LMFDB - Embedded newform 4005.2.a.h.1.1

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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.23607 q^{7} -3.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.23607 q^{7} -3.00000 q^{8} +1.00000 q^{10} -2.47214 q^{11} -2.47214 q^{13} -1.23607 q^{14} -1.00000 q^{16} +2.00000 q^{17} -7.23607 q^{19} -1.00000 q^{20} -2.47214 q^{22} +5.23607 q^{23} +1.00000 q^{25} -2.47214 q^{26} +1.23607 q^{28} +4.47214 q^{29} +3.23607 q^{31} +5.00000 q^{32} +2.00000 q^{34} -1.23607 q^{35} +2.47214 q^{37} -7.23607 q^{38} -3.00000 q^{40} +10.0000 q^{41} +5.23607 q^{43} +2.47214 q^{44} +5.23607 q^{46} -8.00000 q^{47} -5.47214 q^{49} +1.00000 q^{50} +2.47214 q^{52} -2.00000 q^{53} -2.47214 q^{55} +3.70820 q^{56} +4.47214 q^{58} +3.23607 q^{59} +0.472136 q^{61} +3.23607 q^{62} +7.00000 q^{64} -2.47214 q^{65} +6.47214 q^{67} -2.00000 q^{68} -1.23607 q^{70} +14.4721 q^{71} +2.00000 q^{73} +2.47214 q^{74} +7.23607 q^{76} +3.05573 q^{77} +12.9443 q^{79} -1.00000 q^{80} +10.0000 q^{82} +10.7639 q^{83} +2.00000 q^{85} +5.23607 q^{86} +7.41641 q^{88} +1.00000 q^{89} +3.05573 q^{91} -5.23607 q^{92} -8.00000 q^{94} -7.23607 q^{95} -6.00000 q^{97} -5.47214 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^8	$ 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8} + 2 q^{10} + 4 q^{11} + 4 q^{13} + 2 q^{14} - 2 q^{16} + 4 q^{17} - 10 q^{19} - 2 q^{20} + 4 q^{22} + 6 q^{23} + 2 q^{25} + 4 q^{26} - 2 q^{28} + 2 q^{31} + 10 q^{32} + 4 q^{34} + 2 q^{35} - 4 q^{37} - 10 q^{38} - 6 q^{40} + 20 q^{41} + 6 q^{43} - 4 q^{44} + 6 q^{46} - 16 q^{47} - 2 q^{49} + 2 q^{50} - 4 q^{52} - 4 q^{53} + 4 q^{55} - 6 q^{56} + 2 q^{59} - 8 q^{61} + 2 q^{62} + 14 q^{64} + 4 q^{65} + 4 q^{67} - 4 q^{68} + 2 q^{70} + 20 q^{71} + 4 q^{73} - 4 q^{74} + 10 q^{76} + 24 q^{77} + 8 q^{79} - 2 q^{80} + 20 q^{82} + 26 q^{83} + 4 q^{85} + 6 q^{86} - 12 q^{88} + 2 q^{89} + 24 q^{91} - 6 q^{92} - 16 q^{94} - 10 q^{95} - 12 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^8 + 2 * q^10 + 4 * q^11 + 4 * q^13 + 2 * q^14 - 2 * q^16 + 4 * q^17 - 10 * q^19 - 2 * q^20 + 4 * q^22 + 6 * q^23 + 2 * q^25 + 4 * q^26 - 2 * q^28 + 2 * q^31 + 10 * q^32 + 4 * q^34 + 2 * q^35 - 4 * q^37 - 10 * q^38 - 6 * q^40 + 20 * q^41 + 6 * q^43 - 4 * q^44 + 6 * q^46 - 16 * q^47 - 2 * q^49 + 2 * q^50 - 4 * q^52 - 4 * q^53 + 4 * q^55 - 6 * q^56 + 2 * q^59 - 8 * q^61 + 2 * q^62 + 14 * q^64 + 4 * q^65 + 4 * q^67 - 4 * q^68 + 2 * q^70 + 20 * q^71 + 4 * q^73 - 4 * q^74 + 10 * q^76 + 24 * q^77 + 8 * q^79 - 2 * q^80 + 20 * q^82 + 26 * q^83 + 4 * q^85 + 6 * q^86 - 12 * q^88 + 2 * q^89 + 24 * q^91 - 6 * q^92 - 16 * q^94 - 10 * q^95 - 12 * q^97 - 2 * 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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	1.00000	0.316228
$11$	−2.47214	−0.745377	−0.372689	−	0.927957i	$-0.621564\pi$
$11$	−2.47214	−0.745377	−0.372689	+	0.927957i	$0.621564\pi$
$12$	0	0
$13$	−2.47214	−0.685647	−0.342824	−	0.939400i	$-0.611383\pi$
$13$	−2.47214	−0.685647	−0.342824	+	0.939400i	$0.611383\pi$
$14$	−1.23607	−0.330353
$15$	0	0
$16$	−1.00000	−0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−7.23607	−1.66007	−0.830034	−	0.557713i	$-0.811679\pi$
$19$	−7.23607	−1.66007	−0.830034	+	0.557713i	$0.811679\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−2.47214	−0.527061
$23$	5.23607	1.09180	0.545898	−	0.837852i	$-0.316189\pi$
$23$	5.23607	1.09180	0.545898	+	0.837852i	$0.316189\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.47214	−0.484826
$27$	0	0
$28$	1.23607	0.233595
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	3.23607	0.581215	0.290607	−	0.956842i	$-0.406143\pi$
$31$	3.23607	0.581215	0.290607	+	0.956842i	$0.406143\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	2.00000	0.342997
$35$	−1.23607	−0.208934
$36$	0	0
$37$	2.47214	0.406417	0.203208	−	0.979136i	$-0.434863\pi$
$37$	2.47214	0.406417	0.203208	+	0.979136i	$0.434863\pi$
$38$	−7.23607	−1.17385
$39$	0	0
$40$	−3.00000	−0.474342
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	5.23607	0.798493	0.399246	−	0.916844i	$-0.369272\pi$
$43$	5.23607	0.798493	0.399246	+	0.916844i	$0.369272\pi$
$44$	2.47214	0.372689
$45$	0	0
$46$	5.23607	0.772016
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	1.00000	0.141421
$51$	0	0
$52$	2.47214	0.342824
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−2.47214	−0.333343
$56$	3.70820	0.495530
$57$	0	0
$58$	4.47214	0.587220
$59$	3.23607	0.421300	0.210650	−	0.977562i	$-0.432442\pi$
$59$	3.23607	0.421300	0.210650	+	0.977562i	$0.432442\pi$
$60$	0	0
$61$	0.472136	0.0604508	0.0302254	−	0.999543i	$-0.490377\pi$
$61$	0.472136	0.0604508	0.0302254	+	0.999543i	$0.490377\pi$
$62$	3.23607	0.410981
$63$	0	0
$64$	7.00000	0.875000
$65$	−2.47214	−0.306631
$66$	0	0
$67$	6.47214	0.790697	0.395349	−	0.918531i	$-0.370624\pi$
$67$	6.47214	0.790697	0.395349	+	0.918531i	$0.370624\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	−1.23607	−0.147738
