LMFDB - Embedded newform 4005.2.a.v.1.8

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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 58 x^{8} - 202 x^{7} - 95 x^{6} + 432 x^{5} + 4 x^{4} + \cdots - 27 $ x^12 - 3x^11 - 13x^10 + 41x^9 + 58x^8 - 202x^7 - 95x^6 + 432x^5 + 4x^4 - 368x^3 + 94x^2 + 77*x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.02694$ 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)

$f(q)$	$=$	$q+1.02694 q^{2} -0.945401 q^{4} -1.00000 q^{5} +2.58018 q^{7} -3.02474 q^{8} +O(q^{10})$	\(q+1.02694 q^{2} -0.945401 q^{4} -1.00000 q^{5} +2.58018 q^{7} -3.02474 q^{8} -1.02694 q^{10} -3.43937 q^{11} +3.41396 q^{13} +2.64968 q^{14} -1.21541 q^{16} +0.747634 q^{17} -5.35849 q^{19} +0.945401 q^{20} -3.53202 q^{22} +9.33546 q^{23} +1.00000 q^{25} +3.50592 q^{26} -2.43931 q^{28} +0.630020 q^{29} -10.6902 q^{31} +4.80133 q^{32} +0.767772 q^{34} -2.58018 q^{35} -8.33949 q^{37} -5.50283 q^{38} +3.02474 q^{40} +10.6683 q^{41} -4.85201 q^{43} +3.25159 q^{44} +9.58692 q^{46} +0.221731 q^{47} -0.342667 q^{49} +1.02694 q^{50} -3.22756 q^{52} -12.6625 q^{53} +3.43937 q^{55} -7.80438 q^{56} +0.646991 q^{58} +4.16176 q^{59} +1.13273 q^{61} -10.9782 q^{62} +7.36149 q^{64} -3.41396 q^{65} -10.4015 q^{67} -0.706814 q^{68} -2.64968 q^{70} +10.8129 q^{71} -0.703062 q^{73} -8.56413 q^{74} +5.06592 q^{76} -8.87421 q^{77} -6.91213 q^{79} +1.21541 q^{80} +10.9557 q^{82} -13.9744 q^{83} -0.747634 q^{85} -4.98271 q^{86} +10.4032 q^{88} -1.00000 q^{89} +8.80863 q^{91} -8.82575 q^{92} +0.227704 q^{94} +5.35849 q^{95} -14.4939 q^{97} -0.351897 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{2} + 11 q^{4} - 12 q^{5} - 8 q^{7} + 9 q^{8}+O(q^{10}) $ 12 * q + 3 * q^2 + 11 * q^4 - 12 * q^5 - 8 * q^7 + 9 * q^8	$ 12 q + 3 q^{2} + 11 q^{4} - 12 q^{5} - 8 q^{7} + 9 q^{8} - 3 q^{10} - 12 q^{13} - 4 q^{14} + q^{16} - 24 q^{19} - 11 q^{20} - 16 q^{22} + 24 q^{23} + 12 q^{25} + q^{26} - 44 q^{28} - 8 q^{29} - 12 q^{31} + 31 q^{32} - 18 q^{34} + 8 q^{35} - 10 q^{37} - 2 q^{38} - 9 q^{40} + 10 q^{41} - 42 q^{43} - 42 q^{44} - 24 q^{46} + 22 q^{47} - 4 q^{49} + 3 q^{50} - 30 q^{52} - 8 q^{53} - 27 q^{56} - 12 q^{58} - 4 q^{59} - 52 q^{61} + 14 q^{62} + 7 q^{64} + 12 q^{65} - 40 q^{67} - 23 q^{68} + 4 q^{70} - 2 q^{71} - 8 q^{73} - 26 q^{74} - 46 q^{76} + 12 q^{77} - 26 q^{79} - q^{80} - 26 q^{82} + 14 q^{83} - 32 q^{86} - 60 q^{88} - 12 q^{89} - 24 q^{91} + 38 q^{92} - 26 q^{94} + 24 q^{95} - 6 q^{97} - 30 q^{98}+O(q^{100}) $ 12 * q + 3 * q^2 + 11 * q^4 - 12 * q^5 - 8 * q^7 + 9 * q^8 - 3 * q^10 - 12 * q^13 - 4 * q^14 + q^16 - 24 * q^19 - 11 * q^20 - 16 * q^22 + 24 * q^23 + 12 * q^25 + q^26 - 44 * q^28 - 8 * q^29 - 12 * q^31 + 31 * q^32 - 18 * q^34 + 8 * q^35 - 10 * q^37 - 2 * q^38 - 9 * q^40 + 10 * q^41 - 42 * q^43 - 42 * q^44 - 24 * q^46 + 22 * q^47 - 4 * q^49 + 3 * q^50 - 30 * q^52 - 8 * q^53 - 27 * q^56 - 12 * q^58 - 4 * q^59 - 52 * q^61 + 14 * q^62 + 7 * q^64 + 12 * q^65 - 40 * q^67 - 23 * q^68 + 4 * q^70 - 2 * q^71 - 8 * q^73 - 26 * q^74 - 46 * q^76 + 12 * q^77 - 26 * q^79 - q^80 - 26 * q^82 + 14 * q^83 - 32 * q^86 - 60 * q^88 - 12 * q^89 - 24 * q^91 + 38 * q^92 - 26 * q^94 + 24 * q^95 - 6 * q^97 - 30 * 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 $

$2$	1.02694	0.726154	0.363077	−	0.931759i	$-0.381726\pi$
$2$	1.02694	0.726154	0.363077	+	0.931759i	$0.381726\pi$
$3$	0	0
$3$	0	0
$4$	−0.945401	−0.472701
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.58018	0.975217	0.487608	−	0.873062i	$-0.337869\pi$
$7$	2.58018	0.975217	0.487608	+	0.873062i	$0.337869\pi$
$8$	−3.02474	−1.06941
$9$	0	0
$10$	−1.02694	−0.324746
$11$	−3.43937	−1.03701	−0.518505	−	0.855074i	$-0.673511\pi$
$11$	−3.43937	−1.03701	−0.518505	+	0.855074i	$0.673511\pi$
$12$	0	0
$13$	3.41396	0.946862	0.473431	−	0.880831i	$-0.343015\pi$
$13$	3.41396	0.946862	0.473431	+	0.880831i	$0.343015\pi$
$14$	2.64968	0.708157
$15$	0	0
$16$	−1.21541	−0.303854
$17$	0.747634	0.181328	0.0906639	−	0.995882i	$-0.471101\pi$
$17$	0.747634	0.181328	0.0906639	+	0.995882i	$0.471101\pi$
$18$	0	0
$19$	−5.35849	−1.22932	−0.614661	−	0.788792i	$-0.710707\pi$
$19$	−5.35849	−1.22932	−0.614661	+	0.788792i	$0.710707\pi$
$20$	0.945401	0.211398
$21$	0	0
$22$	−3.53202	−0.753029
$23$	9.33546	1.94658	0.973289	−	0.229584i	$-0.0737365\pi$
$23$	9.33546	1.94658	0.973289	+	0.229584i	$0.0737365\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.50592	0.687567
$27$	0	0
$28$	−2.43931	−0.460986
$29$	0.630020	0.116992	0.0584959	−	0.998288i	$-0.481370\pi$
$29$	0.630020	0.116992	0.0584959	+	0.998288i	$0.481370\pi$
$30$	0	0
$31$	−10.6902	−1.92002	−0.960009	−	0.279970i	$-0.909675\pi$
$31$	−10.6902	−1.92002	−0.960009	+	0.279970i	$0.909675\pi$
$32$	4.80133	0.848763
$33$	0	0
$34$	0.767772	0.131672
$35$	−2.58018	−0.436130
$36$	0	0
$37$	−8.33949	−1.37100	−0.685502	−	0.728071i	$-0.740417\pi$
$37$	−8.33949	−1.37100	−0.685502	+	0.728071i	$0.740417\pi$
$38$	−5.50283	−0.892677
$39$	0	0
$40$	3.02474	0.478253
$41$	10.6683	1.66611	0.833053	−	0.553192i	$-0.186591\pi$
$41$	10.6683	1.66611	0.833053	+	0.553192i	$0.186591\pi$
$42$	0	0
$43$	−4.85201	−0.739925	−0.369963	−	0.929047i	$-0.620629\pi$
$43$	−4.85201	−0.739925	−0.369963	+	0.929047i	$0.620629\pi$
$44$	3.25159	0.490195
