q^95 + 14 * q^97 + 31 * 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 $

$2$	2.34292	1.65670	0.828348	−	0.560213i	$-0.189281\pi$
$2$	2.34292	1.65670	0.828348	+	0.560213i	$0.189281\pi$
$3$	0	0
$3$	0	0
$4$	3.48929	1.74464
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.48929	1.69679	0.848396	−	0.529362i	$-0.177569\pi$
$7$	4.48929	1.69679	0.848396	+	0.529362i	$0.177569\pi$
$8$	3.48929	1.23365
$9$	0	0
$10$	−2.34292	−0.740897
$11$	1.14637	0.345642	0.172821	−	0.984953i	$-0.444712\pi$
$11$	1.14637	0.345642	0.172821	+	0.984953i	$0.444712\pi$
$12$	0	0
$13$	0.853635	0.236756	0.118378	−	0.992969i	$-0.462231\pi$
$13$	0.853635	0.236756	0.118378	+	0.992969i	$0.462231\pi$
$14$	10.5181	2.81107
$15$	0	0
$16$	1.19656	0.299139
$17$	−1.34292	−0.325707	−0.162853	−	0.986650i	$-0.552070\pi$
$17$	−1.34292	−0.325707	−0.162853	+	0.986650i	$0.552070\pi$
$18$	0	0
$19$	−3.83221	−0.879170	−0.439585	−	0.898201i	$-0.644874\pi$
$19$	−3.83221	−0.879170	−0.439585	+	0.898201i	$0.644874\pi$
$20$	−3.48929	−0.780229
$21$	0	0
$22$	2.68585	0.572624
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	15.6644	2.96030
$29$	8.02877	1.49091	0.745453	−	0.666559i	$-0.232233\pi$
$29$	8.02877	1.49091	0.745453	+	0.666559i	$0.232233\pi$
$30$	0	0
$31$	2.19656	0.394513	0.197257	−	0.980352i	$-0.436797\pi$
$31$	2.19656	0.394513	0.197257	+	0.980352i	$0.436797\pi$
$32$	−4.17513	−0.738067
$33$	0	0
$34$	−3.14637	−0.539597
$35$	−4.48929	−0.758828
$36$	0	0
$37$	−2.48929	−0.409237	−0.204618	−	0.978842i	$-0.565595\pi$
$37$	−2.48929	−0.409237	−0.204618	+	0.978842i	$0.565595\pi$
$38$	−8.97858	−1.45652
$39$	0	0
$40$	−3.48929	−0.551705
$41$	−11.3001	−1.76478	−0.882388	−	0.470523i	$-0.844065\pi$
$41$	−11.3001	−1.76478	−0.882388	+	0.470523i	$0.844065\pi$
$42$	0	0
$43$	10.6858	1.62958	0.814788	−	0.579759i	$-0.196853\pi$
$43$	10.6858	1.62958	0.814788	+	0.579759i	$0.196853\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−2.34292	−0.345445
$47$	−1.53948	−0.224556	−0.112278	−	0.993677i	$-0.535815\pi$
$47$	−1.53948	−0.224556	−0.112278	+	0.993677i	$0.535815\pi$
$48$	0	0
$49$	13.1537	1.87910
$50$	2.34292	0.331339
$51$	0	0
$52$	2.97858	0.413054
$53$	−4.02877	−0.553394	−0.276697	−	0.960957i	$-0.589240\pi$
$53$	−4.02877	−0.553394	−0.276697	+	0.960957i	$0.589240\pi$
$54$	0	0
$55$	−1.14637	−0.154576
$56$	15.6644	2.09325
$57$	0	0
$58$	18.8108	2.46998
$59$	15.0073	1.95379	0.976895	−	0.213720i	$-0.0685579\pi$
$59$	15.0073	1.95379	0.976895	+	0.213720i	$0.0685579\pi$
$60$	0	0
$61$	−5.83221	−0.746738	−0.373369	−	0.927683i	$-0.621797\pi$
$61$	−5.83221	−0.746738	−0.373369	+	0.927683i	$0.621797\pi$
$62$	5.14637	0.653589
$63$	0	0
$64$	−12.1751	−1.52189
$65$	−0.853635	−0.105880
$66$	0	0
$67$	−11.5682	−1.41329	−0.706643	−	0.707570i	$-0.749791\pi$
$67$	−11.5682	−1.41329	−0.706643	+	0.707570i	$0.749791\pi$
$68$	−4.68585	−0.568242
$69$	0	0
$70$	−10.5181	−1.25715
$71$	−4.32150	−0.512868	−0.256434	−	0.966562i	$-0.582548\pi$
$71$	−4.32150	−0.512868	−0.256434	+	0.966562i	$0.582548\pi$
$72$	0	0
$73$	−13.1035	−1.53365	−0.766825	−	0.641856i	$-0.778165\pi$
$73$	−13.1035	−1.53365	−0.766825	+	0.641856i	$0.778165\pi$
$74$	−5.83221	−0.677981
$75$	0	0
$76$	−13.3717	−1.53384
$77$	5.14637	0.586483
$78$	0	0
$79$	0.585462	0.0658696	0.0329348	−	0.999458i	$-0.489515\pi$
$79$	0.585462	0.0658696	0.0329348	+	0.999458i	$0.489515\pi$
$80$	−1.19656	−0.133779
$81$	0	0
$82$	−26.4752	−2.92370
$83$	−5.63565	−0.618593	−0.309297	−	0.950966i	$-0.600094\pi$
$83$	−5.63565	−0.618593	−0.309297	+	0.950966i	$0.600094\pi$
$84$	0	0
$85$	1.34292	0.145660
$86$	25.0361	2.69971
$87$	0	0
$88$	4.00000	0.426401
$89$	0.100384	0.0106407	0.00532035	−	0.999986i	$-0.498306\pi$
$89$	0.100384	0.0106407	0.00532035	+	0.999986i	$0.498306\pi$
$90$	0	0
$91$	3.83221	0.401725
$92$	−3.48929	−0.363783
$93$	0	0
$94$	−3.60688	−0.372022
$95$	3.83221	0.393177
$96$	0	0
$97$	11.5640	1.17415	0.587075	−	0.809532i	$-0.300279\pi$
$97$	11.5640	1.17415	0.587075	+	0.809532i	$0.300279\pi$
$98$	30.8181	3.11310
$99$	0	0
$100$	3.48929	0.348929
$101$	−11.6932	−1.16352	−0.581758	−	0.813362i	$-0.697635\pi$
$101$	−11.6932	−1.16352	−0.581758	+	0.813362i	$0.697635\pi$
$102$	0	0
$103$	−19.7220	−1.94326	−0.971631	−	0.236501i	$-0.923999\pi$
$103$	−19.7220	−1.94326	−0.971631	+	0.236501i	$0.923999\pi$
$104$	2.97858	0.292074
$105$	0	0
$106$	−9.43910	−0.916806
$107$	−5.44331	−0.526224	−0.263112	−	0.964765i	$-0.584749\pi$
$107$	−5.44331	−0.526224	−0.263112	+	0.964765i	$0.584749\pi$
$108$	0	0
$109$	−12.5181	−1.19901	−0.599506	−	0.800370i	$-0.704636\pi$
$109$	−12.5181	−1.19901	−0.599506	+	0.800370i	$0.704636\pi$
$110$	−2.68585	−0.256085
$111$	0	0
$112$	5.37169	0.507577
$113$	14.1292	1.32916	0.664579	−	0.747218i	$-0.268611\pi$
$113$	14.1292	1.32916	0.664579	+	0.747218i	$0.268611\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	28.0147	2.60110
$117$	0	0