$71$	14.4721	1.71753	0.858763	−	0.512373i	$-0.171233\pi$
$71$	14.4721	1.71753	0.858763	+	0.512373i	$0.171233\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	2.47214	0.287380
$75$	0	0
$76$	7.23607	0.830034
$77$	3.05573	0.348233
$78$	0	0
$79$	12.9443	1.45634	0.728172	−	0.685394i	$-0.240370\pi$
$79$	12.9443	1.45634	0.728172	+	0.685394i	$0.240370\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	10.0000	1.10432
$83$	10.7639	1.18150	0.590748	−	0.806856i	$-0.298833\pi$
$83$	10.7639	1.18150	0.590748	+	0.806856i	$0.298833\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	5.23607	0.564620
$87$	0	0
$88$	7.41641	0.790592
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	3.05573	0.320327
$92$	−5.23607	−0.545898
$93$	0	0
$94$	−8.00000	−0.825137
$95$	−7.23607	−0.742405
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	−5.47214	−0.552769
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−4.47214	−0.444994	−0.222497	−	0.974933i	$-0.571421\pi$
$101$	−4.47214	−0.444994	−0.222497	+	0.974933i	$0.571421\pi$
$102$	0	0
$103$	16.6525	1.64082	0.820409	−	0.571778i	$-0.193746\pi$
$103$	16.6525	1.64082	0.820409	+	0.571778i	$0.193746\pi$
$104$	7.41641	0.727239
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−8.94427	−0.864675	−0.432338	−	0.901712i	$-0.642311\pi$
$107$	−8.94427	−0.864675	−0.432338	+	0.901712i	$0.642311\pi$
$108$	0	0
$109$	0.472136	0.0452224	0.0226112	−	0.999744i	$-0.492802\pi$
$109$	0.472136	0.0452224	0.0226112	+	0.999744i	$0.492802\pi$
$110$	−2.47214	−0.235709
$111$	0	0
$112$	1.23607	0.116797
$113$	−20.9443	−1.97027	−0.985136	−	0.171778i	$-0.945049\pi$
$113$	−20.9443	−1.97027	−0.985136	+	0.171778i	$0.945049\pi$
$114$	0	0
$115$	5.23607	0.488266
$116$	−4.47214	−0.415227
$117$	0	0
$118$	3.23607	0.297904
$119$	−2.47214	−0.226620
$120$	0	0
$121$	−4.88854	−0.444413
$122$	0.472136	0.0427452
$123$	0	0
$124$	−3.23607	−0.290607
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.23607	0.464626	0.232313	−	0.972641i	$-0.425371\pi$
$127$	5.23607	0.464626	0.232313	+	0.972641i	$0.425371\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	−2.47214	−0.216821
$131$	18.4721	1.61392	0.806959	−	0.590607i	$-0.201112\pi$
$131$	18.4721	1.61392	0.806959	+	0.590607i	$0.201112\pi$
$132$	0	0
$133$	8.94427	0.775567
$134$	6.47214	0.559107
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	−2.47214	−0.209684	−0.104842	−	0.994489i	$-0.533434\pi$
$139$	−2.47214	−0.209684	−0.104842	+	0.994489i	$0.533434\pi$
$140$	1.23607	0.104467
$141$	0	0
$142$	14.4721	1.21447
$143$	6.11146	0.511066
$144$	0	0
$145$	4.47214	0.371391
$146$	2.00000	0.165521
$147$	0	0
$148$	−2.47214	−0.203208
$149$	−17.4164	−1.42681	−0.713404	−	0.700753i	$-0.752847\pi$
$149$	−17.4164	−1.42681	−0.713404	+	0.700753i	$0.752847\pi$
$150$	0	0
$151$	−0.763932	−0.0621679	−0.0310840	−	0.999517i	$-0.509896\pi$
$151$	−0.763932	−0.0621679	−0.0310840	+	0.999517i	$0.509896\pi$
$152$	21.7082	1.76077
$153$	0	0
$154$	3.05573	0.246238
$155$	3.23607	0.259927
$156$	0	0
$157$	6.94427	0.554213	0.277107	−	0.960839i	$-0.410624\pi$
$157$	6.94427	0.554213	0.277107	+	0.960839i	$0.410624\pi$
$158$	12.9443	1.02979
$159$	0	0
$160$	5.00000	0.395285
$161$	−6.47214	−0.510076
$162$	0	0
$163$	−6.18034	−0.484082	−0.242041	−	0.970266i	$-0.577817\pi$
$163$	−6.18034	−0.484082	−0.242041	+	0.970266i	$0.577817\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	10.7639	0.835443
$167$	15.4164	1.19296	0.596479	−	0.802629i	$-0.296566\pi$
$167$	15.4164	1.19296	0.596479	+	0.802629i	$0.296566\pi$
$168$	0	0
$169$	−6.88854	−0.529888
$170$	2.00000	0.153393
$171$	0	0
$172$	−5.23607	−0.399246
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−1.23607	−0.0934380
$176$	2.47214	0.186344
$177$	0	0
$178$	1.00000	0.0749532
$179$	10.4721	0.782724	0.391362	−	0.920237i	$-0.372004\pi$
$179$	10.4721	0.782724	0.391362	+	0.920237i	$0.372004\pi$
$180$	0	0
$181$	−3.52786	−0.262224	−0.131112	−	0.991368i	$-0.541855\pi$
$181$	−3.52786	−0.262224	−0.131112	+	0.991368i	$0.541855\pi$
$182$	3.05573	0.226506
$183$	0	0
$184$	−15.7082	−1.15802
$185$	2.47214	0.181755
$186$	0	0
$187$	−4.94427	−0.361561
$188$	8.00000	0.583460
$189$	0	0
$190$	−7.23607	−0.524960
$191$	21.7082	1.57075	0.785375	−	0.619020i	$-0.212470\pi$
$191$	21.7082	1.57075	0.785375	+	0.619020i	$0.212470\pi$
$192$	0	0
$193$	−12.9443	−0.931749	−0.465875	−	0.884851i	$-0.654260\pi$
$193$	−12.9443	−0.931749	−0.465875	+	0.884851i	$0.654260\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	5.47214	0.390867
$197$	−2.47214	−0.176132	−0.0880662	−	0.996115i	$-0.528069\pi$
$197$	−2.47214	−0.176132	−0.0880662	+	0.996115i	$0.528069\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	−4.47214	−0.314658
$203$	−5.52786	−0.387980
$204$	0	0
$205$	10.0000	0.698430
$206$	16.6525	1.16023
$207$	0	0
$208$	2.47214	0.171412
$209$	17.8885	1.23738
$210$	0	0
$211$	4.18034	0.287786	0.143893	−	0.989593i	$-0.454038\pi$
$211$	4.18034	0.287786	0.143893	+	0.989593i	$0.454038\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	−8.94427	−0.611418