$45$	0	0
$46$	9.58692	1.41351
$47$	0.221731	0.0323428	0.0161714	−	0.999869i	$-0.494852\pi$
$47$	0.221731	0.0323428	0.0161714	+	0.999869i	$0.494852\pi$
$48$	0	0
$49$	−0.342667	−0.0489524
$50$	1.02694	0.145231
$51$	0	0
$52$	−3.22756	−0.447582
$53$	−12.6625	−1.73933	−0.869665	−	0.493642i	$-0.835665\pi$
$53$	−12.6625	−1.73933	−0.869665	+	0.493642i	$0.835665\pi$
$54$	0	0
$55$	3.43937	0.463765
$56$	−7.80438	−1.04290
$57$	0	0
$58$	0.646991	0.0849541
$59$	4.16176	0.541815	0.270908	−	0.962605i	$-0.412676\pi$
$59$	4.16176	0.541815	0.270908	+	0.962605i	$0.412676\pi$
$60$	0	0
$61$	1.13273	0.145031	0.0725155	−	0.997367i	$-0.476897\pi$
$61$	1.13273	0.145031	0.0725155	+	0.997367i	$0.476897\pi$
$62$	−10.9782	−1.39423
$63$	0	0
$64$	7.36149	0.920186
$65$	−3.41396	−0.423449
$66$	0	0
$67$	−10.4015	−1.27074	−0.635370	−	0.772208i	$-0.719152\pi$
$67$	−10.4015	−1.27074	−0.635370	+	0.772208i	$0.719152\pi$
$68$	−0.706814	−0.0857137
$69$	0	0
$70$	−2.64968	−0.316698
$71$	10.8129	1.28325	0.641626	−	0.767018i	$-0.278260\pi$
$71$	10.8129	1.28325	0.641626	+	0.767018i	$0.278260\pi$
$72$	0	0
$73$	−0.703062	−0.0822872	−0.0411436	−	0.999153i	$-0.513100\pi$
$73$	−0.703062	−0.0822872	−0.0411436	+	0.999153i	$0.513100\pi$
$74$	−8.56413	−0.995560
$75$	0	0
$76$	5.06592	0.581101
$77$	−8.87421	−1.01131
$78$	0	0
$79$	−6.91213	−0.777675	−0.388838	−	0.921306i	$-0.627123\pi$
$79$	−6.91213	−0.777675	−0.388838	+	0.921306i	$0.627123\pi$
$80$	1.21541	0.135887
$81$	0	0
$82$	10.9557	1.20985
$83$	−13.9744	−1.53389	−0.766943	−	0.641715i	$-0.778223\pi$
$83$	−13.9744	−1.53389	−0.766943	+	0.641715i	$0.778223\pi$
$84$	0	0
$85$	−0.747634	−0.0810922
$86$	−4.98271	−0.537299
$87$	0	0
$88$	10.4032	1.10899
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	8.80863	0.923395
$92$	−8.82575	−0.920148
$93$	0	0
$94$	0.227704	0.0234859
$95$	5.35849	0.549769
$96$	0	0
$97$	−14.4939	−1.47163	−0.735814	−	0.677183i	$-0.763200\pi$
$97$	−14.4939	−1.47163	−0.735814	+	0.677183i	$0.763200\pi$
$98$	−0.351897	−0.0355469
$99$	0	0
$100$	−0.945401	−0.0945401
$101$	−1.23926	−0.123311	−0.0616556	−	0.998097i	$-0.519638\pi$
$101$	−1.23926	−0.123311	−0.0616556	+	0.998097i	$0.519638\pi$
$102$	0	0
$103$	0.359446	0.0354173	0.0177086	−	0.999843i	$-0.494363\pi$
$103$	0.359446	0.0354173	0.0177086	+	0.999843i	$0.494363\pi$
$104$	−10.3263	−1.01258
$105$	0	0
$106$	−13.0036	−1.26302
$107$	1.83003	0.176916	0.0884580	−	0.996080i	$-0.471806\pi$
$107$	1.83003	0.176916	0.0884580	+	0.996080i	$0.471806\pi$
$108$	0	0
$109$	6.43239	0.616111	0.308056	−	0.951368i	$-0.400322\pi$
$109$	6.43239	0.616111	0.308056	+	0.951368i	$0.400322\pi$
$110$	3.53202	0.336765
$111$	0	0
$112$	−3.13599	−0.296323
$113$	−11.4828	−1.08021	−0.540104	−	0.841598i	$-0.681615\pi$
$113$	−11.4828	−1.08021	−0.540104	+	0.841598i	$0.681615\pi$
$114$	0	0
$115$	−9.33546	−0.870536
$116$	−0.595622	−0.0553021
$117$	0	0
$118$	4.27386	0.393441
$119$	1.92903	0.176834
$120$	0	0
$121$	0.829298	0.0753907
$122$	1.16324	0.105315
$123$	0	0
$124$	10.1065	0.907593
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.3676	−1.71860	−0.859298	−	0.511475i	$-0.829099\pi$
$127$	−19.3676	−1.71860	−0.859298	+	0.511475i	$0.829099\pi$
$128$	−2.04287	−0.180566
$129$	0	0
$130$	−3.50592	−0.307489
$131$	−9.28826	−0.811519	−0.405759	−	0.913980i	$-0.632993\pi$
$131$	−9.28826	−0.811519	−0.405759	+	0.913980i	$0.632993\pi$
$132$	0	0
$133$	−13.8259	−1.19885
$134$	−10.6816	−0.922753
$135$	0	0
$136$	−2.26140	−0.193913
$137$	5.11040	0.436611	0.218305	−	0.975881i	$-0.429947\pi$
$137$	5.11040	0.436611	0.218305	+	0.975881i	$0.429947\pi$
$138$	0	0
$139$	−20.0200	−1.69807	−0.849037	−	0.528333i	$-0.822817\pi$
$139$	−20.0200	−1.69807	−0.849037	+	0.528333i	$0.822817\pi$
$140$	2.43931	0.206159
$141$	0	0
$142$	11.1041	0.931838
$143$	−11.7419	−0.981905
$144$	0	0
$145$	−0.630020	−0.0523203
$146$	−0.722000	−0.0597532
$147$	0	0
$148$	7.88417	0.648075
$149$	0.509300	0.0417235	0.0208618	−	0.999782i	$-0.493359\pi$
$149$	0.509300	0.0417235	0.0208618	+	0.999782i	$0.493359\pi$
$150$	0	0
$151$	0.134489	0.0109446	0.00547228	−	0.999985i	$-0.498258\pi$
$151$	0.134489	0.0109446	0.00547228	+	0.999985i	$0.498258\pi$
$152$	16.2080	1.31465
$153$	0	0
$154$	−9.11325	−0.734367
$155$	10.6902	0.858658
$156$	0	0
$157$	3.27404	0.261297	0.130648	−	0.991429i	$-0.458294\pi$
$157$	3.27404	0.261297	0.130648	+	0.991429i	$0.458294\pi$
$158$	−7.09832	−0.564712
$159$	0	0
$160$	−4.80133	−0.379578
$161$	24.0872	1.89834
$162$	0	0
$163$	5.47998	0.429225	0.214612	−	0.976699i	$-0.431151\pi$
$163$	5.47998	0.429225	0.214612	+	0.976699i	$0.431151\pi$
$164$	−10.0858	−0.787570
$165$	0	0
$166$	−14.3508	−1.11384
$167$	−12.6299	−0.977328	−0.488664	−	0.872472i	$-0.662516\pi$
$167$	−12.6299	−0.977328	−0.488664	+	0.872472i	$0.662516\pi$
$168$	0	0
$169$	−1.34489	−0.103453
$170$	−0.767772	−0.0588854
$171$	0	0
$172$	4.58710	0.349763
$173$	4.34868	0.330624	0.165312	−	0.986241i	$-0.447137\pi$
$173$	4.34868	0.330624	0.165312	+	0.986241i	$0.447137\pi$
$174$	0	0
$175$	2.58018	0.195043
$176$	4.18027	0.315099
$177$	0	0
$178$	−1.02694	−0.0769722
$179$	−14.9452	−1.11705	−0.558527	−	0.829486i	$-0.688633\pi$
$179$	−14.9452	−1.11705	−0.558527	+	0.829486i	$0.688633\pi$
$180$	0	0
$181$	12.5561	0.933288	0.466644	−	0.884445i	$-0.345463\pi$