$118$	35.1611	3.23684
$119$	−6.02877	−0.552656
$120$	0	0
$121$	−9.68585	−0.880531
$122$	−13.6644	−1.23712
$123$	0	0
$124$	7.66442	0.688286
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	5.48194	0.486444	0.243222	−	0.969971i	$-0.421796\pi$
$127$	5.48194	0.486444	0.243222	+	0.969971i	$0.421796\pi$
$128$	−20.1751	−1.78325
$129$	0	0
$130$	−2.00000	−0.175412
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−17.2039	−1.49177
$134$	−27.1035	−2.34139
$135$	0	0
$136$	−4.68585	−0.401808
$137$	−11.7648	−1.00514	−0.502568	−	0.864538i	$-0.667611\pi$
$137$	−11.7648	−1.00514	−0.502568	+	0.864538i	$0.667611\pi$
$138$	0	0
$139$	8.15371	0.691589	0.345794	−	0.938310i	$-0.387609\pi$
$139$	8.15371	0.691589	0.345794	+	0.938310i	$0.387609\pi$
$140$	−15.6644	−1.32389
$141$	0	0
$142$	−10.1249	−0.849666
$143$	0.978577	0.0818327
$144$	0	0
$145$	−8.02877	−0.666753
$146$	−30.7005	−2.54079
$147$	0	0
$148$	−8.68585	−0.713972
$149$	3.73183	0.305723	0.152862	−	0.988248i	$-0.451151\pi$
$149$	3.73183	0.305723	0.152862	+	0.988248i	$0.451151\pi$
$150$	0	0
$151$	16.6430	1.35439	0.677194	−	0.735804i	$-0.263196\pi$
$151$	16.6430	1.35439	0.677194	+	0.735804i	$0.263196\pi$
$152$	−13.3717	−1.08459
$153$	0	0
$154$	12.0575	0.971624
$155$	−2.19656	−0.176432
$156$	0	0
$157$	−13.2327	−1.05608	−0.528041	−	0.849219i	$-0.677073\pi$
$157$	−13.2327	−1.05608	−0.528041	+	0.849219i	$0.677073\pi$
$158$	1.37169	0.109126
$159$	0	0
$160$	4.17513	0.330073
$161$	−4.48929	−0.353806
$162$	0	0
$163$	16.3931	1.28401	0.642004	−	0.766701i	$-0.278103\pi$
$163$	16.3931	1.28401	0.642004	+	0.766701i	$0.278103\pi$
$164$	−39.4292	−3.07891
$165$	0	0
$166$	−13.2039	−1.02482
$167$	−3.04598	−0.235705	−0.117853	−	0.993031i	$-0.537601\pi$
$167$	−3.04598	−0.235705	−0.117853	+	0.993031i	$0.537601\pi$
$168$	0	0
$169$	−12.2713	−0.943947
$170$	3.14637	0.241315
$171$	0	0
$172$	37.2860	2.84303
$173$	24.9357	1.89583	0.947914	−	0.318526i	$-0.103188\pi$
$173$	24.9357	1.89583	0.947914	+	0.318526i	$0.103188\pi$
$174$	0	0
$175$	4.48929	0.339358
$176$	1.37169	0.103395
$177$	0	0
$178$	0.235192	0.0176284
$179$	6.68585	0.499724	0.249862	−	0.968282i	$-0.419615\pi$
$179$	6.68585	0.499724	0.249862	+	0.968282i	$0.419615\pi$
$180$	0	0
$181$	−20.6002	−1.53120	−0.765599	−	0.643318i	$-0.777557\pi$
$181$	−20.6002	−1.53120	−0.765599	+	0.643318i	$0.777557\pi$
$182$	8.97858	0.665536
$183$	0	0
$184$	−3.48929	−0.257234
$185$	2.48929	0.183016
$186$	0	0
$187$	−1.53948	−0.112578
$188$	−5.37169	−0.391771
$189$	0	0
$190$	8.97858	0.651374
$191$	−4.95402	−0.358460	−0.179230	−	0.983807i	$-0.557361\pi$
$191$	−4.95402	−0.358460	−0.179230	+	0.983807i	$0.557361\pi$
$192$	0	0
$193$	6.05754	0.436031	0.218016	−	0.975945i	$-0.430042\pi$
$193$	6.05754	0.436031	0.218016	+	0.975945i	$0.430042\pi$
$194$	27.0937	1.94521
$195$	0	0
$196$	45.8971	3.27836
$197$	24.4078	1.73898	0.869492	−	0.493947i	$-0.164446\pi$
$197$	24.4078	1.73898	0.869492	+	0.493947i	$0.164446\pi$
$198$	0	0
$199$	−3.85677	−0.273399	−0.136700	−	0.990613i	$-0.543650\pi$
$199$	−3.85677	−0.273399	−0.136700	+	0.990613i	$0.543650\pi$
$200$	3.48929	0.246730
$201$	0	0
$202$	−27.3963	−1.92759
$203$	36.0435	2.52976
$204$	0	0
$205$	11.3001	0.789232
$206$	−46.2070	−3.21940
$207$	0	0
$208$	1.02142	0.0708229
$209$	−4.39312	−0.303878
$210$	0	0
$211$	−0.824865	−0.0567861	−0.0283930	−	0.999597i	$-0.509039\pi$
$211$	−0.824865	−0.0567861	−0.0283930	+	0.999597i	$0.509039\pi$
$212$	−14.0575	−0.965476
$213$	0	0
$214$	−12.7533	−0.871794
$215$	−10.6858	−0.728769
$216$	0	0
$217$	9.86098	0.669407
$218$	−29.3288	−1.98640
$219$	0	0
$220$	−4.00000	−0.269680
$221$	−1.14637	−0.0771129
$222$	0	0
$223$	5.56404	0.372596	0.186298	−	0.982493i	$-0.440351\pi$
$223$	5.56404	0.372596	0.186298	+	0.982493i	$0.440351\pi$
$224$	−18.7434	−1.25235
$225$	0	0
$226$	33.1035	2.20201
$227$	−16.5855	−1.10082	−0.550408	−	0.834896i	$-0.685528\pi$
$227$	−16.5855	−1.10082	−0.550408	+	0.834896i	$0.685528\pi$
$228$	0	0
$229$	6.64300	0.438982	0.219491	−	0.975615i	$-0.429560\pi$
$229$	6.64300	0.438982	0.219491	+	0.975615i	$0.429560\pi$
$230$	2.34292	0.154488
$231$	0	0
$232$	28.0147	1.83925
$233$	14.3931	0.942924	0.471462	−	0.881886i	$-0.343726\pi$
$233$	14.3931	0.942924	0.471462	+	0.881886i	$0.343726\pi$
$234$	0	0
$235$	1.53948	0.100425
$236$	52.3650	3.40867
$237$	0	0
$238$	−14.1249	−0.915584
$239$	13.1077	0.847869	0.423934	−	0.905693i	$-0.360649\pi$
$239$	13.1077	0.847869	0.423934	+	0.905693i	$0.360649\pi$
$240$	0	0
$241$	12.5181	0.806359	0.403179	−	0.915121i	$-0.367905\pi$
$241$	12.5181	0.806359	0.403179	+	0.915121i	$0.367905\pi$
$242$	−22.6932	−1.45877
$243$	0	0
$244$	−20.3503	−1.30279
$245$	−13.1537	−0.840360
$246$	0	0
$247$	−3.27131	−0.208148
$248$	7.66442	0.486691
$249$	0	0
$250$	−2.34292	−0.148179
$251$	26.3503	1.66321	0.831607	−	0.555364i	$-0.187421\pi$
$251$	26.3503	1.66321	0.831607	+	0.555364i	$0.187421\pi$