$215$	5.23607	0.357097
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0.472136	0.0319771
$219$	0	0
$220$	2.47214	0.166671
$221$	−4.94427	−0.332588
$222$	0	0
$223$	−4.94427	−0.331093	−0.165546	−	0.986202i	$-0.552939\pi$
$223$	−4.94427	−0.331093	−0.165546	+	0.986202i	$0.552939\pi$
$224$	−6.18034	−0.412941
$225$	0	0
$226$	−20.9443	−1.39319
$227$	19.4164	1.28871	0.644356	−	0.764726i	$-0.277125\pi$
$227$	19.4164	1.28871	0.644356	+	0.764726i	$0.277125\pi$
$228$	0	0
$229$	16.4721	1.08851	0.544255	−	0.838920i	$-0.316813\pi$
$229$	16.4721	1.08851	0.544255	+	0.838920i	$0.316813\pi$
$230$	5.23607	0.345256
$231$	0	0
$232$	−13.4164	−0.880830
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	−3.23607	−0.210650
$237$	0	0
$238$	−2.47214	−0.160245
$239$	−21.7082	−1.40419	−0.702093	−	0.712085i	$-0.747751\pi$
$239$	−21.7082	−1.40419	−0.702093	+	0.712085i	$0.747751\pi$
$240$	0	0
$241$	1.05573	0.0680054	0.0340027	−	0.999422i	$-0.489175\pi$
$241$	1.05573	0.0680054	0.0340027	+	0.999422i	$0.489175\pi$
$242$	−4.88854	−0.314247
$243$	0	0
$244$	−0.472136	−0.0302254
$245$	−5.47214	−0.349602
$246$	0	0
$247$	17.8885	1.13822
$248$	−9.70820	−0.616472
$249$	0	0
$250$	1.00000	0.0632456
$251$	26.4721	1.67091	0.835453	−	0.549562i	$-0.185205\pi$
$251$	26.4721	1.67091	0.835453	+	0.549562i	$0.185205\pi$
$252$	0	0
$253$	−12.9443	−0.813799
$254$	5.23607	0.328540
$255$	0	0
$256$	−17.0000	−1.06250
$257$	5.05573	0.315368	0.157684	−	0.987490i	$-0.449597\pi$
$257$	5.05573	0.315368	0.157684	+	0.987490i	$0.449597\pi$
$258$	0	0
$259$	−3.05573	−0.189874
$260$	2.47214	0.153315
$261$	0	0
$262$	18.4721	1.14121
$263$	22.8328	1.40793	0.703966	−	0.710234i	$-0.251411\pi$
$263$	22.8328	1.40793	0.703966	+	0.710234i	$0.251411\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	8.94427	0.548408
$267$	0	0
$268$	−6.47214	−0.395349
$269$	1.41641	0.0863599	0.0431800	−	0.999067i	$-0.486251\pi$
$269$	1.41641	0.0863599	0.0431800	+	0.999067i	$0.486251\pi$
$270$	0	0
$271$	−17.5279	−1.06474	−0.532371	−	0.846511i	$-0.678699\pi$
$271$	−17.5279	−1.06474	−0.532371	+	0.846511i	$0.678699\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−8.00000	−0.483298
$275$	−2.47214	−0.149075
$276$	0	0
$277$	14.9443	0.897914	0.448957	−	0.893553i	$-0.351796\pi$
$277$	14.9443	0.897914	0.448957	+	0.893553i	$0.351796\pi$
$278$	−2.47214	−0.148269
$279$	0	0
$280$	3.70820	0.221608
$281$	−18.9443	−1.13012	−0.565060	−	0.825050i	$-0.691147\pi$
$281$	−18.9443	−1.13012	−0.565060	+	0.825050i	$0.691147\pi$
$282$	0	0
$283$	8.94427	0.531682	0.265841	−	0.964017i	$-0.414350\pi$
$283$	8.94427	0.531682	0.265841	+	0.964017i	$0.414350\pi$
$284$	−14.4721	−0.858763
$285$	0	0
$286$	6.11146	0.361378
$287$	−12.3607	−0.729628
$288$	0	0
$289$	−13.0000	−0.764706
$290$	4.47214	0.262613
$291$	0	0
$292$	−2.00000	−0.117041
$293$	11.4164	0.666954	0.333477	−	0.942758i	$-0.391778\pi$
$293$	11.4164	0.666954	0.333477	+	0.942758i	$0.391778\pi$
$294$	0	0
$295$	3.23607	0.188411
$296$	−7.41641	−0.431070
$297$	0	0
$298$	−17.4164	−1.00891
$299$	−12.9443	−0.748587
$300$	0	0
$301$	−6.47214	−0.373048
$302$	−0.763932	−0.0439593
$303$	0	0
$304$	7.23607	0.415017
$305$	0.472136	0.0270344
$306$	0	0
$307$	11.4164	0.651569	0.325784	−	0.945444i	$-0.394372\pi$
$307$	11.4164	0.651569	0.325784	+	0.945444i	$0.394372\pi$
$308$	−3.05573	−0.174116
$309$	0	0
$310$	3.23607	0.183796
$311$	−11.4164	−0.647365	−0.323683	−	0.946166i	$-0.604921\pi$
$311$	−11.4164	−0.647365	−0.323683	+	0.946166i	$0.604921\pi$
$312$	0	0
$313$	17.8885	1.01112	0.505560	−	0.862791i	$-0.331286\pi$
$313$	17.8885	1.01112	0.505560	+	0.862791i	$0.331286\pi$
$314$	6.94427	0.391888
$315$	0	0
$316$	−12.9443	−0.728172
$317$	6.94427	0.390029	0.195015	−	0.980800i	$-0.437525\pi$
$317$	6.94427	0.390029	0.195015	+	0.980800i	$0.437525\pi$
$318$	0	0
$319$	−11.0557	−0.619002
$320$	7.00000	0.391312
$321$	0	0
$322$	−6.47214	−0.360678
$323$	−14.4721	−0.805251
$324$	0	0
$325$	−2.47214	−0.137129
$326$	−6.18034	−0.342297
$327$	0	0
$328$	−30.0000	−1.65647
$329$	9.88854	0.545173
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−10.7639	−0.590748
$333$	0	0
$334$	15.4164	0.843548
$335$	6.47214	0.353611
$336$	0	0
$337$	−0.944272	−0.0514378	−0.0257189	−	0.999669i	$-0.508187\pi$
$337$	−0.944272	−0.0514378	−0.0257189	+	0.999669i	$0.508187\pi$
$338$	−6.88854	−0.374687
$339$	0	0
$340$	−2.00000	−0.108465
$341$	−8.00000	−0.433224
$342$	0	0
$343$	15.4164	0.832408
$344$	−15.7082	−0.846930
$345$	0	0
$346$	6.00000	0.322562
$347$	18.8328	1.01100	0.505499	−	0.862827i	$-0.331308\pi$
$347$	18.8328	1.01100	0.505499	+	0.862827i	$0.331308\pi$
$348$	0	0
$349$	34.3607	1.83929	0.919643	−	0.392756i	$-0.128478\pi$
$349$	34.3607	1.83929	0.919643	+	0.392756i	$0.128478\pi$
$350$	−1.23607	−0.0660706
$351$	0	0
$352$	−12.3607	−0.658826
$353$	−32.9443	−1.75345	−0.876723	−	0.480995i	$-0.840276\pi$
$353$	−32.9443	−1.75345	−0.876723	+	0.480995i	$0.840276\pi$
$354$	0	0
$355$	14.4721	0.768101