$181$	12.5561	0.933288	0.466644	+	0.884445i	$0.345463\pi$
$182$	9.04590	0.670527
$183$	0	0
$184$	−28.2373	−2.08168
$185$	8.33949	0.613132
$186$	0	0
$187$	−2.57139	−0.188039
$188$	−0.209625	−0.0152885
$189$	0	0
$190$	5.50283	0.399217
$191$	7.82928	0.566507	0.283254	−	0.959045i	$-0.408586\pi$
$191$	7.82928	0.566507	0.283254	+	0.959045i	$0.408586\pi$
$192$	0	0
$193$	20.7311	1.49226	0.746128	−	0.665802i	$-0.231911\pi$
$193$	20.7311	1.49226	0.746128	+	0.665802i	$0.231911\pi$
$194$	−14.8843	−1.06863
$195$	0	0
$196$	0.323957	0.0231398
$197$	3.84740	0.274116	0.137058	−	0.990563i	$-0.456235\pi$
$197$	3.84740	0.274116	0.137058	+	0.990563i	$0.456235\pi$
$198$	0	0
$199$	−23.7892	−1.68637	−0.843184	−	0.537624i	$-0.819322\pi$
$199$	−23.7892	−1.68637	−0.843184	+	0.537624i	$0.819322\pi$
$200$	−3.02474	−0.213881
$201$	0	0
$202$	−1.27264	−0.0895429
$203$	1.62557	0.114092
$204$	0	0
$205$	−10.6683	−0.745106
$206$	0.369128	0.0257184
$207$	0	0
$208$	−4.14937	−0.287707
$209$	18.4298	1.27482
$210$	0	0
$211$	13.1407	0.904642	0.452321	−	0.891855i	$-0.350596\pi$
$211$	13.1407	0.904642	0.452321	+	0.891855i	$0.350596\pi$
$212$	11.9712	0.822183
$213$	0	0
$214$	1.87933	0.128468
$215$	4.85201	0.330905
$216$	0	0
$217$	−27.5827	−1.87243
$218$	6.60566	0.447392
$219$	0	0
$220$	−3.25159	−0.219222
$221$	2.55239	0.171692
$222$	0	0
$223$	−10.6852	−0.715535	−0.357767	−	0.933811i	$-0.616462\pi$
$223$	−10.6852	−0.715535	−0.357767	+	0.933811i	$0.616462\pi$
$224$	12.3883	0.827728
$225$	0	0
$226$	−11.7921	−0.784397
$227$	3.07941	0.204388	0.102194	−	0.994765i	$-0.467414\pi$
$227$	3.07941	0.204388	0.102194	+	0.994765i	$0.467414\pi$
$228$	0	0
$229$	−4.61159	−0.304742	−0.152371	−	0.988323i	$-0.548691\pi$
$229$	−4.61159	−0.304742	−0.152371	+	0.988323i	$0.548691\pi$
$230$	−9.58692	−0.632143
$231$	0	0
$232$	−1.90565	−0.125112
$233$	12.4879	0.818111	0.409056	−	0.912509i	$-0.365858\pi$
$233$	12.4879	0.818111	0.409056	+	0.912509i	$0.365858\pi$
$234$	0	0
$235$	−0.221731	−0.0144641
$236$	−3.93453	−0.256116
$237$	0	0
$238$	1.98099	0.128409
$239$	−19.1239	−1.23702	−0.618510	−	0.785777i	$-0.712263\pi$
$239$	−19.1239	−1.23702	−0.618510	+	0.785777i	$0.712263\pi$
$240$	0	0
$241$	22.5585	1.45312	0.726560	−	0.687103i	$-0.241118\pi$
$241$	22.5585	1.45312	0.726560	+	0.687103i	$0.241118\pi$
$242$	0.851637	0.0547453
$243$	0	0
$244$	−1.07088	−0.0685562
$245$	0.342667	0.0218922
$246$	0	0
$247$	−18.2937	−1.16400
$248$	32.3351	2.05328
$249$	0	0
$250$	−1.02694	−0.0649492
$251$	−16.6638	−1.05181	−0.525905	−	0.850544i	$-0.676273\pi$
$251$	−16.6638	−1.05181	−0.525905	+	0.850544i	$0.676273\pi$
$252$	0	0
$253$	−32.1081	−2.01862
$254$	−19.8893	−1.24797
$255$	0	0
$256$	−16.8209	−1.05130
$257$	9.95210	0.620795	0.310397	−	0.950607i	$-0.399538\pi$
$257$	9.95210	0.620795	0.310397	+	0.950607i	$0.399538\pi$
$258$	0	0
$259$	−21.5174	−1.33703
$260$	3.22756	0.200165
$261$	0	0
$262$	−9.53845	−0.589287
$263$	10.2412	0.631498	0.315749	−	0.948843i	$-0.397744\pi$
$263$	10.2412	0.631498	0.315749	+	0.948843i	$0.397744\pi$
$264$	0	0
$265$	12.6625	0.777852
$266$	−14.1983	−0.870553
$267$	0	0
$268$	9.83355	0.600680
$269$	−18.0397	−1.09990	−0.549949	−	0.835198i	$-0.685353\pi$
$269$	−18.0397	−1.09990	−0.549949	+	0.835198i	$0.685353\pi$
$270$	0	0
$271$	29.2066	1.77417	0.887086	−	0.461604i	$-0.152726\pi$
$271$	29.2066	1.77417	0.887086	+	0.461604i	$0.152726\pi$
$272$	−0.908684	−0.0550971
$273$	0	0
$274$	5.24805	0.317047
$275$	−3.43937	−0.207402
$276$	0	0
$277$	11.5867	0.696175	0.348088	−	0.937462i	$-0.386831\pi$
$277$	11.5867	0.696175	0.348088	+	0.937462i	$0.386831\pi$
$278$	−20.5593	−1.23306
$279$	0	0
$280$	7.80438	0.466401
$281$	−15.9089	−0.949044	−0.474522	−	0.880244i	$-0.657379\pi$
$281$	−15.9089	−0.949044	−0.474522	+	0.880244i	$0.657379\pi$
$282$	0	0
$283$	1.96247	0.116657	0.0583285	−	0.998297i	$-0.481423\pi$
$283$	1.96247	0.116657	0.0583285	+	0.998297i	$0.481423\pi$
$284$	−10.2225	−0.606594
$285$	0	0
$286$	−12.0582	−0.713014
$287$	27.5261	1.62482
$288$	0	0
$289$	−16.4410	−0.967120
$290$	−0.646991	−0.0379926
$291$	0	0
$292$	0.664676	0.0388972
$293$	−15.2736	−0.892293	−0.446147	−	0.894960i	$-0.647204\pi$
$293$	−15.2736	−0.892293	−0.446147	+	0.894960i	$0.647204\pi$
$294$	0	0
$295$	−4.16176	−0.242307
$296$	25.2248	1.46616
$297$	0	0
$298$	0.523019	0.0302977
$299$	31.8709	1.84314
$300$	0	0
$301$	−12.5191	−0.721587
$302$	0.138112	0.00794743
$303$	0	0
$304$	6.51278	0.373534
$305$	−1.13273	−0.0648598
$306$	0	0
$307$	3.74776	0.213896	0.106948	−	0.994265i	$-0.465892\pi$
$307$	3.74776	0.213896	0.106948	+	0.994265i	$0.465892\pi$
$308$	8.38969	0.478047
$309$	0	0
$310$	10.9782	0.623518
$311$	5.19154	0.294385	0.147193	−	0.989108i	$-0.452976\pi$
$311$	5.19154	0.294385	0.147193	+	0.989108i	$0.452976\pi$
$312$	0	0
$313$	8.87974	0.501913	0.250956	−	0.967998i	$-0.419255\pi$
$313$	8.87974	0.501913	0.250956	+	0.967998i	$0.419255\pi$
$314$	3.36223	0.189741
$315$	0	0
$316$	6.53474	0.367608
$317$	−29.6573	−1.66572	−0.832859	−	0.553484i	$-0.813298\pi$
$317$	−29.6573	−1.66572	−0.832859	+	0.553484i	$0.813298\pi$
$318$	0	0
$319$	−2.16688	−0.121322
$320$	−7.36149	−0.411520
$321$	0	0
$322$	24.7360	1.37848
$323$	−4.00619	−0.222910
$324$	0	0
$325$	3.41396	0.189372
$326$	5.62759	0.311683
$327$	0	0