$252$	0	0
$253$	−1.14637	−0.0720714
$254$	12.8438	0.805890
$255$	0	0
$256$	−22.9185	−1.43241
$257$	27.0116	1.68493	0.842467	−	0.538747i	$-0.181102\pi$
$257$	27.0116	1.68493	0.842467	+	0.538747i	$0.181102\pi$
$258$	0	0
$259$	−11.1751	−0.694389
$260$	−2.97858	−0.184724
$261$	0	0
$262$	0	0
$263$	−21.8855	−1.34952	−0.674760	−	0.738037i	$-0.735753\pi$
$263$	−21.8855	−1.34952	−0.674760	+	0.738037i	$0.735753\pi$
$264$	0	0
$265$	4.02877	0.247485
$266$	−40.3074	−2.47141
$267$	0	0
$268$	−40.3650	−2.46568
$269$	−3.69319	−0.225178	−0.112589	−	0.993642i	$-0.535914\pi$
$269$	−3.69319	−0.225178	−0.112589	+	0.993642i	$0.535914\pi$
$270$	0	0
$271$	0.288520	0.0175264	0.00876318	−	0.999962i	$-0.497211\pi$
$271$	0.288520	0.0175264	0.00876318	+	0.999962i	$0.497211\pi$
$272$	−1.60688	−0.0974317
$273$	0	0
$274$	−27.5640	−1.66520
$275$	1.14637	0.0691284
$276$	0	0
$277$	21.3288	1.28153	0.640763	−	0.767739i	$-0.278618\pi$
$277$	21.3288	1.28153	0.640763	+	0.767739i	$0.278618\pi$
$278$	19.1035	1.14575
$279$	0	0
$280$	−15.6644	−0.936128
$281$	26.6676	1.59085	0.795427	−	0.606050i	$-0.207247\pi$
$281$	26.6676	1.59085	0.795427	+	0.606050i	$0.207247\pi$
$282$	0	0
$283$	25.9185	1.54070	0.770348	−	0.637624i	$-0.220082\pi$
$283$	25.9185	1.54070	0.770348	+	0.637624i	$0.220082\pi$
$284$	−15.0790	−0.894772
$285$	0	0
$286$	2.29273	0.135572
$287$	−50.7293	−2.99446
$288$	0	0
$289$	−15.1966	−0.893915
$290$	−18.8108	−1.10461
$291$	0	0
$292$	−45.7220	−2.67568
$293$	−13.6503	−0.797462	−0.398731	−	0.917068i	$-0.630549\pi$
$293$	−13.6503	−0.797462	−0.398731	+	0.917068i	$0.630549\pi$
$294$	0	0
$295$	−15.0073	−0.873761
$296$	−8.68585	−0.504855
$297$	0	0
$298$	8.74338	0.506491
$299$	−0.853635	−0.0493670
$300$	0	0
$301$	47.9718	2.76505
$302$	38.9933	2.24381
$303$	0	0
$304$	−4.58546	−0.262994
$305$	5.83221	0.333951
$306$	0	0
$307$	−9.78937	−0.558709	−0.279354	−	0.960188i	$-0.590120\pi$
$307$	−9.78937	−0.558709	−0.279354	+	0.960188i	$0.590120\pi$
$308$	17.9572	1.02320
$309$	0	0
$310$	−5.14637	−0.292294
$311$	3.32885	0.188762	0.0943808	−	0.995536i	$-0.469913\pi$
$311$	3.32885	0.188762	0.0943808	+	0.995536i	$0.469913\pi$
$312$	0	0
$313$	−18.8824	−1.06730	−0.533648	−	0.845707i	$-0.679179\pi$
$313$	−18.8824	−1.06730	−0.533648	+	0.845707i	$0.679179\pi$
$314$	−31.0031	−1.74961
$315$	0	0
$316$	2.04285	0.114919
$317$	8.06740	0.453111	0.226555	−	0.973998i	$-0.427254\pi$
$317$	8.06740	0.453111	0.226555	+	0.973998i	$0.427254\pi$
$318$	0	0
$319$	9.20390	0.515320
$320$	12.1751	0.680611
$321$	0	0
$322$	−10.5181	−0.586148
$323$	5.14637	0.286351
$324$	0	0
$325$	0.853635	0.0473511
$326$	38.4078	2.12721
$327$	0	0
$328$	−39.4292	−2.17712
$329$	−6.91117	−0.381025
$330$	0	0
$331$	−26.6388	−1.46420	−0.732100	−	0.681197i	$-0.761460\pi$
$331$	−26.6388	−1.46420	−0.732100	+	0.681197i	$0.761460\pi$
$332$	−19.6644	−1.07923
$333$	0	0
$334$	−7.13650	−0.390492
$335$	11.5682	0.632041
$336$	0	0
$337$	−8.54262	−0.465346	−0.232673	−	0.972555i	$-0.574747\pi$
$337$	−8.54262	−0.465346	−0.232673	+	0.972555i	$0.574747\pi$
$338$	−28.7507	−1.56383
$339$	0	0
$340$	4.68585	0.254126
$341$	2.51806	0.136360
$342$	0	0
$343$	27.6258	1.49165
$344$	37.2860	2.01033
$345$	0	0
$346$	58.4225	3.14081
$347$	16.2499	0.872340	0.436170	−	0.899864i	$-0.356335\pi$
$347$	16.2499	0.872340	0.436170	+	0.899864i	$0.356335\pi$
$348$	0	0
$349$	23.1898	1.24132	0.620662	−	0.784079i	$-0.286864\pi$
$349$	23.1898	1.24132	0.620662	+	0.784079i	$0.286864\pi$
$350$	10.5181	0.562214
$351$	0	0
$352$	−4.78623	−0.255107
$353$	7.48194	0.398224	0.199112	−	0.979977i	$-0.436194\pi$
$353$	7.48194	0.398224	0.199112	+	0.979977i	$0.436194\pi$
$354$	0	0
$355$	4.32150	0.229361
$356$	0.350269	0.0185642
$357$	0	0
$358$	15.6644	0.827890
$359$	−1.87506	−0.0989617	−0.0494809	−	0.998775i	$-0.515757\pi$
$359$	−1.87506	−0.0989617	−0.0494809	+	0.998775i	$0.515757\pi$
$360$	0	0
$361$	−4.31415	−0.227061
$362$	−48.2646	−2.53673
$363$	0	0
$364$	13.3717	0.700867
$365$	13.1035	0.685870
$366$	0	0
$367$	−4.68164	−0.244379	−0.122190	−	0.992507i	$-0.538992\pi$
$367$	−4.68164	−0.244379	−0.122190	+	0.992507i	$0.538992\pi$
$368$	−1.19656	−0.0623749
$369$	0	0
$370$	5.83221	0.303202
$371$	−18.0863	−0.938994
$372$	0	0
$373$	18.6430	0.965298	0.482649	−	0.875814i	$-0.339675\pi$
$373$	18.6430	0.965298	0.482649	+	0.875814i	$0.339675\pi$
$374$	−3.60688	−0.186508
$375$	0	0
$376$	−5.37169	−0.277024
$377$	6.85363	0.352980
$378$	0	0
$379$	29.8715	1.53439	0.767197	−	0.641412i	$-0.221651\pi$
$379$	29.8715	1.53439	0.767197	+	0.641412i	$0.221651\pi$
$380$	13.3717	0.685953
$381$	0	0
$382$	−11.6069	−0.593860
$383$	−20.5714	−1.05115	−0.525574	−	0.850748i	$-0.676150\pi$
$383$	−20.5714	−1.05115	−0.525574	+	0.850748i	$0.676150\pi$
$384$	0	0
$385$	−5.14637	−0.262283
$386$	14.1923	0.722371
$387$	0	0
$388$	40.3503	2.04847
$389$	2.33558	0.118418	0.0592092	−	0.998246i	$-0.481142\pi$