$356$	−1.00000	−0.0529999
$357$	0	0
$358$	10.4721	0.553470
$359$	−1.70820	−0.0901556	−0.0450778	−	0.998983i	$-0.514354\pi$
$359$	−1.70820	−0.0901556	−0.0450778	+	0.998983i	$0.514354\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	−3.52786	−0.185420
$363$	0	0
$364$	−3.05573	−0.160164
$365$	2.00000	0.104685
$366$	0	0
$367$	−25.8885	−1.35137	−0.675685	−	0.737190i	$-0.736152\pi$
$367$	−25.8885	−1.35137	−0.675685	+	0.737190i	$0.736152\pi$
$368$	−5.23607	−0.272949
$369$	0	0
$370$	2.47214	0.128520
$371$	2.47214	0.128347
$372$	0	0
$373$	−28.8328	−1.49291	−0.746453	−	0.665438i	$-0.768245\pi$
$373$	−28.8328	−1.49291	−0.746453	+	0.665438i	$0.768245\pi$
$374$	−4.94427	−0.255662
$375$	0	0
$376$	24.0000	1.23771
$377$	−11.0557	−0.569399
$378$	0	0
$379$	−36.5410	−1.87699	−0.938493	−	0.345298i	$-0.887778\pi$
$379$	−36.5410	−1.87699	−0.938493	+	0.345298i	$0.887778\pi$
$380$	7.23607	0.371202
$381$	0	0
$382$	21.7082	1.11069
$383$	−5.23607	−0.267551	−0.133775	−	0.991012i	$-0.542710\pi$
$383$	−5.23607	−0.267551	−0.133775	+	0.991012i	$0.542710\pi$
$384$	0	0
$385$	3.05573	0.155734
$386$	−12.9443	−0.658846
$387$	0	0
$388$	6.00000	0.304604
$389$	21.4164	1.08585	0.542927	−	0.839780i	$-0.317316\pi$
$389$	21.4164	1.08585	0.542927	+	0.839780i	$0.317316\pi$
$390$	0	0
$391$	10.4721	0.529599
$392$	16.4164	0.829154
$393$	0	0
$394$	−2.47214	−0.124544
$395$	12.9443	0.651297
$396$	0	0
$397$	−1.52786	−0.0766813	−0.0383406	−	0.999265i	$-0.512207\pi$
$397$	−1.52786	−0.0766813	−0.0383406	+	0.999265i	$0.512207\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−7.52786	−0.375924	−0.187962	−	0.982176i	$-0.560188\pi$
$401$	−7.52786	−0.375924	−0.187962	+	0.982176i	$0.560188\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	4.47214	0.222497
$405$	0	0
$406$	−5.52786	−0.274343
$407$	−6.11146	−0.302934
$408$	0	0
$409$	−21.4164	−1.05897	−0.529487	−	0.848318i	$-0.677615\pi$
$409$	−21.4164	−1.05897	−0.529487	+	0.848318i	$0.677615\pi$
$410$	10.0000	0.493865
$411$	0	0
$412$	−16.6525	−0.820409
$413$	−4.00000	−0.196827
$414$	0	0
$415$	10.7639	0.528381
$416$	−12.3607	−0.606032
$417$	0	0
$418$	17.8885	0.874957
$419$	18.2918	0.893613	0.446806	−	0.894631i	$-0.352561\pi$
$419$	18.2918	0.893613	0.446806	+	0.894631i	$0.352561\pi$
$420$	0	0
$421$	31.3050	1.52571	0.762855	−	0.646570i	$-0.223797\pi$
$421$	31.3050	1.52571	0.762855	+	0.646570i	$0.223797\pi$
$422$	4.18034	0.203496
$423$	0	0
$424$	6.00000	0.291386
$425$	2.00000	0.0970143
$426$	0	0
$427$	−0.583592	−0.0282420
$428$	8.94427	0.432338
$429$	0	0
$430$	5.23607	0.252506
$431$	13.1246	0.632190	0.316095	−	0.948727i	$-0.397628\pi$
$431$	13.1246	0.632190	0.316095	+	0.948727i	$0.397628\pi$
$432$	0	0
$433$	−28.9443	−1.39097	−0.695486	−	0.718539i	$-0.744811\pi$
$433$	−28.9443	−1.39097	−0.695486	+	0.718539i	$0.744811\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−0.472136	−0.0226112
$437$	−37.8885	−1.81245
$438$	0	0
$439$	−32.1803	−1.53588	−0.767942	−	0.640519i	$-0.778719\pi$
$439$	−32.1803	−1.53588	−0.767942	+	0.640519i	$0.778719\pi$
$440$	7.41641	0.353563
$441$	0	0
$442$	−4.94427	−0.235175
$443$	21.8885	1.03996	0.519978	−	0.854180i	$-0.325940\pi$
$443$	21.8885	1.03996	0.519978	+	0.854180i	$0.325940\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−4.94427	−0.234118
$447$	0	0
$448$	−8.65248	−0.408791
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	−24.7214	−1.16408
$452$	20.9443	0.985136
$453$	0	0
$454$	19.4164	0.911257
$455$	3.05573	0.143255
$456$	0	0
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	16.4721	0.769692
$459$	0	0
$460$	−5.23607	−0.244133
$461$	−22.3607	−1.04144	−0.520720	−	0.853727i	$-0.674337\pi$
$461$	−22.3607	−1.04144	−0.520720	+	0.853727i	$0.674337\pi$
$462$	0	0
$463$	−17.8885	−0.831351	−0.415676	−	0.909513i	$-0.636455\pi$
$463$	−17.8885	−0.831351	−0.415676	+	0.909513i	$0.636455\pi$
$464$	−4.47214	−0.207614
$465$	0	0
$466$	26.0000	1.20443
$467$	−24.3607	−1.12728	−0.563639	−	0.826021i	$-0.690599\pi$
$467$	−24.3607	−1.12728	−0.563639	+	0.826021i	$0.690599\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	−8.00000	−0.369012
$471$	0	0
$472$	−9.70820	−0.446856
$473$	−12.9443	−0.595178
$474$	0	0
$475$	−7.23607	−0.332014
$476$	2.47214	0.113310
$477$	0	0
$478$	−21.7082	−0.992910
$479$	−12.9443	−0.591439	−0.295719	−	0.955275i	$-0.595559\pi$
$479$	−12.9443	−0.591439	−0.295719	+	0.955275i	$0.595559\pi$
$480$	0	0
$481$	−6.11146	−0.278658
$482$	1.05573	0.0480871
$483$	0	0
$484$	4.88854	0.222207
$485$	−6.00000	−0.272446
$486$	0	0
$487$	41.3050	1.87171	0.935853	−	0.352391i	$-0.114631\pi$
$487$	41.3050	1.87171	0.935853	+	0.352391i	$0.114631\pi$
$488$	−1.41641	−0.0641178
$489$	0	0
$490$	−5.47214	−0.247206
$491$	19.5967	0.884389	0.442194	−	0.896919i	$-0.354200\pi$
$491$	19.5967	0.884389	0.442194	+	0.896919i	$0.354200\pi$
$492$	0	0
$493$	8.94427	0.402830
$494$	17.8885	0.804844
$495$	0	0
$496$	−3.23607	−0.145304
$497$	−17.8885	−0.802411
$498$	0	0
$499$	−13.1246	−0.587538	−0.293769	−	0.955876i	$-0.594910\pi$