$328$	−32.2688	−1.78175
$329$	0.572107	0.0315413
$330$	0	0
$331$	14.7946	0.813186	0.406593	−	0.913609i	$-0.366717\pi$
$331$	14.7946	0.813186	0.406593	+	0.913609i	$0.366717\pi$
$332$	13.2114	0.725069
$333$	0	0
$334$	−12.9701	−0.709690
$335$	10.4015	0.568292
$336$	0	0
$337$	0.519562	0.0283023	0.0141512	−	0.999900i	$-0.495495\pi$
$337$	0.519562	0.0283023	0.0141512	+	0.999900i	$0.495495\pi$
$338$	−1.38112	−0.0751229
$339$	0	0
$340$	0.706814	0.0383323
$341$	36.7676	1.99108
$342$	0	0
$343$	−18.9454	−1.02296
$344$	14.6761	0.791281
$345$	0	0
$346$	4.46582	0.240084
$347$	10.3953	0.558049	0.279024	−	0.960284i	$-0.409989\pi$
$347$	10.3953	0.558049	0.279024	+	0.960284i	$0.409989\pi$
$348$	0	0
$349$	11.6831	0.625383	0.312692	−	0.949855i	$-0.398769\pi$
$349$	11.6831	0.625383	0.312692	+	0.949855i	$0.398769\pi$
$350$	2.64968	0.141631
$351$	0	0
$352$	−16.5136	−0.880176
$353$	11.9701	0.637106	0.318553	−	0.947905i	$-0.396803\pi$
$353$	11.9701	0.637106	0.318553	+	0.947905i	$0.396803\pi$
$354$	0	0
$355$	−10.8129	−0.573888
$356$	0.945401	0.0501062
$357$	0	0
$358$	−15.3477	−0.811154
$359$	8.69215	0.458754	0.229377	−	0.973338i	$-0.426331\pi$
$359$	8.69215	0.458754	0.229377	+	0.973338i	$0.426331\pi$
$360$	0	0
$361$	9.71340	0.511231
$362$	12.8943	0.677711
$363$	0	0
$364$	−8.32769	−0.436489
$365$	0.703062	0.0368000
$366$	0	0
$367$	35.1036	1.83239	0.916197	−	0.400729i	$-0.131243\pi$
$367$	35.1036	1.83239	0.916197	+	0.400729i	$0.131243\pi$
$368$	−11.3465	−0.591475
$369$	0	0
$370$	8.56413	0.445228
$371$	−32.6716	−1.69622
$372$	0	0
$373$	−31.9943	−1.65660	−0.828300	−	0.560285i	$-0.810692\pi$
$373$	−31.9943	−1.65660	−0.828300	+	0.560285i	$0.810692\pi$
$374$	−2.64066	−0.136545
$375$	0	0
$376$	−0.670679	−0.0345876
$377$	2.15086	0.110775
$378$	0	0
$379$	5.31368	0.272946	0.136473	−	0.990644i	$-0.456423\pi$
$379$	5.31368	0.272946	0.136473	+	0.990644i	$0.456423\pi$
$380$	−5.06592	−0.259876
$381$	0	0
$382$	8.04018	0.411371
$383$	21.0826	1.07727	0.538635	−	0.842539i	$-0.318940\pi$
$383$	21.0826	1.07727	0.538635	+	0.842539i	$0.318940\pi$
$384$	0	0
$385$	8.87421	0.452272
$386$	21.2895	1.08361
$387$	0	0
$388$	13.7025	0.695640
$389$	−18.2192	−0.923749	−0.461875	−	0.886945i	$-0.652823\pi$
$389$	−18.2192	−0.923749	−0.461875	+	0.886945i	$0.652823\pi$
$390$	0	0
$391$	6.97950	0.352969
$392$	1.03648	0.0523500
$393$	0	0
$394$	3.95103	0.199050
$395$	6.91213	0.347787
$396$	0	0
$397$	−28.7516	−1.44300	−0.721501	−	0.692414i	$-0.756547\pi$
$397$	−28.7516	−1.44300	−0.721501	+	0.692414i	$0.756547\pi$
$398$	−24.4300	−1.22456
$399$	0	0
$400$	−1.21541	−0.0607707
$401$	17.9467	0.896216	0.448108	−	0.893979i	$-0.352098\pi$
$401$	17.9467	0.896216	0.448108	+	0.893979i	$0.352098\pi$
$402$	0	0
$403$	−36.4959	−1.81799
$404$	1.17160	0.0582893
$405$	0	0
$406$	1.66935	0.0828486
$407$	28.6826	1.42175
$408$	0	0
$409$	21.1329	1.04496	0.522478	−	0.852653i	$-0.325008\pi$
$409$	21.1329	1.04496	0.522478	+	0.852653i	$0.325008\pi$
$410$	−10.9557	−0.541061
$411$	0	0
$412$	−0.339821	−0.0167418
$413$	10.7381	0.528387
$414$	0	0
$415$	13.9744	0.685975
$416$	16.3915	0.803661
$417$	0	0
$418$	18.9263	0.925715
$419$	−23.1353	−1.13023	−0.565116	−	0.825011i	$-0.691169\pi$
$419$	−23.1353	−1.13023	−0.565116	+	0.825011i	$0.691169\pi$
$420$	0	0
$421$	−2.04245	−0.0995432	−0.0497716	−	0.998761i	$-0.515849\pi$
$421$	−2.04245	−0.0995432	−0.0497716	+	0.998761i	$0.515849\pi$
$422$	13.4947	0.656910
$423$	0	0
$424$	38.3008	1.86005
$425$	0.747634	0.0362656
$426$	0	0
$427$	2.92264	0.141437
$428$	−1.73012	−0.0836283
$429$	0	0
$430$	4.98271	0.240288
$431$	−20.7971	−1.00176	−0.500882	−	0.865516i	$-0.666991\pi$
$431$	−20.7971	−1.00176	−0.500882	+	0.865516i	$0.666991\pi$
$432$	0	0
$433$	−28.3246	−1.36119	−0.680596	−	0.732659i	$-0.738279\pi$
$433$	−28.3246	−1.36119	−0.680596	+	0.732659i	$0.738279\pi$
$434$	−28.3256	−1.35967
$435$	0	0
$436$	−6.08119	−0.291236
$437$	−50.0239	−2.39297
$438$	0	0
$439$	8.98698	0.428925	0.214463	−	0.976732i	$-0.431200\pi$
$439$	8.98698	0.428925	0.214463	+	0.976732i	$0.431200\pi$
$440$	−10.4032	−0.495954
$441$	0	0
$442$	2.62114	0.124675
$443$	−16.1261	−0.766173	−0.383086	−	0.923713i	$-0.625139\pi$
$443$	−16.1261	−0.766173	−0.383086	+	0.923713i	$0.625139\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−10.9730	−0.519588
$447$	0	0
$448$	18.9940	0.897381
$449$	7.36613	0.347629	0.173815	−	0.984778i	$-0.444391\pi$
$449$	7.36613	0.347629	0.173815	+	0.984778i	$0.444391\pi$
$450$	0	0
$451$	−36.6922	−1.72777
$452$	10.8558	0.510615
$453$	0	0
$454$	3.16236	0.148417
$455$	−8.80863	−0.412955
$456$	0	0
$457$	−26.0167	−1.21701	−0.608504	−	0.793551i	$-0.708230\pi$
$457$	−26.0167	−1.21701	−0.608504	+	0.793551i	$0.708230\pi$
$458$	−4.73581	−0.221290
$459$	0	0
$460$	8.82575	0.411503
$461$	−39.4229	−1.83611	−0.918053	−	0.396458i	$-0.870239\pi$
$461$	−39.4229	−1.83611	−0.918053	+	0.396458i	$0.870239\pi$
$462$	0	0
$463$	−6.57702	−0.305660	−0.152830	−	0.988253i	$-0.548839\pi$
$463$	−6.57702	−0.305660	−0.152830	+	0.988253i	$0.548839\pi$
$464$	−0.765736	−0.0355484
$465$	0	0
$466$	12.8243	0.594075
$467$	32.5531	1.50638	0.753189	−	0.657805i	$-0.228515\pi$
$467$	32.5531	1.50638	0.753189	+	0.657805i	$0.228515\pi$
$468$	0	0
$469$	−26.8376	−1.23925
$470$	−0.227704	−0.0105032