$389$	2.33558	0.118418	0.0592092	+	0.998246i	$0.481142\pi$
$390$	0	0
$391$	1.34292	0.0679145
$392$	45.8971	2.31815
$393$	0	0
$394$	57.1856	2.88097
$395$	−0.585462	−0.0294578
$396$	0	0
$397$	29.1365	1.46232	0.731160	−	0.682207i	$-0.238980\pi$
$397$	29.1365	1.46232	0.731160	+	0.682207i	$0.238980\pi$
$398$	−9.03612	−0.452940
$399$	0	0
$400$	1.19656	0.0598279
$401$	14.8353	0.740842	0.370421	−	0.928864i	$-0.379213\pi$
$401$	14.8353	0.740842	0.370421	+	0.928864i	$0.379213\pi$
$402$	0	0
$403$	1.87506	0.0934033
$404$	−40.8009	−2.02992
$405$	0	0
$406$	84.4471	4.19104
$407$	−2.85363	−0.141449
$408$	0	0
$409$	−38.0189	−1.87991	−0.939957	−	0.341293i	$-0.889135\pi$
$409$	−38.0189	−1.87991	−0.939957	+	0.341293i	$0.889135\pi$
$410$	26.4752	1.30752
$411$	0	0
$412$	−68.8156	−3.39030
$413$	67.3723	3.31517
$414$	0	0
$415$	5.63565	0.276643
$416$	−3.56404	−0.174741
$417$	0	0
$418$	−10.2927	−0.503434
$419$	−21.8469	−1.06729	−0.533646	−	0.845708i	$-0.679178\pi$
$419$	−21.8469	−1.06729	−0.533646	+	0.845708i	$0.679178\pi$
$420$	0	0
$421$	21.4391	1.04488	0.522439	−	0.852677i	$-0.325022\pi$
$421$	21.4391	1.04488	0.522439	+	0.852677i	$0.325022\pi$
$422$	−1.93260	−0.0940773
$423$	0	0
$424$	−14.0575	−0.682694
$425$	−1.34292	−0.0651413
$426$	0	0
$427$	−26.1825	−1.26706
$428$	−18.9933	−0.918074
$429$	0	0
$430$	−25.0361	−1.20735
$431$	−29.8223	−1.43649	−0.718246	−	0.695789i	$-0.755055\pi$
$431$	−29.8223	−1.43649	−0.718246	+	0.695789i	$0.755055\pi$
$432$	0	0
$433$	−0.925249	−0.0444647	−0.0222323	−	0.999753i	$-0.507077\pi$
$433$	−0.925249	−0.0444647	−0.0222323	+	0.999753i	$0.507077\pi$
$434$	23.1035	1.10900
$435$	0	0
$436$	−43.6791	−2.09185
$437$	3.83221	0.183320
$438$	0	0
$439$	−21.6791	−1.03469	−0.517344	−	0.855778i	$-0.673079\pi$
$439$	−21.6791	−1.03469	−0.517344	+	0.855778i	$0.673079\pi$
$440$	−4.00000	−0.190693
$441$	0	0
$442$	−2.68585	−0.127753
$443$	−5.20390	−0.247245	−0.123622	−	0.992329i	$-0.539451\pi$
$443$	−5.20390	−0.247245	−0.123622	+	0.992329i	$0.539451\pi$
$444$	0	0
$445$	−0.100384	−0.00475867
$446$	13.0361	0.617278
$447$	0	0
$448$	−54.6577	−2.58233
$449$	10.8578	0.512413	0.256207	−	0.966622i	$-0.417527\pi$
$449$	10.8578	0.512413	0.256207	+	0.966622i	$0.417527\pi$
$450$	0	0
$451$	−12.9540	−0.609981
$452$	49.3007	2.31891
$453$	0	0
$454$	−38.8585	−1.82372
$455$	−3.83221	−0.179657
$456$	0	0
$457$	27.2755	1.27589	0.637947	−	0.770080i	$-0.279784\pi$
$457$	27.2755	1.27589	0.637947	+	0.770080i	$0.279784\pi$
$458$	15.5640	0.727260
$459$	0	0
$460$	3.48929	0.162689
$461$	−20.2070	−0.941136	−0.470568	−	0.882364i	$-0.655951\pi$
$461$	−20.2070	−0.941136	−0.470568	+	0.882364i	$0.655951\pi$
$462$	0	0
$463$	8.03298	0.373324	0.186662	−	0.982424i	$-0.440233\pi$
$463$	8.03298	0.373324	0.186662	+	0.982424i	$0.440233\pi$
$464$	9.60688	0.445988
$465$	0	0
$466$	33.7220	1.56214
$467$	19.4580	0.900409	0.450204	−	0.892926i	$-0.351351\pi$
$467$	19.4580	0.900409	0.450204	+	0.892926i	$0.351351\pi$
$468$	0	0
$469$	−51.9332	−2.39805
$470$	3.60688	0.166373
$471$	0	0
$472$	52.3650	2.41029
$473$	12.2499	0.563250
$474$	0	0
$475$	−3.83221	−0.175834
$476$	−21.0361	−0.964189
$477$	0	0
$478$	30.7104	1.40466
$479$	−28.9540	−1.32294	−0.661471	−	0.749970i	$-0.730068\pi$
$479$	−28.9540	−1.32294	−0.661471	+	0.749970i	$0.730068\pi$
$480$	0	0
$481$	−2.12494	−0.0968890
$482$	29.3288	1.33589
$483$	0	0
$484$	−33.7967	−1.53621
$485$	−11.5640	−0.525096
$486$	0	0
$487$	20.8108	0.943027	0.471513	−	0.881859i	$-0.343708\pi$
$487$	20.8108	0.943027	0.471513	+	0.881859i	$0.343708\pi$
$488$	−20.3503	−0.921213
$489$	0	0
$490$	−30.8181	−1.39222
$491$	−35.1140	−1.58467	−0.792336	−	0.610084i	$-0.791135\pi$
$491$	−35.1140	−1.58467	−0.792336	+	0.610084i	$0.791135\pi$
$492$	0	0
$493$	−10.7820	−0.485598
$494$	−7.66442	−0.344839
$495$	0	0
$496$	2.62831	0.118014
$497$	−19.4005	−0.870230
$498$	0	0
$499$	−7.31836	−0.327615	−0.163807	−	0.986492i	$-0.552378\pi$
$499$	−7.31836	−0.327615	−0.163807	+	0.986492i	$0.552378\pi$
$500$	−3.48929	−0.156046
$501$	0	0
$502$	61.7367	2.75544
$503$	22.0006	0.980959	0.490479	−	0.871453i	$-0.336822\pi$
$503$	22.0006	0.980959	0.490479	+	0.871453i	$0.336822\pi$
$504$	0	0
$505$	11.6932	0.520340
$506$	−2.68585	−0.119400
$507$	0	0
$508$	19.1281	0.848671
$509$	35.4783	1.57255	0.786275	−	0.617877i	$-0.212007\pi$
$509$	35.4783	1.57255	0.786275	+	0.617877i	$0.212007\pi$
$510$	0	0
$511$	−58.8255	−2.60229
$512$	−13.3461	−0.589818
$513$	0	0
$514$	63.2860	2.79143
$515$	19.7220	0.869053
$516$	0	0
$517$	−1.76481	−0.0776161
$518$	−26.1825	−1.15039
$519$	0	0
$520$	−2.97858	−0.130619
$521$	−21.4966	−0.941785	−0.470892	−	0.882191i	$-0.656068\pi$
$521$	−21.4966	−0.941785	−0.470892	+	0.882191i	$0.656068\pi$
$522$	0	0
$523$	−11.3288	−0.495376	−0.247688	−	0.968840i	$-0.579671\pi$
$523$	−11.3288	−0.495376	−0.247688	+	0.968840i	$0.579671\pi$
$524$	0	0
$525$	0	0