$499$	−13.1246	−0.587538	−0.293769	+	0.955876i	$0.594910\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	26.4721	1.18151
$503$	0.875388	0.0390316	0.0195158	−	0.999810i	$-0.493788\pi$
$503$	0.875388	0.0390316	0.0195158	+	0.999810i	$0.493788\pi$
$504$	0	0
$505$	−4.47214	−0.199007
$506$	−12.9443	−0.575443
$507$	0	0
$508$	−5.23607	−0.232313
$509$	16.4721	0.730115	0.365057	−	0.930985i	$-0.381049\pi$
$509$	16.4721	0.730115	0.365057	+	0.930985i	$0.381049\pi$
$510$	0	0
$511$	−2.47214	−0.109361
$512$	−11.0000	−0.486136
$513$	0	0
$514$	5.05573	0.222999
$515$	16.6525	0.733796
$516$	0	0
$517$	19.7771	0.869795
$518$	−3.05573	−0.134261
$519$	0	0
$520$	7.41641	0.325231
$521$	1.05573	0.0462523	0.0231261	−	0.999733i	$-0.492638\pi$
$521$	1.05573	0.0462523	0.0231261	+	0.999733i	$0.492638\pi$
$522$	0	0
$523$	−17.5279	−0.766440	−0.383220	−	0.923657i	$-0.625185\pi$
$523$	−17.5279	−0.766440	−0.383220	+	0.923657i	$0.625185\pi$
$524$	−18.4721	−0.806959
$525$	0	0
$526$	22.8328	0.995558
$527$	6.47214	0.281931
$528$	0	0
$529$	4.41641	0.192018
$530$	−2.00000	−0.0868744
$531$	0	0
$532$	−8.94427	−0.387783
$533$	−24.7214	−1.07080
$534$	0	0
$535$	−8.94427	−0.386695
$536$	−19.4164	−0.838661
$537$	0	0
$538$	1.41641	0.0610657
$539$	13.5279	0.582686
$540$	0	0
$541$	25.4164	1.09274	0.546368	−	0.837545i	$-0.316010\pi$
$541$	25.4164	1.09274	0.546368	+	0.837545i	$0.316010\pi$
$542$	−17.5279	−0.752886
$543$	0	0
$544$	10.0000	0.428746
$545$	0.472136	0.0202241
$546$	0	0
$547$	−12.0689	−0.516028	−0.258014	−	0.966141i	$-0.583068\pi$
$547$	−12.0689	−0.516028	−0.258014	+	0.966141i	$0.583068\pi$
$548$	8.00000	0.341743
$549$	0	0
$550$	−2.47214	−0.105412
$551$	−32.3607	−1.37861
$552$	0	0
$553$	−16.0000	−0.680389
$554$	14.9443	0.634921
$555$	0	0
$556$	2.47214	0.104842
$557$	13.5279	0.573194	0.286597	−	0.958051i	$-0.407476\pi$
$557$	13.5279	0.573194	0.286597	+	0.958051i	$0.407476\pi$
$558$	0	0
$559$	−12.9443	−0.547484
$560$	1.23607	0.0522334
$561$	0	0
$562$	−18.9443	−0.799116
$563$	−4.87539	−0.205473	−0.102737	−	0.994709i	$-0.532760\pi$
$563$	−4.87539	−0.205473	−0.102737	+	0.994709i	$0.532760\pi$
$564$	0	0
$565$	−20.9443	−0.881132
$566$	8.94427	0.375956
$567$	0	0
$568$	−43.4164	−1.82171
$569$	−34.0000	−1.42535	−0.712677	−	0.701492i	$-0.752517\pi$
$569$	−34.0000	−1.42535	−0.712677	+	0.701492i	$0.752517\pi$
$570$	0	0
$571$	−18.2918	−0.765488	−0.382744	−	0.923854i	$-0.625021\pi$
$571$	−18.2918	−0.765488	−0.382744	+	0.923854i	$0.625021\pi$
$572$	−6.11146	−0.255533
$573$	0	0
$574$	−12.3607	−0.515925
$575$	5.23607	0.218359
$576$	0	0
$577$	34.8328	1.45011	0.725055	−	0.688691i	$-0.241815\pi$
$577$	34.8328	1.45011	0.725055	+	0.688691i	$0.241815\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	−4.47214	−0.185695
$581$	−13.3050	−0.551982
$582$	0	0
$583$	4.94427	0.204771
$584$	−6.00000	−0.248282
$585$	0	0
$586$	11.4164	0.471607
$587$	32.3607	1.33567	0.667834	−	0.744310i	$-0.267222\pi$
$587$	32.3607	1.33567	0.667834	+	0.744310i	$0.267222\pi$
$588$	0	0
$589$	−23.4164	−0.964856
$590$	3.23607	0.133227
$591$	0	0
$592$	−2.47214	−0.101604
$593$	−4.94427	−0.203037	−0.101518	−	0.994834i	$-0.532370\pi$
$593$	−4.94427	−0.203037	−0.101518	+	0.994834i	$0.532370\pi$
$594$	0	0
$595$	−2.47214	−0.101348
$596$	17.4164	0.713404
$597$	0	0
$598$	−12.9443	−0.529331
$599$	−23.5967	−0.964137	−0.482068	−	0.876134i	$-0.660114\pi$
$599$	−23.5967	−0.964137	−0.482068	+	0.876134i	$0.660114\pi$
$600$	0	0
$601$	−3.88854	−0.158617	−0.0793085	−	0.996850i	$-0.525271\pi$
$601$	−3.88854	−0.158617	−0.0793085	+	0.996850i	$0.525271\pi$
$602$	−6.47214	−0.263785
$603$	0	0
$604$	0.763932	0.0310840
$605$	−4.88854	−0.198748
$606$	0	0
$607$	−5.52786	−0.224369	−0.112185	−	0.993687i	$-0.535785\pi$
$607$	−5.52786	−0.224369	−0.112185	+	0.993687i	$0.535785\pi$
$608$	−36.1803	−1.46731
$609$	0	0
$610$	0.472136	0.0191162
$611$	19.7771	0.800095
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	11.4164	0.460729
$615$	0	0
$616$	−9.16718	−0.369356
$617$	−27.7771	−1.11826	−0.559132	−	0.829079i	$-0.688865\pi$
$617$	−27.7771	−1.11826	−0.559132	+	0.829079i	$0.688865\pi$
$618$	0	0
$619$	29.8885	1.20132	0.600661	−	0.799504i	$-0.294904\pi$
$619$	29.8885	1.20132	0.600661	+	0.799504i	$0.294904\pi$
$620$	−3.23607	−0.129964
$621$	0	0
$622$	−11.4164	−0.457756
$623$	−1.23607	−0.0495220
$624$	0	0
$625$	1.00000	0.0400000
$626$	17.8885	0.714970
$627$	0	0
$628$	−6.94427	−0.277107
$629$	4.94427	0.197141
$630$	0	0
$631$	−25.5279	−1.01625	−0.508124	−	0.861284i	$-0.669661\pi$
$631$	−25.5279	−1.01625	−0.508124	+	0.861284i	$0.669661\pi$
$632$	−38.8328	−1.54469
$633$	0	0
$634$	6.94427	0.275792
$635$	5.23607	0.207787
$636$	0	0
$637$	13.5279	0.535993
$638$	−11.0557	−0.437700
$639$	0	0
$640$	−3.00000	−0.118585
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	0	0
$643$	24.3607	0.960691	0.480346	−	0.877079i	$-0.340511\pi$
$643$	24.3607	0.960691	0.480346	+	0.877079i	$0.340511\pi$
$644$	6.47214	0.255038
$645$	0	0
$646$	−14.4721	−0.569399