$471$	0	0
$472$	−12.5882	−0.579421
$473$	16.6879	0.767310
$474$	0	0
$475$	−5.35849	−0.245864
$476$	−1.82371	−0.0835895
$477$	0	0
$478$	−19.6390	−0.898267
$479$	20.8369	0.952064	0.476032	−	0.879428i	$-0.342075\pi$
$479$	20.8369	0.952064	0.476032	+	0.879428i	$0.342075\pi$
$480$	0	0
$481$	−28.4707	−1.29815
$482$	23.1661	1.05519
$483$	0	0
$484$	−0.784019	−0.0356372
$485$	14.4939	0.658132
$486$	0	0
$487$	14.1889	0.642960	0.321480	−	0.946916i	$-0.395820\pi$
$487$	14.1889	0.642960	0.321480	+	0.946916i	$0.395820\pi$
$488$	−3.42621	−0.155097
$489$	0	0
$490$	0.351897	0.0158971
$491$	−0.869053	−0.0392198	−0.0196099	−	0.999808i	$-0.506242\pi$
$491$	−0.869053	−0.0392198	−0.0196099	+	0.999808i	$0.506242\pi$
$492$	0	0
$493$	0.471024	0.0212139
$494$	−18.7864	−0.845241
$495$	0	0
$496$	12.9930	0.583404
$497$	27.8992	1.25145
$498$	0	0
$499$	−5.23759	−0.234467	−0.117233	−	0.993104i	$-0.537403\pi$
$499$	−5.23759	−0.234467	−0.117233	+	0.993104i	$0.537403\pi$
$500$	0.945401	0.0422796
$501$	0	0
$502$	−17.1127	−0.763775
$503$	−9.18893	−0.409714	−0.204857	−	0.978792i	$-0.565673\pi$
$503$	−9.18893	−0.409714	−0.204857	+	0.978792i	$0.565673\pi$
$504$	0	0
$505$	1.23926	0.0551465
$506$	−32.9730	−1.46583
$507$	0	0
$508$	18.3102	0.812382
$509$	8.49107	0.376360	0.188180	−	0.982135i	$-0.439741\pi$
$509$	8.49107	0.376360	0.188180	+	0.982135i	$0.439741\pi$
$510$	0	0
$511$	−1.81403	−0.0802479
$512$	−13.1882	−0.582843
$513$	0	0
$514$	10.2202	0.450793
$515$	−0.359446	−0.0158391
$516$	0	0
$517$	−0.762617	−0.0335398
$518$	−22.0970	−0.970887
$519$	0	0
$520$	10.3263	0.452840
$521$	18.8682	0.826633	0.413316	−	0.910588i	$-0.364370\pi$
$521$	18.8682	0.826633	0.413316	+	0.910588i	$0.364370\pi$
$522$	0	0
$523$	34.8252	1.52280	0.761400	−	0.648282i	$-0.224512\pi$
$523$	34.8252	1.52280	0.761400	+	0.648282i	$0.224512\pi$
$524$	8.78113	0.383605
$525$	0	0
$526$	10.5170	0.458565
$527$	−7.99235	−0.348152
$528$	0	0
$529$	64.1508	2.78916
$530$	13.0036	0.564840
$531$	0	0
$532$	13.0710	0.566699
$533$	36.4211	1.57757
$534$	0	0
$535$	−1.83003	−0.0791193
$536$	31.4617	1.35894
$537$	0	0
$538$	−18.5256	−0.798695
$539$	1.17856	0.0507641
$540$	0	0
$541$	43.0509	1.85090	0.925452	−	0.378865i	$-0.123686\pi$
$541$	43.0509	1.85090	0.925452	+	0.378865i	$0.123686\pi$
$542$	29.9933	1.28832
$543$	0	0
$544$	3.58963	0.153904
$545$	−6.43239	−0.275533
$546$	0	0
$547$	31.6240	1.35214	0.676072	−	0.736835i	$-0.263681\pi$
$547$	31.6240	1.35214	0.676072	+	0.736835i	$0.263681\pi$
$548$	−4.83138	−0.206386
$549$	0	0
$550$	−3.53202	−0.150606
$551$	−3.37596	−0.143821
$552$	0	0
$553$	−17.8345	−0.758402
$554$	11.8988	0.505530
$555$	0	0
$556$	18.9269	0.802681
$557$	39.9873	1.69432	0.847158	−	0.531341i	$-0.178312\pi$
$557$	39.9873	1.69432	0.847158	+	0.531341i	$0.178312\pi$
$558$	0	0
$559$	−16.5646	−0.700607
$560$	3.13599	0.132520
$561$	0	0
$562$	−16.3374	−0.689152
$563$	−10.0422	−0.423227	−0.211613	−	0.977353i	$-0.567872\pi$
$563$	−10.0422	−0.423227	−0.211613	+	0.977353i	$0.567872\pi$
$564$	0	0
$565$	11.4828	0.483084
$566$	2.01534	0.0847109
$567$	0	0
$568$	−32.7061	−1.37232
$569$	−16.1008	−0.674982	−0.337491	−	0.941329i	$-0.609578\pi$
$569$	−16.1008	−0.674982	−0.337491	+	0.941329i	$0.609578\pi$
$570$	0	0
$571$	0.878670	0.0367712	0.0183856	−	0.999831i	$-0.494147\pi$
$571$	0.878670	0.0367712	0.0183856	+	0.999831i	$0.494147\pi$
$572$	11.1008	0.464147
$573$	0	0
$574$	28.2676	1.17987
$575$	9.33546	0.389316
$576$	0	0
$577$	−9.47221	−0.394333	−0.197167	−	0.980370i	$-0.563174\pi$
$577$	−9.47221	−0.394333	−0.197167	+	0.980370i	$0.563174\pi$
$578$	−16.8839	−0.702278
$579$	0	0
$580$	0.595622	0.0247318
$581$	−36.0564	−1.49587
$582$	0	0
$583$	43.5511	1.80370
$584$	2.12658	0.0879985
$585$	0	0
$586$	−15.6850	−0.647942
$587$	43.6472	1.80151	0.900757	−	0.434323i	$-0.143012\pi$
$587$	43.6472	1.80151	0.900757	+	0.434323i	$0.143012\pi$
$588$	0	0
$589$	57.2833	2.36032
$590$	−4.27386	−0.175952
$591$	0	0
$592$	10.1359	0.416585
$593$	−6.64796	−0.272999	−0.136499	−	0.990640i	$-0.543585\pi$
$593$	−6.64796	−0.272999	−0.136499	+	0.990640i	$0.543585\pi$
$594$	0	0
$595$	−1.92903	−0.0790825
$596$	−0.481493	−0.0197227
$597$	0	0
$598$	32.7294	1.33840
$599$	−15.2897	−0.624721	−0.312361	−	0.949964i	$-0.601120\pi$
$599$	−15.2897	−0.624721	−0.312361	+	0.949964i	$0.601120\pi$
$600$	0	0
$601$	1.09542	0.0446831	0.0223415	−	0.999750i	$-0.492888\pi$
$601$	1.09542	0.0446831	0.0223415	+	0.999750i	$0.492888\pi$
$602$	−12.8563	−0.523983
$603$	0	0
$604$	−0.127146	−0.00517350
$605$	−0.829298	−0.0337158
$606$	0	0
$607$	5.15786	0.209351	0.104676	−	0.994506i	$-0.466620\pi$
$607$	5.15786	0.209351	0.104676	+	0.994506i	$0.466620\pi$
$608$	−25.7279	−1.04340
$609$	0	0
$610$	−1.16324	−0.0470982
$611$	0.756981	0.0306242
$612$	0	0
$613$	25.2363	1.01928	0.509642	−	0.860387i	$-0.329778\pi$
$613$	25.2363	1.01928	0.509642	+	0.860387i	$0.329778\pi$
$614$	3.84871	0.155321
$615$	0	0
$616$	26.8422	1.08150
$617$	0.353769	0.0142422	0.00712110	−	0.999975i	$-0.497733\pi$
$617$	0.353769	0.0142422	0.00712110	+	0.999975i	$0.497733\pi$
$618$	0	0
$619$	1.52817	0.0614225	0.0307113	−	0.999528i	$-0.490223\pi$
$619$	1.52817	0.0614225	0.0307113	+	0.999528i	$0.490223\pi$
$620$	−10.1065	−0.405888
$621$	0	0
$622$	5.33139	0.213769
$623$	−2.58018	−0.103373