$526$	−51.2761	−2.23575
$527$	−2.94981	−0.128496
$528$	0	0
$529$	1.00000	0.0434783
$530$	9.43910	0.410008
$531$	0	0
$532$	−60.0294	−2.60260
$533$	−9.64614	−0.417821
$534$	0	0
$535$	5.44331	0.235335
$536$	−40.3650	−1.74350
$537$	0	0
$538$	−8.65287	−0.373052
$539$	15.0790	0.649497
$540$	0	0
$541$	−7.67912	−0.330151	−0.165075	−	0.986281i	$-0.552787\pi$
$541$	−7.67912	−0.330151	−0.165075	+	0.986281i	$0.552787\pi$
$542$	0.675981	0.0290358
$543$	0	0
$544$	5.60688	0.240393
$545$	12.5181	0.536215
$546$	0	0
$547$	31.5296	1.34811	0.674054	−	0.738682i	$-0.264551\pi$
$547$	31.5296	1.34811	0.674054	+	0.738682i	$0.264551\pi$
$548$	−41.0508	−1.75360
$549$	0	0
$550$	2.68585	0.114525
$551$	−30.7679	−1.31076
$552$	0	0
$553$	2.62831	0.111767
$554$	49.9718	2.12310
$555$	0	0
$556$	28.4507	1.20658
$557$	10.1292	0.429186	0.214593	−	0.976704i	$-0.431157\pi$
$557$	10.1292	0.429186	0.214593	+	0.976704i	$0.431157\pi$
$558$	0	0
$559$	9.12181	0.385811
$560$	−5.37169	−0.226995
$561$	0	0
$562$	62.4800	2.63556
$563$	−40.7146	−1.71592	−0.857958	−	0.513719i	$-0.828267\pi$
$563$	−40.7146	−1.71592	−0.857958	+	0.513719i	$0.828267\pi$
$564$	0	0
$565$	−14.1292	−0.594418
$566$	60.7251	2.55247
$567$	0	0
$568$	−15.0790	−0.632699
$569$	−1.66442	−0.0697763	−0.0348881	−	0.999391i	$-0.511107\pi$
$569$	−1.66442	−0.0697763	−0.0348881	+	0.999391i	$0.511107\pi$
$570$	0	0
$571$	−41.8469	−1.75124	−0.875619	−	0.483002i	$-0.839546\pi$
$571$	−41.8469	−1.75124	−0.875619	+	0.483002i	$0.839546\pi$
$572$	3.41454	0.142769
$573$	0	0
$574$	−118.855	−4.96091
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	21.6069	0.899506	0.449753	−	0.893153i	$-0.351512\pi$
$577$	21.6069	0.899506	0.449753	+	0.893153i	$0.351512\pi$
$578$	−35.6044	−1.48095
$579$	0	0
$580$	−28.0147	−1.16325
$581$	−25.3001	−1.04962
$582$	0	0
$583$	−4.61844	−0.191276
$584$	−45.7220	−1.89199
$585$	0	0
$586$	−31.9817	−1.32115
$587$	−0.393115	−0.0162256	−0.00811280	−	0.999967i	$-0.502582\pi$
$587$	−0.393115	−0.0162256	−0.00811280	+	0.999967i	$0.502582\pi$
$588$	0	0
$589$	−8.41767	−0.346844
$590$	−35.1611	−1.44756
$591$	0	0
$592$	−2.97858	−0.122419
$593$	8.31729	0.341550	0.170775	−	0.985310i	$-0.445373\pi$
$593$	8.31729	0.341550	0.170775	+	0.985310i	$0.445373\pi$
$594$	0	0
$595$	6.02877	0.247155
$596$	13.0214	0.533378
$597$	0	0
$598$	−2.00000	−0.0817861
$599$	−9.03612	−0.369206	−0.184603	−	0.982813i	$-0.559100\pi$
$599$	−9.03612	−0.369206	−0.184603	+	0.982813i	$0.559100\pi$
$600$	0	0
$601$	−27.6602	−1.12828	−0.564142	−	0.825678i	$-0.690793\pi$
$601$	−27.6602	−1.12828	−0.564142	+	0.825678i	$0.690793\pi$
$602$	112.394	4.58085
$603$	0	0
$604$	58.0722	2.36293
$605$	9.68585	0.393786
$606$	0	0
$607$	29.6461	1.20330	0.601650	−	0.798760i	$-0.294510\pi$
$607$	29.6461	1.20330	0.601650	+	0.798760i	$0.294510\pi$
$608$	16.0000	0.648886
$609$	0	0
$610$	13.6644	0.553256
$611$	−1.31415	−0.0531650
$612$	0	0
$613$	39.4868	1.59486	0.797428	−	0.603414i	$-0.206193\pi$
$613$	39.4868	1.59486	0.797428	+	0.603414i	$0.206193\pi$
$614$	−22.9357	−0.925611
$615$	0	0
$616$	17.9572	0.723514
$617$	−16.8641	−0.678924	−0.339462	−	0.940620i	$-0.610245\pi$
$617$	−16.8641	−0.678924	−0.339462	+	0.940620i	$0.610245\pi$
$618$	0	0
$619$	2.74338	0.110266	0.0551330	−	0.998479i	$-0.482442\pi$
$619$	2.74338	0.110266	0.0551330	+	0.998479i	$0.482442\pi$
$620$	−7.66442	−0.307811
$621$	0	0
$622$	7.79923	0.312721
$623$	0.450654	0.0180551
$624$	0	0
$625$	1.00000	0.0400000
$626$	−44.2400	−1.76819
$627$	0	0
$628$	−46.1726	−1.84249
$629$	3.34292	0.133291
$630$	0	0
$631$	−9.93260	−0.395410	−0.197705	−	0.980262i	$-0.563349\pi$
$631$	−9.93260	−0.395410	−0.197705	+	0.980262i	$0.563349\pi$
$632$	2.04285	0.0812600
$633$	0	0
$634$	18.9013	0.750667
$635$	−5.48194	−0.217544
$636$	0	0
$637$	11.2285	0.444888
$638$	21.5640	0.853728
$639$	0	0
$640$	20.1751	0.797492
$641$	10.6184	0.419403	0.209702	−	0.977765i	$-0.432751\pi$
$641$	10.6184	0.419403	0.209702	+	0.977765i	$0.432751\pi$
$642$	0	0
$643$	20.5959	0.812225	0.406112	−	0.913823i	$-0.366884\pi$
$643$	20.5959	0.812225	0.406112	+	0.913823i	$0.366884\pi$
$644$	−15.6644	−0.617265
$645$	0	0
$646$	12.0575	0.474398
$647$	6.71883	0.264144	0.132072	−	0.991240i	$-0.457837\pi$
$647$	6.71883	0.264144	0.132072	+	0.991240i	$0.457837\pi$
$648$	0	0
$649$	17.2039	0.675312
$650$	2.00000	0.0784465
$651$	0	0
$652$	57.2003	2.24014
$653$	4.07583	0.159499	0.0797497	−	0.996815i	$-0.474588\pi$
$653$	4.07583	0.159499	0.0797497	+	0.996815i	$0.474588\pi$
$654$	0	0
$655$	0	0
$656$	−13.5212	−0.527914
$657$	0	0
$658$	−16.1923	−0.631243
$659$	29.0607	1.13204	0.566022	−	0.824390i	$-0.308482\pi$
$659$	29.0607	1.13204	0.566022	+	0.824390i	$0.308482\pi$
$660$	0	0
$661$	31.2369	1.21497	0.607487	−	0.794330i	$-0.292178\pi$
$661$	31.2369	1.21497	0.607487	+	0.794330i	$0.292178\pi$
$662$	−62.4126	−2.42574
$663$	0	0
$664$	−19.6644	−0.763128