$647$	0.291796	0.0114717	0.00573584	−	0.999984i	$-0.498174\pi$
$647$	0.291796	0.0114717	0.00573584	+	0.999984i	$0.498174\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	−2.47214	−0.0969651
$651$	0	0
$652$	6.18034	0.242041
$653$	−31.4164	−1.22942	−0.614710	−	0.788754i	$-0.710727\pi$
$653$	−31.4164	−1.22942	−0.614710	+	0.788754i	$0.710727\pi$
$654$	0	0
$655$	18.4721	0.721766
$656$	−10.0000	−0.390434
$657$	0	0
$658$	9.88854	0.385496
$659$	18.4721	0.719572	0.359786	−	0.933035i	$-0.382850\pi$
$659$	18.4721	0.719572	0.359786	+	0.933035i	$0.382850\pi$
$660$	0	0
$661$	37.4164	1.45533	0.727665	−	0.685933i	$-0.240606\pi$
$661$	37.4164	1.45533	0.727665	+	0.685933i	$0.240606\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	−32.2918	−1.25316
$665$	8.94427	0.346844
$666$	0	0
$667$	23.4164	0.906687
$668$	−15.4164	−0.596479
$669$	0	0
$670$	6.47214	0.250040
$671$	−1.16718	−0.0450586
$672$	0	0
$673$	3.88854	0.149892	0.0749462	−	0.997188i	$-0.476122\pi$
$673$	3.88854	0.149892	0.0749462	+	0.997188i	$0.476122\pi$
$674$	−0.944272	−0.0363720
$675$	0	0
$676$	6.88854	0.264944
$677$	13.3050	0.511351	0.255675	−	0.966763i	$-0.417702\pi$
$677$	13.3050	0.511351	0.255675	+	0.966763i	$0.417702\pi$
$678$	0	0
$679$	7.41641	0.284616
$680$	−6.00000	−0.230089
$681$	0	0
$682$	−8.00000	−0.306336
$683$	2.18034	0.0834284	0.0417142	−	0.999130i	$-0.486718\pi$
$683$	2.18034	0.0834284	0.0417142	+	0.999130i	$0.486718\pi$
$684$	0	0
$685$	−8.00000	−0.305664
$686$	15.4164	0.588601
$687$	0	0
$688$	−5.23607	−0.199623
$689$	4.94427	0.188362
$690$	0	0
$691$	13.5279	0.514624	0.257312	−	0.966328i	$-0.417163\pi$
$691$	13.5279	0.514624	0.257312	+	0.966328i	$0.417163\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	18.8328	0.714884
$695$	−2.47214	−0.0937735
$696$	0	0
$697$	20.0000	0.757554
$698$	34.3607	1.30057
$699$	0	0
$700$	1.23607	0.0467190
$701$	37.7771	1.42682	0.713410	−	0.700746i	$-0.247150\pi$
$701$	37.7771	1.42682	0.713410	+	0.700746i	$0.247150\pi$
$702$	0	0
$703$	−17.8885	−0.674679
$704$	−17.3050	−0.652205
$705$	0	0
$706$	−32.9443	−1.23987
$707$	5.52786	0.207897
$708$	0	0
$709$	−22.3607	−0.839773	−0.419886	−	0.907577i	$-0.637930\pi$
$709$	−22.3607	−0.839773	−0.419886	+	0.907577i	$0.637930\pi$
$710$	14.4721	0.543130
$711$	0	0
$712$	−3.00000	−0.112430
$713$	16.9443	0.634568
$714$	0	0
$715$	6.11146	0.228556
$716$	−10.4721	−0.391362
$717$	0	0
$718$	−1.70820	−0.0637496
$719$	−10.8754	−0.405584	−0.202792	−	0.979222i	$-0.565001\pi$
$719$	−10.8754	−0.405584	−0.202792	+	0.979222i	$0.565001\pi$
$720$	0	0
$721$	−20.5836	−0.766573
$722$	33.3607	1.24156
$723$	0	0
$724$	3.52786	0.131112
$725$	4.47214	0.166091
$726$	0	0
$727$	−3.70820	−0.137530	−0.0687648	−	0.997633i	$-0.521906\pi$
$727$	−3.70820	−0.137530	−0.0687648	+	0.997633i	$0.521906\pi$
$728$	−9.16718	−0.339758
$729$	0	0
$730$	2.00000	0.0740233
$731$	10.4721	0.387326
$732$	0	0
$733$	−42.9443	−1.58618	−0.793091	−	0.609103i	$-0.791530\pi$
$733$	−42.9443	−1.58618	−0.793091	+	0.609103i	$0.791530\pi$
$734$	−25.8885	−0.955564
$735$	0	0
$736$	26.1803	0.965020
$737$	−16.0000	−0.589368
$738$	0	0
$739$	−42.0689	−1.54753	−0.773764	−	0.633473i	$-0.781629\pi$
$739$	−42.0689	−1.54753	−0.773764	+	0.633473i	$0.781629\pi$
$740$	−2.47214	−0.0908775
$741$	0	0
$742$	2.47214	0.0907550
$743$	−10.1803	−0.373480	−0.186740	−	0.982409i	$-0.559792\pi$
$743$	−10.1803	−0.373480	−0.186740	+	0.982409i	$0.559792\pi$
$744$	0	0
$745$	−17.4164	−0.638088
$746$	−28.8328	−1.05564
$747$	0	0
$748$	4.94427	0.180780
$749$	11.0557	0.403968
$750$	0	0
$751$	14.4721	0.528096	0.264048	−	0.964510i	$-0.414942\pi$
$751$	14.4721	0.528096	0.264048	+	0.964510i	$0.414942\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	−11.0557	−0.402626
$755$	−0.763932	−0.0278023
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	−36.5410	−1.32723
$759$	0	0
$760$	21.7082	0.787439
$761$	−4.47214	−0.162115	−0.0810574	−	0.996709i	$-0.525830\pi$
$761$	−4.47214	−0.162115	−0.0810574	+	0.996709i	$0.525830\pi$
$762$	0	0
$763$	−0.583592	−0.0211275
$764$	−21.7082	−0.785375
$765$	0	0
$766$	−5.23607	−0.189187
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−44.4721	−1.60371	−0.801853	−	0.597521i	$-0.796152\pi$
$769$	−44.4721	−1.60371	−0.801853	+	0.597521i	$0.796152\pi$
$770$	3.05573	0.110121
$771$	0	0
$772$	12.9443	0.465875
$773$	36.3607	1.30780	0.653901	−	0.756580i	$-0.273131\pi$
$773$	36.3607	1.30780	0.653901	+	0.756580i	$0.273131\pi$
$774$	0	0
$775$	3.23607	0.116243
$776$	18.0000	0.646162
$777$	0	0
$778$	21.4164	0.767815
$779$	−72.3607	−2.59259
$780$	0	0
$781$	−35.7771	−1.28020
$782$	10.4721	0.374483
$783$	0	0
$784$	5.47214	0.195433
$785$	6.94427	0.247852
$786$	0	0
$787$	−34.5410	−1.23125	−0.615627	−	0.788038i	$-0.711097\pi$
$787$	−34.5410	−1.23125	−0.615627	+	0.788038i	$0.711097\pi$
$788$	2.47214	0.0880662
$789$	0	0
$790$	12.9443	0.460537
$791$	25.8885	0.920491
$792$	0	0
$793$	−1.16718	−0.0414479
$794$	−1.52786	−0.0542219
$795$	0	0
$796$	8.00000	0.283552