$624$	0	0
$625$	1.00000	0.0400000
$626$	9.11893	0.364466
$627$	0	0
$628$	−3.09528	−0.123515
$629$	−6.23489	−0.248601
$630$	0	0
$631$	26.0670	1.03771	0.518856	−	0.854862i	$-0.326358\pi$
$631$	26.0670	1.03771	0.518856	+	0.854862i	$0.326358\pi$
$632$	20.9074	0.831652
$633$	0	0
$634$	−30.4562	−1.20957
$635$	19.3676	0.768580
$636$	0	0
$637$	−1.16985	−0.0463511
$638$	−2.22524	−0.0880982
$639$	0	0
$640$	2.04287	0.0807517
$641$	37.7869	1.49249	0.746247	−	0.665669i	$-0.231854\pi$
$641$	37.7869	1.49249	0.746247	+	0.665669i	$0.231854\pi$
$642$	0	0
$643$	−5.10991	−0.201515	−0.100758	−	0.994911i	$-0.532127\pi$
$643$	−5.10991	−0.201515	−0.100758	+	0.994911i	$0.532127\pi$
$644$	−22.7720	−0.897344
$645$	0	0
$646$	−4.11410	−0.161867
$647$	5.60327	0.220287	0.110144	−	0.993916i	$-0.464869\pi$
$647$	5.60327	0.220287	0.110144	+	0.993916i	$0.464869\pi$
$648$	0	0
$649$	−14.3139	−0.561868
$650$	3.50592	0.137513
$651$	0	0
$652$	−5.18078	−0.202895
$653$	27.5203	1.07695	0.538476	−	0.842641i	$-0.319000\pi$
$653$	27.5203	1.07695	0.538476	+	0.842641i	$0.319000\pi$
$654$	0	0
$655$	9.28826	0.362922
$656$	−12.9664	−0.506253
$657$	0	0
$658$	0.587517	0.0229038
$659$	19.5617	0.762016	0.381008	−	0.924572i	$-0.375577\pi$
$659$	19.5617	0.762016	0.381008	+	0.924572i	$0.375577\pi$
$660$	0	0
$661$	−46.4222	−1.80561	−0.902806	−	0.430047i	$-0.858497\pi$
$661$	−46.4222	−1.80561	−0.902806	+	0.430047i	$0.858497\pi$
$662$	15.1931	0.590498
$663$	0	0
$664$	42.2688	1.64035
$665$	13.8259	0.536144
$666$	0	0
$667$	5.88153	0.227734
$668$	11.9403	0.461983
$669$	0	0
$670$	10.6816	0.412668
$671$	−3.89588	−0.150399
$672$	0	0
$673$	28.7048	1.10649	0.553244	−	0.833019i	$-0.313390\pi$
$673$	28.7048	1.10649	0.553244	+	0.833019i	$0.313390\pi$
$674$	0.533557	0.0205518
$675$	0	0
$676$	1.27146	0.0489024
$677$	30.4780	1.17136	0.585682	−	0.810541i	$-0.300827\pi$
$677$	30.4780	1.17136	0.585682	+	0.810541i	$0.300827\pi$
$678$	0	0
$679$	−37.3968	−1.43516
$680$	2.26140	0.0867206
$681$	0	0
$682$	37.7580	1.44583
$683$	−21.0963	−0.807226	−0.403613	−	0.914930i	$-0.632246\pi$
$683$	−21.0963	−0.807226	−0.403613	+	0.914930i	$0.632246\pi$
$684$	0	0
$685$	−5.11040	−0.195258
$686$	−19.4557	−0.742823
$687$	0	0
$688$	5.89721	0.224829
$689$	−43.2293	−1.64691
$690$	0	0
$691$	24.4658	0.930725	0.465362	−	0.885120i	$-0.345924\pi$
$691$	24.4658	0.930725	0.465362	+	0.885120i	$0.345924\pi$
$692$	−4.11125	−0.156286
$693$	0	0
$694$	10.6753	0.405229
$695$	20.0200	0.759402
$696$	0	0
$697$	7.97597	0.302111
$698$	11.9978	0.454125
$699$	0	0
$700$	−2.43931	−0.0921971
$701$	−11.4953	−0.434170	−0.217085	−	0.976153i	$-0.569655\pi$
$701$	−11.4953	−0.434170	−0.217085	+	0.976153i	$0.569655\pi$
$702$	0	0
$703$	44.6871	1.68541
$704$	−25.3189	−0.954242
$705$	0	0
$706$	12.2926	0.462637
$707$	−3.19752	−0.120255
$708$	0	0
$709$	−19.4126	−0.729054	−0.364527	−	0.931193i	$-0.618769\pi$
$709$	−19.4126	−0.729054	−0.364527	+	0.931193i	$0.618769\pi$
$710$	−11.1041	−0.416731
$711$	0	0
$712$	3.02474	0.113357
$713$	−99.7980	−3.73746
$714$	0	0
$715$	11.7419	0.439121
$716$	14.1292	0.528032
$717$	0	0
$718$	8.92629	0.333126
$719$	21.1113	0.787317	0.393659	−	0.919257i	$-0.371209\pi$
$719$	21.1113	0.787317	0.393659	+	0.919257i	$0.371209\pi$
$720$	0	0
$721$	0.927436	0.0345395
$722$	9.97504	0.371233
$723$	0	0
$724$	−11.8706	−0.441166
$725$	0.630020	0.0233984
$726$	0	0
$727$	−42.6780	−1.58284	−0.791420	−	0.611273i	$-0.790658\pi$
$727$	−42.6780	−1.58284	−0.791420	+	0.611273i	$0.790658\pi$
$728$	−26.6438	−0.987485
$729$	0	0
$730$	0.722000	0.0267224
$731$	−3.62753	−0.134169
$732$	0	0
$733$	1.27363	0.0470428	0.0235214	−	0.999723i	$-0.492512\pi$
$733$	1.27363	0.0470428	0.0235214	+	0.999723i	$0.492512\pi$
$734$	36.0492	1.33060
$735$	0	0
$736$	44.8226	1.65218
$737$	35.7745	1.31777
$738$	0	0
$739$	−33.6104	−1.23638	−0.618189	−	0.786030i	$-0.712133\pi$
$739$	−33.6104	−1.23638	−0.618189	+	0.786030i	$0.712133\pi$
$740$	−7.88417	−0.289828
$741$	0	0
$742$	−33.5516	−1.23172
$743$	−7.00792	−0.257096	−0.128548	−	0.991703i	$-0.541032\pi$
$743$	−7.00792	−0.257096	−0.128548	+	0.991703i	$0.541032\pi$
$744$	0	0
$745$	−0.509300	−0.0186593
$746$	−32.8561	−1.20295
$747$	0	0
$748$	2.43100	0.0888860
$749$	4.72182	0.172532
$750$	0	0
$751$	−8.55775	−0.312277	−0.156138	−	0.987735i	$-0.549905\pi$
$751$	−8.55775	−0.312277	−0.156138	+	0.987735i	$0.549905\pi$
$752$	−0.269495	−0.00982748
$753$	0	0
$754$	2.20880	0.0804397
$755$	−0.134489	−0.00489455
$756$	0	0
$757$	−7.42031	−0.269696	−0.134848	−	0.990866i	$-0.543055\pi$
$757$	−7.42031	−0.269696	−0.134848	+	0.990866i	$0.543055\pi$
$758$	5.45681	0.198200
$759$	0	0
$760$	−16.2080	−0.587927
$761$	24.5803	0.891034	0.445517	−	0.895274i	$-0.353020\pi$
$761$	24.5803	0.891034	0.445517	+	0.895274i	$0.353020\pi$
$762$	0	0
$763$	16.5967	0.600842
$764$	−7.40181	−0.267788
$765$	0	0
$766$	21.6505	0.782264
$767$	14.2081	0.513024
$768$	0	0
$769$	−8.88149	−0.320275	−0.160137	−	0.987095i	$-0.551194\pi$
$769$	−8.88149	−0.320275	−0.160137	+	0.987095i	$0.551194\pi$
$770$	9.11325	0.328419
$771$	0	0
$772$	−19.5992	−0.705390
$773$	23.9252	0.860531	0.430266	−	0.902702i	$-0.358420\pi$
$773$	23.9252	0.860531	0.430266	+	0.902702i	$0.358420\pi$
$774$	0	0
$775$	−10.6902	−0.384003
$776$	43.8402	1.57377
$777$	0	0