$665$	17.2039	0.667139
$666$	0	0
$667$	−8.02877	−0.310875
$668$	−10.6283	−0.411222
$669$	0	0
$670$	27.1035	1.04710
$671$	−6.68585	−0.258104
$672$	0	0
$673$	9.74652	0.375701	0.187850	−	0.982198i	$-0.439848\pi$
$673$	9.74652	0.375701	0.187850	+	0.982198i	$0.439848\pi$
$674$	−20.0147	−0.770937
$675$	0	0
$676$	−42.8181	−1.64685
$677$	−6.80031	−0.261357	−0.130679	−	0.991425i	$-0.541716\pi$
$677$	−6.80031	−0.261357	−0.130679	+	0.991425i	$0.541716\pi$
$678$	0	0
$679$	51.9143	1.99229
$680$	4.68585	0.179694
$681$	0	0
$682$	5.89962	0.225908
$683$	41.5029	1.58806	0.794032	−	0.607876i	$-0.207978\pi$
$683$	41.5029	1.58806	0.794032	+	0.607876i	$0.207978\pi$
$684$	0	0
$685$	11.7648	0.449510
$686$	64.7251	2.47122
$687$	0	0
$688$	12.7862	0.487470
$689$	−3.43910	−0.131019
$690$	0	0
$691$	−31.2713	−1.18962	−0.594808	−	0.803868i	$-0.702772\pi$
$691$	−31.2713	−1.18962	−0.594808	+	0.803868i	$0.702772\pi$
$692$	87.0080	3.30755
$693$	0	0
$694$	38.0722	1.44520
$695$	−8.15371	−0.309288
$696$	0	0
$697$	15.1751	0.574799
$698$	54.3320	2.05650
$699$	0	0
$700$	15.6644	0.592060
$701$	6.02456	0.227544	0.113772	−	0.993507i	$-0.463707\pi$
$701$	6.02456	0.227544	0.113772	+	0.993507i	$0.463707\pi$
$702$	0	0
$703$	9.53948	0.359788
$704$	−13.9572	−0.526030
$705$	0	0
$706$	17.5296	0.659736
$707$	−52.4941	−1.97424
$708$	0	0
$709$	37.6974	1.41576	0.707878	−	0.706335i	$-0.249653\pi$
$709$	37.6974	1.41576	0.707878	+	0.706335i	$0.249653\pi$
$710$	10.1249	0.379982
$711$	0	0
$712$	0.350269	0.0131269
$713$	−2.19656	−0.0822617
$714$	0	0
$715$	−0.978577	−0.0365967
$716$	23.3288	0.871840
$717$	0	0
$718$	−4.39312	−0.163950
$719$	−19.8708	−0.741058	−0.370529	−	0.928821i	$-0.620824\pi$
$719$	−19.8708	−0.741058	−0.370529	+	0.928821i	$0.620824\pi$
$720$	0	0
$721$	−88.5376	−3.29731
$722$	−10.1077	−0.376171
$723$	0	0
$724$	−71.8799	−2.67139
$725$	8.02877	0.298181
$726$	0	0
$727$	15.8757	0.588796	0.294398	−	0.955683i	$-0.404881\pi$
$727$	15.8757	0.588796	0.294398	+	0.955683i	$0.404881\pi$
$728$	13.3717	0.495588
$729$	0	0
$730$	30.7005	1.13628
$731$	−14.3503	−0.530764
$732$	0	0
$733$	37.1751	1.37309	0.686547	−	0.727085i	$-0.259125\pi$
$733$	37.1751	1.37309	0.686547	+	0.727085i	$0.259125\pi$
$734$	−10.9687	−0.404863
$735$	0	0
$736$	4.17513	0.153898
$737$	−13.2614	−0.488492
$738$	0	0
$739$	44.0189	1.61926	0.809631	−	0.586940i	$-0.199667\pi$
$739$	44.0189	1.61926	0.809631	+	0.586940i	$0.199667\pi$
$740$	8.68585	0.319298
$741$	0	0
$742$	−42.3748	−1.55563
$743$	−51.5296	−1.89044	−0.945219	−	0.326437i	$-0.894152\pi$
$743$	−51.5296	−1.89044	−0.945219	+	0.326437i	$0.894152\pi$
$744$	0	0
$745$	−3.73183	−0.136724
$746$	43.6791	1.59921
$747$	0	0
$748$	−5.37169	−0.196409
$749$	−24.4366	−0.892893
$750$	0	0
$751$	−3.91790	−0.142966	−0.0714832	−	0.997442i	$-0.522773\pi$
$751$	−3.91790	−0.142966	−0.0714832	+	0.997442i	$0.522773\pi$
$752$	−1.84208	−0.0671736
$753$	0	0
$754$	16.0575	0.584781
$755$	−16.6430	−0.605701
$756$	0	0
$757$	25.7606	0.936285	0.468142	−	0.883653i	$-0.344923\pi$
$757$	25.7606	0.936285	0.468142	+	0.883653i	$0.344923\pi$
$758$	69.9865	2.54203
$759$	0	0
$760$	13.3717	0.485042
$761$	15.7507	0.570964	0.285482	−	0.958384i	$-0.407846\pi$
$761$	15.7507	0.570964	0.285482	+	0.958384i	$0.407846\pi$
$762$	0	0
$763$	−56.1972	−2.03447
$764$	−17.2860	−0.625386
$765$	0	0
$766$	−48.1972	−1.74143
$767$	12.8108	0.462571
$768$	0	0
$769$	−19.0031	−0.685271	−0.342635	−	0.939468i	$-0.611320\pi$
$769$	−19.0031	−0.685271	−0.342635	+	0.939468i	$0.611320\pi$
$770$	−12.0575	−0.434524
$771$	0	0
$772$	21.1365	0.760719
$773$	41.7795	1.50270	0.751352	−	0.659902i	$-0.229402\pi$
$773$	41.7795	1.50270	0.751352	+	0.659902i	$0.229402\pi$
$774$	0	0
$775$	2.19656	0.0789027
$776$	40.3503	1.44849
$777$	0	0
$778$	5.47208	0.196183
$779$	43.3043	1.55154
$780$	0	0
$781$	−4.95402	−0.177269
$782$	3.14637	0.112514
$783$	0	0
$784$	15.7392	0.562113
$785$	13.2327	0.472294
$786$	0	0
$787$	−21.8610	−0.779260	−0.389630	−	0.920972i	$-0.627397\pi$
$787$	−21.8610	−0.779260	−0.389630	+	0.920972i	$0.627397\pi$
$788$	85.1659	3.03391
$789$	0	0
$790$	−1.37169	−0.0488026
$791$	63.4298	2.25531
$792$	0	0
$793$	−4.97858	−0.176794
$794$	68.2646	2.42262
$795$	0	0
$796$	−13.4574	−0.476984
$797$	27.8280	0.985718	0.492859	−	0.870109i	$-0.335952\pi$
$797$	27.8280	0.985718	0.492859	+	0.870109i	$0.335952\pi$
$798$	0	0
$799$	2.06740	0.0731395
$800$	−4.17513	−0.147613
$801$	0	0
$802$	34.7581	1.22735
$803$	−15.0214	−0.530095
$804$	0	0
$805$	4.48929	0.158227
$806$	4.39312	0.154741
$807$	0	0
$808$	−40.8009	−1.43537
$809$	21.0565	0.740306	0.370153	−	0.928971i	$-0.379305\pi$
$809$	21.0565	0.740306	0.370153	+	0.928971i	$0.379305\pi$
$810$	0	0
$811$	−16.8249	−0.590801	−0.295400	−	0.955374i	$-0.595453\pi$
$811$	−16.8249	−0.590801	−0.295400	+	0.955374i	$0.595453\pi$
$812$	125.766	4.41352
$813$	0	0
$814$	−6.68585	−0.234339
$815$	−16.3931	−0.574226