$797$	0.111456	0.00394798	0.00197399	−	0.999998i	$-0.499372\pi$
$797$	0.111456	0.00394798	0.00197399	+	0.999998i	$0.499372\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	5.00000	0.176777
$801$	0	0
$802$	−7.52786	−0.265818
$803$	−4.94427	−0.174480
$804$	0	0
$805$	−6.47214	−0.228113
$806$	−8.00000	−0.281788
$807$	0	0
$808$	13.4164	0.471988
$809$	−41.7771	−1.46880	−0.734402	−	0.678715i	$-0.762537\pi$
$809$	−41.7771	−1.46880	−0.734402	+	0.678715i	$0.762537\pi$
$810$	0	0
$811$	−37.8885	−1.33045	−0.665223	−	0.746644i	$-0.731664\pi$
$811$	−37.8885	−1.33045	−0.665223	+	0.746644i	$0.731664\pi$
$812$	5.52786	0.193990
$813$	0	0
$814$	−6.11146	−0.214206
$815$	−6.18034	−0.216488
$816$	0	0
$817$	−37.8885	−1.32555
$818$	−21.4164	−0.748807
$819$	0	0
$820$	−10.0000	−0.349215
$821$	4.47214	0.156079	0.0780393	−	0.996950i	$-0.475134\pi$
$821$	4.47214	0.156079	0.0780393	+	0.996950i	$0.475134\pi$
$822$	0	0
$823$	42.4721	1.48049	0.740243	−	0.672340i	$-0.234711\pi$
$823$	42.4721	1.48049	0.740243	+	0.672340i	$0.234711\pi$
$824$	−49.9574	−1.74035
$825$	0	0
$826$	−4.00000	−0.139178
$827$	23.1246	0.804122	0.402061	−	0.915613i	$-0.368294\pi$
$827$	23.1246	0.804122	0.402061	+	0.915613i	$0.368294\pi$
$828$	0	0
$829$	40.2492	1.39791	0.698957	−	0.715164i	$-0.253648\pi$
$829$	40.2492	1.39791	0.698957	+	0.715164i	$0.253648\pi$
$830$	10.7639	0.373622
$831$	0	0
$832$	−17.3050	−0.599941
$833$	−10.9443	−0.379197
$834$	0	0
$835$	15.4164	0.533507
$836$	−17.8885	−0.618688
$837$	0	0
$838$	18.2918	0.631880
$839$	17.7082	0.611355	0.305678	−	0.952135i	$-0.401117\pi$
$839$	17.7082	0.611355	0.305678	+	0.952135i	$0.401117\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	31.3050	1.07884
$843$	0	0
$844$	−4.18034	−0.143893
$845$	−6.88854	−0.236973
$846$	0	0
$847$	6.04257	0.207625
$848$	2.00000	0.0686803
$849$	0	0
$850$	2.00000	0.0685994
$851$	12.9443	0.443724
$852$	0	0
$853$	26.4721	0.906389	0.453194	−	0.891412i	$-0.350284\pi$
$853$	26.4721	0.906389	0.453194	+	0.891412i	$0.350284\pi$
$854$	−0.583592	−0.0199701
$855$	0	0
$856$	26.8328	0.917127
$857$	−4.00000	−0.136637	−0.0683187	−	0.997664i	$-0.521763\pi$
$857$	−4.00000	−0.136637	−0.0683187	+	0.997664i	$0.521763\pi$
$858$	0	0
$859$	−5.12461	−0.174849	−0.0874247	−	0.996171i	$-0.527864\pi$
$859$	−5.12461	−0.174849	−0.0874247	+	0.996171i	$0.527864\pi$
$860$	−5.23607	−0.178548
$861$	0	0
$862$	13.1246	0.447026
$863$	39.4853	1.34409	0.672047	−	0.740508i	$-0.265415\pi$
$863$	39.4853	1.34409	0.672047	+	0.740508i	$0.265415\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	−28.9443	−0.983566
$867$	0	0
$868$	4.00000	0.135769
$869$	−32.0000	−1.08553
$870$	0	0
$871$	−16.0000	−0.542139
$872$	−1.41641	−0.0479656
$873$	0	0
$874$	−37.8885	−1.28160
$875$	−1.23607	−0.0417867
$876$	0	0
$877$	−50.4721	−1.70432	−0.852161	−	0.523279i	$-0.824709\pi$
$877$	−50.4721	−1.70432	−0.852161	+	0.523279i	$0.824709\pi$
$878$	−32.1803	−1.08603
$879$	0	0
$880$	2.47214	0.0833357
$881$	16.4721	0.554960	0.277480	−	0.960731i	$-0.410501\pi$
$881$	16.4721	0.554960	0.277480	+	0.960731i	$0.410501\pi$
$882$	0	0
$883$	52.0689	1.75226	0.876129	−	0.482077i	$-0.160118\pi$
$883$	52.0689	1.75226	0.876129	+	0.482077i	$0.160118\pi$
$884$	4.94427	0.166294
$885$	0	0
$886$	21.8885	0.735360
$887$	−38.5410	−1.29408	−0.647040	−	0.762456i	$-0.723994\pi$
$887$	−38.5410	−1.29408	−0.647040	+	0.762456i	$0.723994\pi$
$888$	0	0
$889$	−6.47214	−0.217068
$890$	1.00000	0.0335201
$891$	0	0
$892$	4.94427	0.165546
$893$	57.8885	1.93717
$894$	0	0
$895$	10.4721	0.350045
$896$	3.70820	0.123882
$897$	0	0
$898$	14.0000	0.467186
$899$	14.4721	0.482673
$900$	0	0
$901$	−4.00000	−0.133259
$902$	−24.7214	−0.823131
$903$	0	0
$904$	62.8328	2.08979
$905$	−3.52786	−0.117270
$906$	0	0
$907$	46.4721	1.54308	0.771541	−	0.636180i	$-0.219486\pi$
$907$	46.4721	1.54308	0.771541	+	0.636180i	$0.219486\pi$
$908$	−19.4164	−0.644356
$909$	0	0
$910$	3.05573	0.101296
$911$	25.5279	0.845776	0.422888	−	0.906182i	$-0.361016\pi$
$911$	25.5279	0.845776	0.422888	+	0.906182i	$0.361016\pi$
$912$	0	0
$913$	−26.6099	−0.880659
$914$	28.0000	0.926158
$915$	0	0
$916$	−16.4721	−0.544255
$917$	−22.8328	−0.754006
$918$	0	0
$919$	−28.5410	−0.941481	−0.470741	−	0.882272i	$-0.656013\pi$
$919$	−28.5410	−0.941481	−0.470741	+	0.882272i	$0.656013\pi$
$920$	−15.7082	−0.517884
$921$	0	0
$922$	−22.3607	−0.736410
$923$	−35.7771	−1.17762
$924$	0	0
$925$	2.47214	0.0812833
$926$	−17.8885	−0.587854
$927$	0	0
$928$	22.3607	0.734025
$929$	−59.8885	−1.96488	−0.982440	−	0.186580i	$-0.940260\pi$
$929$	−59.8885	−1.96488	−0.982440	+	0.186580i	$0.940260\pi$
$930$	0	0
$931$	39.5967	1.29773
$932$	−26.0000	−0.851658
$933$	0	0
$934$	−24.3607	−0.797106
$935$	−4.94427	−0.161695
$936$	0	0
$937$	−35.8885	−1.17243	−0.586214	−	0.810156i	$-0.699382\pi$
$937$	−35.8885	−1.17243	−0.586214	+	0.810156i	$0.699382\pi$
$938$	−8.00000	−0.261209
$939$	0	0
$940$	8.00000	0.260931
$941$	28.2492	0.920898	0.460449	−	0.887686i	$-0.347688\pi$
$941$	28.2492	0.920898	0.460449	+	0.887686i	$0.347688\pi$