$778$	−18.7099	−0.670784
$779$	−57.1659	−2.04818
$780$	0	0
$781$	−37.1895	−1.33075
$782$	7.16751	0.256309
$783$	0	0
$784$	0.416482	0.0148744
$785$	−3.27404	−0.116855
$786$	0	0
$787$	−18.7332	−0.667768	−0.333884	−	0.942614i	$-0.608359\pi$
$787$	−18.7332	−0.667768	−0.333884	+	0.942614i	$0.608359\pi$
$788$	−3.63733	−0.129575
$789$	0	0
$790$	7.09832	0.252547
$791$	−29.6276	−1.05344
$792$	0	0
$793$	3.86709	0.137324
$794$	−29.5261	−1.04784
$795$	0	0
$796$	22.4903	0.797148
$797$	−30.4779	−1.07958	−0.539792	−	0.841799i	$-0.681497\pi$
$797$	−30.4779	−1.07958	−0.539792	+	0.841799i	$0.681497\pi$
$798$	0	0
$799$	0.165774	0.00586465
$800$	4.80133	0.169753
$801$	0	0
$802$	18.4301	0.650791
$803$	2.41809	0.0853327
$804$	0	0
$805$	−24.0872	−0.848961
$806$	−37.4790	−1.32014
$807$	0	0
$808$	3.74845	0.131870
$809$	14.1591	0.497806	0.248903	−	0.968528i	$-0.419930\pi$
$809$	14.1591	0.497806	0.248903	+	0.968528i	$0.419930\pi$
$810$	0	0
$811$	24.1079	0.846543	0.423272	−	0.906003i	$-0.360882\pi$
$811$	24.1079	0.846543	0.423272	+	0.906003i	$0.360882\pi$
$812$	−1.53681	−0.0539315
$813$	0	0
$814$	29.4553	1.03241
$815$	−5.47998	−0.191955
$816$	0	0
$817$	25.9995	0.909606
$818$	21.7022	0.758799
$819$	0	0
$820$	10.0858	0.352212
$821$	3.57462	0.124755	0.0623775	−	0.998053i	$-0.480132\pi$
$821$	3.57462	0.124755	0.0623775	+	0.998053i	$0.480132\pi$
$822$	0	0
$823$	−38.1738	−1.33065	−0.665327	−	0.746552i	$-0.731708\pi$
$823$	−38.1738	−1.33065	−0.665327	+	0.746552i	$0.731708\pi$
$824$	−1.08723	−0.0378755
$825$	0	0
$826$	11.0273	0.383690
$827$	20.3517	0.707699	0.353850	−	0.935302i	$-0.384873\pi$
$827$	20.3517	0.707699	0.353850	+	0.935302i	$0.384873\pi$
$828$	0	0
$829$	30.8702	1.07217	0.536084	−	0.844165i	$-0.319903\pi$
$829$	30.8702	1.07217	0.536084	+	0.844165i	$0.319903\pi$
$830$	14.3508	0.498123
$831$	0	0
$832$	25.1318	0.871289
$833$	−0.256189	−0.00887642
$834$	0	0
$835$	12.6299	0.437074
$836$	−17.4236	−0.602608
$837$	0	0
$838$	−23.7585	−0.820723
$839$	−1.34419	−0.0464067	−0.0232034	−	0.999731i	$-0.507387\pi$
$839$	−1.34419	−0.0464067	−0.0232034	+	0.999731i	$0.507387\pi$
$840$	0	0
$841$	−28.6031	−0.986313
$842$	−2.09747	−0.0722837
$843$	0	0
$844$	−12.4232	−0.427625
$845$	1.34489	0.0462657
$846$	0	0
$847$	2.13974	0.0735223
$848$	15.3902	0.528502
$849$	0	0
$850$	0.767772	0.0263344
$851$	−77.8530	−2.66877
$852$	0	0
$853$	−34.0459	−1.16571	−0.582855	−	0.812576i	$-0.698064\pi$
$853$	−34.0459	−1.16571	−0.582855	+	0.812576i	$0.698064\pi$
$854$	3.00137	0.102705
$855$	0	0
$856$	−5.53538	−0.189195
$857$	−28.9125	−0.987631	−0.493816	−	0.869567i	$-0.664398\pi$
$857$	−28.9125	−0.987631	−0.493816	+	0.869567i	$0.664398\pi$
$858$	0	0
$859$	−12.9294	−0.441144	−0.220572	−	0.975371i	$-0.570792\pi$
$859$	−12.9294	−0.441144	−0.220572	+	0.975371i	$0.570792\pi$
$860$	−4.58710	−0.156419
$861$	0	0
$862$	−21.3574	−0.727434
$863$	16.0880	0.547642	0.273821	−	0.961781i	$-0.411712\pi$
$863$	16.0880	0.547642	0.273821	+	0.961781i	$0.411712\pi$
$864$	0	0
$865$	−4.34868	−0.147860
$866$	−29.0875	−0.988435
$867$	0	0
$868$	26.0767	0.885100
$869$	23.7734	0.806457
$870$	0	0
$871$	−35.5101	−1.20322
$872$	−19.4563	−0.658874
$873$	0	0
$874$	−51.3714	−1.73766
$875$	−2.58018	−0.0872260
$876$	0	0
$877$	39.2005	1.32371	0.661854	−	0.749633i	$-0.269770\pi$
$877$	39.2005	1.32371	0.661854	+	0.749633i	$0.269770\pi$
$878$	9.22906	0.311466
$879$	0	0
$880$	−4.18027	−0.140917
$881$	37.9154	1.27740	0.638702	−	0.769454i	$-0.279472\pi$
$881$	37.9154	1.27740	0.638702	+	0.769454i	$0.279472\pi$
$882$	0	0
$883$	−46.9451	−1.57983	−0.789915	−	0.613216i	$-0.789875\pi$
$883$	−46.9451	−1.57983	−0.789915	+	0.613216i	$0.789875\pi$
$884$	−2.41303	−0.0811590
$885$	0	0
$886$	−16.5605	−0.556359
$887$	6.37871	0.214176	0.107088	−	0.994250i	$-0.465847\pi$
$887$	6.37871	0.214176	0.107088	+	0.994250i	$0.465847\pi$
$888$	0	0
$889$	−49.9719	−1.67600
$890$	1.02694	0.0344230
$891$	0	0
$892$	10.1018	0.338234
$893$	−1.18814	−0.0397597
$894$	0	0
$895$	14.9452	0.499562
$896$	−5.27098	−0.176091
$897$	0	0
$898$	7.56455	0.252432
$899$	−6.73504	−0.224626
$900$	0	0
$901$	−9.46692	−0.315389
$902$	−37.6806	−1.25463
$903$	0	0
$904$	34.7324	1.15518
$905$	−12.5561	−0.417379
$906$	0	0
$907$	14.1935	0.471286	0.235643	−	0.971840i	$-0.424280\pi$
$907$	14.1935	0.471286	0.235643	+	0.971840i	$0.424280\pi$
$908$	−2.91128	−0.0966142
$909$	0	0
$910$	−9.04590	−0.299869
$911$	−58.6271	−1.94240	−0.971202	−	0.238259i	$-0.923423\pi$
$911$	−58.6271	−1.94240	−0.971202	+	0.238259i	$0.923423\pi$
$912$	0	0
$913$	48.0631	1.59066
$914$	−26.7175	−0.883735
$915$	0	0
$916$	4.35980	0.144052
$917$	−23.9654	−0.791407
$918$	0	0
$919$	−12.1429	−0.400556	−0.200278	−	0.979739i	$-0.564185\pi$
$919$	−12.1429	−0.400556	−0.200278	+	0.979739i	$0.564185\pi$
$920$	28.2373	0.930957
$921$	0	0
$922$	−40.4848	−1.33330
$923$	36.9147	1.21506
$924$	0	0
$925$	−8.33949	−0.274201
$926$	−6.75418	−0.221956
$927$	0	0
$928$	3.02493	0.0992983
$929$	56.4685	1.85267	0.926336	−	0.376697i	$-0.122940\pi$
$929$	56.4685	1.85267	0.926336	+	0.376697i	$0.122940\pi$
$930$	0	0
$931$	1.83617	0.0601782
$932$	−11.8061	−0.386722
$933$	0	0
$934$	33.4300	1.09386
$935$	2.57139	0.0840935
$936$	0	0
$937$	−48.9205	−1.59816	−0.799081	−	0.601224i	$-0.794680\pi$