$816$	0	0
$817$	−40.9504	−1.43267
$818$	−89.0754	−3.11445
$819$	0	0
$820$	39.4292	1.37693
$821$	2.59388	0.0905272	0.0452636	−	0.998975i	$-0.485587\pi$
$821$	2.59388	0.0905272	0.0452636	+	0.998975i	$0.485587\pi$
$822$	0	0
$823$	0.786230	0.0274063	0.0137031	−	0.999906i	$-0.495638\pi$
$823$	0.786230	0.0274063	0.0137031	+	0.999906i	$0.495638\pi$
$824$	−68.8156	−2.39731
$825$	0	0
$826$	157.848	5.49224
$827$	−10.2211	−0.355423	−0.177712	−	0.984083i	$-0.556869\pi$
$827$	−10.2211	−0.355423	−0.177712	+	0.984083i	$0.556869\pi$
$828$	0	0
$829$	−8.97437	−0.311693	−0.155846	−	0.987781i	$-0.549810\pi$
$829$	−8.97437	−0.311693	−0.155846	+	0.987781i	$0.549810\pi$
$830$	13.2039	0.458314
$831$	0	0
$832$	−10.3931	−0.360316
$833$	−17.6644	−0.612036
$834$	0	0
$835$	3.04598	0.105411
$836$	−15.3288	−0.530159
$837$	0	0
$838$	−51.1856	−1.76818
$839$	−19.6069	−0.676905	−0.338452	−	0.940984i	$-0.609903\pi$
$839$	−19.6069	−0.676905	−0.338452	+	0.940984i	$0.609903\pi$
$840$	0	0
$841$	35.4611	1.22280
$842$	50.2302	1.73105
$843$	0	0
$844$	−2.87819	−0.0990715
$845$	12.2713	0.422146
$846$	0	0
$847$	−43.4826	−1.49408
$848$	−4.82065	−0.165542
$849$	0	0
$850$	−3.14637	−0.107919
$851$	2.48929	0.0853317
$852$	0	0
$853$	−14.3650	−0.491847	−0.245923	−	0.969289i	$-0.579091\pi$
$853$	−14.3650	−0.491847	−0.245923	+	0.969289i	$0.579091\pi$
$854$	−61.3435	−2.09913
$855$	0	0
$856$	−18.9933	−0.649177
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	−4.85425	−0.165625	−0.0828125	−	0.996565i	$-0.526390\pi$
$859$	−4.85425	−0.165625	−0.0828125	+	0.996565i	$0.526390\pi$
$860$	−37.2860	−1.27144
$861$	0	0
$862$	−69.8715	−2.37983
$863$	−41.8139	−1.42336	−0.711681	−	0.702503i	$-0.752066\pi$
$863$	−41.8139	−1.42336	−0.711681	+	0.702503i	$0.752066\pi$
$864$	0	0
$865$	−24.9357	−0.847840
$866$	−2.16779	−0.0736644
$867$	0	0
$868$	34.4078	1.16788
$869$	0.671153	0.0227673
$870$	0	0
$871$	−9.87506	−0.334604
$872$	−43.6791	−1.47916
$873$	0	0
$874$	8.97858	0.303705
$875$	−4.48929	−0.151766
$876$	0	0
$877$	−1.27973	−0.0432134	−0.0216067	−	0.999767i	$-0.506878\pi$
$877$	−1.27973	−0.0432134	−0.0216067	+	0.999767i	$0.506878\pi$
$878$	−50.7925	−1.71416
$879$	0	0
$880$	−1.37169	−0.0462397
$881$	−52.9687	−1.78456	−0.892281	−	0.451481i	$-0.850896\pi$
$881$	−52.9687	−1.78456	−0.892281	+	0.451481i	$0.850896\pi$
$882$	0	0
$883$	−17.5479	−0.590534	−0.295267	−	0.955415i	$-0.595409\pi$
$883$	−17.5479	−0.590534	−0.295267	+	0.955415i	$0.595409\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−12.1923	−0.409610
$887$	11.2713	0.378453	0.189227	−	0.981933i	$-0.439402\pi$
$887$	11.2713	0.378453	0.189227	+	0.981933i	$0.439402\pi$
$888$	0	0
$889$	24.6100	0.825394
$890$	−0.235192	−0.00788367
$891$	0	0
$892$	19.4145	0.650047
$893$	5.89962	0.197423
$894$	0	0
$895$	−6.68585	−0.223483
$896$	−90.5720	−3.02580
$897$	0	0
$898$	25.4391	0.848914
$899$	17.6357	0.588182
$900$	0	0
$901$	5.41033	0.180244
$902$	−30.3503	−1.01055
$903$	0	0
$904$	49.3007	1.63972
$905$	20.6002	0.684772
$906$	0	0
$907$	27.2045	0.903311	0.451656	−	0.892192i	$-0.350834\pi$
$907$	27.2045	0.903311	0.451656	+	0.892192i	$0.350834\pi$
$908$	−57.8715	−1.92053
$909$	0	0
$910$	−8.97858	−0.297637
$911$	3.41454	0.113129	0.0565643	−	0.998399i	$-0.481985\pi$
$911$	3.41454	0.113129	0.0565643	+	0.998399i	$0.481985\pi$
$912$	0	0
$913$	−6.46052	−0.213812
$914$	63.9044	2.11377
$915$	0	0
$916$	23.1793	0.765867
$917$	0	0
$918$	0	0
$919$	20.4998	0.676225	0.338113	−	0.941106i	$-0.390212\pi$
$919$	20.4998	0.676225	0.338113	+	0.941106i	$0.390212\pi$
$920$	3.48929	0.115038
$921$	0	0
$922$	−47.3435	−1.55918
$923$	−3.68898	−0.121424
$924$	0	0
$925$	−2.48929	−0.0818473
$926$	18.8207	0.618485
$927$	0	0
$928$	−33.5212	−1.10039
$929$	−38.2640	−1.25540	−0.627700	−	0.778455i	$-0.716004\pi$
$929$	−38.2640	−1.25540	−0.627700	+	0.778455i	$0.716004\pi$
$930$	0	0
$931$	−50.4078	−1.65205
$932$	50.2217	1.64507
$933$	0	0
$934$	45.5886	1.49170
$935$	1.53948	0.0503464
$936$	0	0
$937$	35.6791	1.16559	0.582793	−	0.812621i	$-0.301960\pi$
$937$	35.6791	1.16559	0.582793	+	0.812621i	$0.301960\pi$
$938$	−121.676	−3.97285
$939$	0	0
$940$	5.37169	0.175205
$941$	−15.2321	−0.496551	−0.248275	−	0.968689i	$-0.579864\pi$
$941$	−15.2321	−0.496551	−0.248275	+	0.968689i	$0.579864\pi$
$942$	0	0
$943$	11.3001	0.367981
$944$	17.9572	0.584456
$945$	0	0
$946$	28.7005	0.933135
$947$	−6.29273	−0.204486	−0.102243	−	0.994759i	$-0.532602\pi$
$947$	−6.29273	−0.204486	−0.102243	+	0.994759i	$0.532602\pi$
$948$	0	0
$949$	−11.1856	−0.363100
$950$	−8.97858	−0.291304
$951$	0	0
$952$	−21.0361	−0.681784
$953$	−20.3418	−0.658937	−0.329469	−	0.944167i	$-0.606870\pi$
$953$	−20.3418	−0.658937	−0.329469	+	0.944167i	$0.606870\pi$
$954$	0	0
$955$	4.95402	0.160308
$956$	45.7367	1.47923
$957$	0	0
$958$	−67.8370	−2.19172
$959$	−52.8156	−1.70551
$960$	0	0
$961$	−26.1751	−0.844359