$942$	0	0
$943$	52.3607	1.70510
$944$	−3.23607	−0.105325
$945$	0	0
$946$	−12.9443	−0.420855
$947$	−24.3607	−0.791616	−0.395808	−	0.918333i	$-0.629535\pi$
$947$	−24.3607	−0.791616	−0.395808	+	0.918333i	$0.629535\pi$
$948$	0	0
$949$	−4.94427	−0.160498
$950$	−7.23607	−0.234769
$951$	0	0
$952$	7.41641	0.240367
$953$	14.1115	0.457115	0.228557	−	0.973530i	$-0.426599\pi$
$953$	14.1115	0.457115	0.228557	+	0.973530i	$0.426599\pi$
$954$	0	0
$955$	21.7082	0.702461
$956$	21.7082	0.702093
$957$	0	0
$958$	−12.9443	−0.418210
$959$	9.88854	0.319318
$960$	0	0
$961$	−20.5279	−0.662189
$962$	−6.11146	−0.197041
$963$	0	0
$964$	−1.05573	−0.0340027
$965$	−12.9443	−0.416691
$966$	0	0
$967$	18.7639	0.603407	0.301704	−	0.953402i	$-0.402445\pi$
$967$	18.7639	0.603407	0.301704	+	0.953402i	$0.402445\pi$
$968$	14.6656	0.471371
$969$	0	0
$970$	−6.00000	−0.192648
$971$	−18.1115	−0.581224	−0.290612	−	0.956841i	$-0.593859\pi$
$971$	−18.1115	−0.581224	−0.290612	+	0.956841i	$0.593859\pi$
$972$	0	0
$973$	3.05573	0.0979621
$974$	41.3050	1.32350
$975$	0	0
$976$	−0.472136	−0.0151127
$977$	14.0000	0.447900	0.223950	−	0.974601i	$-0.428105\pi$
$977$	14.0000	0.447900	0.223950	+	0.974601i	$0.428105\pi$
$978$	0	0
$979$	−2.47214	−0.0790098
$980$	5.47214	0.174801
$981$	0	0
$982$	19.5967	0.625357
$983$	−11.0557	−0.352623	−0.176311	−	0.984334i	$-0.556417\pi$
$983$	−11.0557	−0.352623	−0.176311	+	0.984334i	$0.556417\pi$
$984$	0	0
$985$	−2.47214	−0.0787688
$986$	8.94427	0.284844
$987$	0	0
$988$	−17.8885	−0.569110
$989$	27.4164	0.871791
$990$	0	0
$991$	−29.3475	−0.932255	−0.466127	−	0.884718i	$-0.654351\pi$
$991$	−29.3475	−0.932255	−0.466127	+	0.884718i	$0.654351\pi$
$992$	16.1803	0.513726
$993$	0	0
$994$	−17.8885	−0.567390
$995$	−8.00000	−0.253617
$996$	0	0
$997$	21.0557	0.666842	0.333421	−	0.942778i	$-0.391797\pi$
$997$	21.0557	0.666842	0.333421	+	0.942778i	$0.391797\pi$
$998$	−13.1246	−0.415452
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.h.1.1		2
3.2	odd	2		445.2.a.a.1.2	✓	2
12.11	even	2		7120.2.a.w.1.1		2
15.14	odd	2		2225.2.a.e.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.a.1.2	✓	2	3.2	odd	2
2225.2.a.e.1.1		2	15.14	odd	2
4005.2.a.h.1.1		2	1.1	even	1	trivial
7120.2.a.w.1.1		2	12.11	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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.23607 q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.23607 q^{7} -3.00000 q^{8} +1.00000 q^{10} -2.47214 q^{11} -2.47214 q^{13} -1.23607 q^{14} -1.00000 q^{16} +2.00000 q^{17} -7.23607 q^{19} -1.00000 q^{20} -2.47214 q^{22} +5.23607 q^{23} +1.00000 q^{25} -2.47214 q^{26} +1.23607 q^{28} +4.47214 q^{29} +3.23607 q^{31} +5.00000 q^{32} +2.00000 q^{34} -1.23607 q^{35} +2.47214 q^{37} -7.23607 q^{38} -3.00000 q^{40} +10.0000 q^{41} +5.23607 q^{43} +2.47214 q^{44} +5.23607 q^{46} -8.00000 q^{47} -5.47214 q^{49} +1.00000 q^{50} +2.47214 q^{52} -2.00000 q^{53} -2.47214 q^{55} +3.70820 q^{56} +4.47214 q^{58} +3.23607 q^{59} +0.472136 q^{61} +3.23607 q^{62} +7.00000 q^{64} -2.47214 q^{65} +6.47214 q^{67} -2.00000 q^{68} -1.23607 q^{70} +14.4721 q^{71} +2.00000 q^{73} +2.47214 q^{74} +7.23607 q^{76} +3.05573 q^{77} +12.9443 q^{79} -1.00000 q^{80} +10.0000 q^{82} +10.7639 q^{83} +2.00000 q^{85} +5.23607 q^{86} +7.41641 q^{88} +1.00000 q^{89} +3.05573 q^{91} -5.23607 q^{92} -8.00000 q^{94} -7.23607 q^{95} -6.00000 q^{97} -5.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^8	\( 2 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{8} + 2 q^{10} + 4 q^{11} + 4 q^{13} + 2 q^{14} - 2 q^{16} + 4 q^{17} - 10 q^{19} - 2 q^{20} + 4 q^{22} + 6 q^{23} + 2 q^{25} + 4 q^{26} - 2 q^{28} + 2 q^{31} + 10 q^{32} + 4 q^{34} + 2 q^{35} - 4 q^{37} - 10 q^{38} - 6 q^{40} + 20 q^{41} + 6 q^{43} - 4 q^{44} + 6 q^{46} - 16 q^{47} - 2 q^{49} + 2 q^{50} - 4 q^{52} - 4 q^{53} + 4 q^{55} - 6 q^{56} + 2 q^{59} - 8 q^{61} + 2 q^{62} + 14 q^{64} + 4 q^{65} + 4 q^{67} - 4 q^{68} + 2 q^{70} + 20 q^{71} + 4 q^{73} - 4 q^{74} + 10 q^{76} + 24 q^{77} + 8 q^{79} - 2 q^{80} + 20 q^{82} + 26 q^{83} + 4 q^{85} + 6 q^{86} - 12 q^{88} + 2 q^{89} + 24 q^{91} - 6 q^{92} - 16 q^{94} - 10 q^{95} - 12 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^8 + 2 * q^10 + 4 * q^11 + 4 * q^13 + 2 * q^14 - 2 * q^16 + 4 * q^17 - 10 * q^19 - 2 * q^20 + 4 * q^22 + 6 * q^23 + 2 * q^25 + 4 * q^26 - 2 * q^28 + 2 * q^31 + 10 * q^32 + 4 * q^34 + 2 * q^35 - 4 * q^37 - 10 * q^38 - 6 * q^40 + 20 * q^41 + 6 * q^43 - 4 * q^44 + 6 * q^46 - 16 * q^47 - 2 * q^49 + 2 * q^50 - 4 * q^52 - 4 * q^53 + 4 * q^55 - 6 * q^56 + 2 * q^59 - 8 * q^61 + 2 * q^62 + 14 * q^64 + 4 * q^65 + 4 * q^67 - 4 * q^68 + 2 * q^70 + 20 * q^71 + 4 * q^73 - 4 * q^74 + 10 * q^76 + 24 * q^77 + 8 * q^79 - 2 * q^80 + 20 * q^82 + 26 * q^83 + 4 * q^85 + 6 * q^86 - 12 * q^88 + 2 * q^89 + 24 * q^91 - 6 * q^92 - 16 * q^94 - 10 * q^95 - 12 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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