$937$	−48.9205	−1.59816	−0.799081	+	0.601224i	$0.794680\pi$
$938$	−27.5606	−0.899884
$939$	0	0
$940$	0.209625	0.00683721
$941$	−56.5237	−1.84262	−0.921309	−	0.388830i	$-0.872879\pi$
$941$	−56.5237	−1.84262	−0.921309	+	0.388830i	$0.872879\pi$
$942$	0	0
$943$	99.5934	3.24321
$944$	−5.05826	−0.164632
$945$	0	0
$946$	17.1374	0.557185
$947$	−12.1785	−0.395748	−0.197874	−	0.980228i	$-0.563404\pi$
$947$	−12.1785	−0.395748	−0.197874	+	0.980228i	$0.563404\pi$
$948$	0	0
$949$	−2.40022	−0.0779146
$950$	−5.50283	−0.178535
$951$	0	0
$952$	−5.83481	−0.189107
$953$	50.2787	1.62869	0.814344	−	0.580383i	$-0.197097\pi$
$953$	50.2787	1.62869	0.814344	+	0.580383i	$0.197097\pi$
$954$	0	0
$955$	−7.82928	−0.253350
$956$	18.0797	0.584740
$957$	0	0
$958$	21.3982	0.691345
$959$	13.1858	0.425790
$960$	0	0
$961$	83.2805	2.68647
$962$	−29.2376	−0.942658
$963$	0	0
$964$	−21.3268	−0.686891
$965$	−20.7311	−0.667357
$966$	0	0
$967$	−32.2406	−1.03679	−0.518395	−	0.855142i	$-0.673470\pi$
$967$	−32.2406	−1.03679	−0.518395	+	0.855142i	$0.673470\pi$
$968$	−2.50841	−0.0806234
$969$	0	0
$970$	14.8843	0.477905
$971$	5.44527	0.174747	0.0873736	−	0.996176i	$-0.472153\pi$
$971$	5.44527	0.174747	0.0873736	+	0.996176i	$0.472153\pi$
$972$	0	0
$973$	−51.6552	−1.65599
$974$	14.5711	0.466888
$975$	0	0
$976$	−1.37673	−0.0440682
$977$	−9.50778	−0.304181	−0.152090	−	0.988367i	$-0.548600\pi$
$977$	−9.50778	−0.304181	−0.152090	+	0.988367i	$0.548600\pi$
$978$	0	0
$979$	3.43937	0.109923
$980$	−0.323957	−0.0103484
$981$	0	0
$982$	−0.892463	−0.0284796
$983$	−28.5271	−0.909872	−0.454936	−	0.890524i	$-0.650338\pi$
$983$	−28.5271	−0.909872	−0.454936	+	0.890524i	$0.650338\pi$
$984$	0	0
$985$	−3.84740	−0.122588
$986$	0.483712	0.0154045
$987$	0	0
$988$	17.2948	0.550222
$989$	−45.2958	−1.44032
$990$	0	0
$991$	−54.2885	−1.72453	−0.862266	−	0.506455i	$-0.830956\pi$
$991$	−54.2885	−1.72453	−0.862266	+	0.506455i	$0.830956\pi$
$992$	−51.3272	−1.62964
$993$	0	0
$994$	28.6507	0.908744
$995$	23.7892	0.754167
$996$	0	0
$997$	−12.2788	−0.388874	−0.194437	−	0.980915i	$-0.562288\pi$
$997$	−12.2788	−0.388874	−0.194437	+	0.980915i	$0.562288\pi$
$998$	−5.37867	−0.170259
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.v.1.8	yes	12
3.2	odd	2		4005.2.a.u.1.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.u.1.5	✓	12	3.2	odd	2
4005.2.a.v.1.8	yes	12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.02694 q^{2} -0.945401 q^{4} -1.00000 q^{5} +2.58018 q^{7} -3.02474 q^{8} +O(q^{10})\)	\(q+1.02694 q^{2} -0.945401 q^{4} -1.00000 q^{5} +2.58018 q^{7} -3.02474 q^{8} -1.02694 q^{10} -3.43937 q^{11} +3.41396 q^{13} +2.64968 q^{14} -1.21541 q^{16} +0.747634 q^{17} -5.35849 q^{19} +0.945401 q^{20} -3.53202 q^{22} +9.33546 q^{23} +1.00000 q^{25} +3.50592 q^{26} -2.43931 q^{28} +0.630020 q^{29} -10.6902 q^{31} +4.80133 q^{32} +0.767772 q^{34} -2.58018 q^{35} -8.33949 q^{37} -5.50283 q^{38} +3.02474 q^{40} +10.6683 q^{41} -4.85201 q^{43} +3.25159 q^{44} +9.58692 q^{46} +0.221731 q^{47} -0.342667 q^{49} +1.02694 q^{50} -3.22756 q^{52} -12.6625 q^{53} +3.43937 q^{55} -7.80438 q^{56} +0.646991 q^{58} +4.16176 q^{59} +1.13273 q^{61} -10.9782 q^{62} +7.36149 q^{64} -3.41396 q^{65} -10.4015 q^{67} -0.706814 q^{68} -2.64968 q^{70} +10.8129 q^{71} -0.703062 q^{73} -8.56413 q^{74} +5.06592 q^{76} -8.87421 q^{77} -6.91213 q^{79} +1.21541 q^{80} +10.9557 q^{82} -13.9744 q^{83} -0.747634 q^{85} -4.98271 q^{86} +10.4032 q^{88} -1.00000 q^{89} +8.80863 q^{91} -8.82575 q^{92} +0.227704 q^{94} +5.35849 q^{95} -14.4939 q^{97} -0.351897 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{2} + 11 q^{4} - 12 q^{5} - 8 q^{7} + 9 q^{8}+O(q^{10}) \) 12 * q + 3 * q^2 + 11 * q^4 - 12 * q^5 - 8 * q^7 + 9 * q^8	\( 12 q + 3 q^{2} + 11 q^{4} - 12 q^{5} - 8 q^{7} + 9 q^{8} - 3 q^{10} - 12 q^{13} - 4 q^{14} + q^{16} - 24 q^{19} - 11 q^{20} - 16 q^{22} + 24 q^{23} + 12 q^{25} + q^{26} - 44 q^{28} - 8 q^{29} - 12 q^{31} + 31 q^{32} - 18 q^{34} + 8 q^{35} - 10 q^{37} - 2 q^{38} - 9 q^{40} + 10 q^{41} - 42 q^{43} - 42 q^{44} - 24 q^{46} + 22 q^{47} - 4 q^{49} + 3 q^{50} - 30 q^{52} - 8 q^{53} - 27 q^{56} - 12 q^{58} - 4 q^{59} - 52 q^{61} + 14 q^{62} + 7 q^{64} + 12 q^{65} - 40 q^{67} - 23 q^{68} + 4 q^{70} - 2 q^{71} - 8 q^{73} - 26 q^{74} - 46 q^{76} + 12 q^{77} - 26 q^{79} - q^{80} - 26 q^{82} + 14 q^{83} - 32 q^{86} - 60 q^{88} - 12 q^{89} - 24 q^{91} + 38 q^{92} - 26 q^{94} + 24 q^{95} - 6 q^{97} - 30 q^{98}+O(q^{100}) \) 12 * q + 3 * q^2 + 11 * q^4 - 12 * q^5 - 8 * q^7 + 9 * q^8 - 3 * q^10 - 12 * q^13 - 4 * q^14 + q^16 - 24 * q^19 - 11 * q^20 - 16 * q^22 + 24 * q^23 + 12 * q^25 + q^26 - 44 * q^28 - 8 * q^29 - 12 * q^31 + 31 * q^32 - 18 * q^34 + 8 * q^35 - 10 * q^37 - 2 * q^38 - 9 * q^40 + 10 * q^41 - 42 * q^43 - 42 * q^44 - 24 * q^46 + 22 * q^47 - 4 * q^49 + 3 * q^50 - 30 * q^52 - 8 * q^53 - 27 * q^56 - 12 * q^58 - 4 * q^59 - 52 * q^61 + 14 * q^62 + 7 * q^64 + 12 * q^65 - 40 * q^67 - 23 * q^68 + 4 * q^70 - 2 * q^71 - 8 * q^73 - 26 * q^74 - 46 * q^76 + 12 * q^77 - 26 * q^79 - q^80 - 26 * q^82 + 14 * q^83 - 32 * q^86 - 60 * q^88 - 12 * q^89 - 24 * q^91 + 38 * q^92 - 26 * q^94 + 24 * q^95 - 6 * q^97 - 30 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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