$962$	−4.97858	−0.160516
$963$	0	0
$964$	43.6791	1.40681
$965$	−6.05754	−0.194999
$966$	0	0
$967$	−37.6461	−1.21062	−0.605309	−	0.795991i	$-0.706950\pi$
$967$	−37.6461	−1.21062	−0.605309	+	0.795991i	$0.706950\pi$
$968$	−33.7967	−1.08627
$969$	0	0
$970$	−27.0937	−0.869925
$971$	29.6216	0.950602	0.475301	−	0.879823i	$-0.342339\pi$
$971$	29.6216	0.950602	0.475301	+	0.879823i	$0.342339\pi$
$972$	0	0
$973$	36.6044	1.17348
$974$	48.7581	1.56231
$975$	0	0
$976$	−6.97858	−0.223379
$977$	−21.0158	−0.672354	−0.336177	−	0.941799i	$-0.609134\pi$
$977$	−21.0158	−0.672354	−0.336177	+	0.941799i	$0.609134\pi$
$978$	0	0
$979$	0.115077	0.00367788
$980$	−45.8971	−1.46613
$981$	0	0
$982$	−82.2694	−2.62532
$983$	27.5647	0.879176	0.439588	−	0.898200i	$-0.355124\pi$
$983$	27.5647	0.879176	0.439588	+	0.898200i	$0.355124\pi$
$984$	0	0
$985$	−24.4078	−0.777697
$986$	−25.2614	−0.804488
$987$	0	0
$988$	−11.4145	−0.363145
$989$	−10.6858	−0.339790
$990$	0	0
$991$	−20.1537	−0.640204	−0.320102	−	0.947383i	$-0.603717\pi$
$991$	−20.1537	−0.640204	−0.320102	+	0.947383i	$0.603717\pi$
$992$	−9.17092	−0.291177
$993$	0	0
$994$	−45.4538	−1.44171
$995$	3.85677	0.122268
$996$	0	0
$997$	14.9295	0.472821	0.236410	−	0.971653i	$-0.424029\pi$
$997$	14.9295	0.472821	0.236410	+	0.971653i	$0.424029\pi$
$998$	−17.1464	−0.542759
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1035.2.a.n.1.3		3
3.2	odd	2		345.2.a.j.1.1	✓	3
5.4	even	2		5175.2.a.br.1.1		3
12.11	even	2		5520.2.a.by.1.1		3
15.2	even	4		1725.2.b.u.1174.1		6
15.8	even	4		1725.2.b.u.1174.6		6
15.14	odd	2		1725.2.a.bi.1.3		3
69.68	even	2		7935.2.a.u.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.j.1.1	✓	3	3.2	odd	2
1035.2.a.n.1.3		3	1.1	even	1	trivial
1725.2.a.bi.1.3		3	15.14	odd	2
1725.2.b.u.1174.1		6	15.2	even	4
1725.2.b.u.1174.6		6	15.8	even	4
5175.2.a.br.1.1		3	5.4	even	2
5520.2.a.by.1.1		3	12.11	even	2
7935.2.a.u.1.1		3	69.68	even	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:	\( N \)	\(=\)	\( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1035.a (trivial)

\(f(q)\)	\(=\)	\(q+2.34292 q^{2} +3.48929 q^{4} -1.00000 q^{5} +4.48929 q^{7} +3.48929 q^{8} +O(q^{10})\)	\(q+2.34292 q^{2} +3.48929 q^{4} -1.00000 q^{5} +4.48929 q^{7} +3.48929 q^{8} -2.34292 q^{10} +1.14637 q^{11} +0.853635 q^{13} +10.5181 q^{14} +1.19656 q^{16} -1.34292 q^{17} -3.83221 q^{19} -3.48929 q^{20} +2.68585 q^{22} -1.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +15.6644 q^{28} +8.02877 q^{29} +2.19656 q^{31} -4.17513 q^{32} -3.14637 q^{34} -4.48929 q^{35} -2.48929 q^{37} -8.97858 q^{38} -3.48929 q^{40} -11.3001 q^{41} +10.6858 q^{43} +4.00000 q^{44} -2.34292 q^{46} -1.53948 q^{47} +13.1537 q^{49} +2.34292 q^{50} +2.97858 q^{52} -4.02877 q^{53} -1.14637 q^{55} +15.6644 q^{56} +18.8108 q^{58} +15.0073 q^{59} -5.83221 q^{61} +5.14637 q^{62} -12.1751 q^{64} -0.853635 q^{65} -11.5682 q^{67} -4.68585 q^{68} -10.5181 q^{70} -4.32150 q^{71} -13.1035 q^{73} -5.83221 q^{74} -13.3717 q^{76} +5.14637 q^{77} +0.585462 q^{79} -1.19656 q^{80} -26.4752 q^{82} -5.63565 q^{83} +1.34292 q^{85} +25.0361 q^{86} +4.00000 q^{88} +0.100384 q^{89} +3.83221 q^{91} -3.48929 q^{92} -3.60688 q^{94} +3.83221 q^{95} +11.5640 q^{97} +30.8181 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + q^2 + 3 * q^4 - 3 * q^5 + 6 * q^7 + 3 * q^8	\( 3 q + q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{7} + 3 q^{8} - q^{10} + 2 q^{11} + 4 q^{13} + 6 q^{14} - q^{16} + 2 q^{17} + 2 q^{19} - 3 q^{20} - 4 q^{22} - 3 q^{23} + 3 q^{25} + 6 q^{26} + 20 q^{28} + 6 q^{29} + 2 q^{31} + 7 q^{32} - 8 q^{34} - 6 q^{35} - 12 q^{38} - 3 q^{40} + 2 q^{41} + 20 q^{43} + 12 q^{44} - q^{46} + 6 q^{47} + 5 q^{49} + q^{50} - 6 q^{52} + 6 q^{53} - 2 q^{55} + 20 q^{56} + 28 q^{58} + 12 q^{59} - 4 q^{61} + 14 q^{62} - 17 q^{64} - 4 q^{65} - 6 q^{67} - 2 q^{68} - 6 q^{70} + 8 q^{71} - 8 q^{73} - 4 q^{74} - 16 q^{76} + 14 q^{77} - 4 q^{79} + q^{80} - 24 q^{82} - 8 q^{83} - 2 q^{85} + 24 q^{86} + 12 q^{88} - 6 q^{89} - 2 q^{91} - 3 q^{92} - 20 q^{94} - 2 q^{95} + 14 q^{97} + 31 q^{98}+O(q^{100}) \) 3 * q + q^2 + 3 * q^4 - 3 * q^5 + 6 * q^7 + 3 * q^8 - q^10 + 2 * q^11 + 4 * q^13 + 6 * q^14 - q^16 + 2 * q^17 + 2 * q^19 - 3 * q^20 - 4 * q^22 - 3 * q^23 + 3 * q^25 + 6 * q^26 + 20 * q^28 + 6 * q^29 + 2 * q^31 + 7 * q^32 - 8 * q^34 - 6 * q^35 - 12 * q^38 - 3 * q^40 + 2 * q^41 + 20 * q^43 + 12 * q^44 - q^46 + 6 * q^47 + 5 * q^49 + q^50 - 6 * q^52 + 6 * q^53 - 2 * q^55 + 20 * q^56 + 28 * q^58 + 12 * q^59 - 4 * q^61 + 14 * q^62 - 17 * q^64 - 4 * q^65 - 6 * q^67 - 2 * q^68 - 6 * q^70 + 8 * q^71 - 8 * q^73 - 4 * q^74 - 16 * q^76 + 14 * q^77 - 4 * q^79 + q^80 - 24 * q^82 - 8 * q^83 - 2 * q^85 + 24 * q^86 + 12 * q^88 - 6 * q^89 - 2 * q^91 - 3 * q^92 - 20 * q^94 - 2 * q^95 + 14 * q^97 + 31 * q^98

Introduction

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