LMFDB - Embedded newform 4005.2.a.g.1.2

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(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{5} +6.56155 q^{8} +O(q^{10})$	$q+2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{5} +6.56155 q^{8} -2.56155 q^{10} -4.00000 q^{11} +2.00000 q^{13} +7.68466 q^{16} +1.12311 q^{17} +3.12311 q^{19} -4.56155 q^{20} -10.2462 q^{22} +5.56155 q^{23} +1.00000 q^{25} +5.12311 q^{26} +2.68466 q^{29} +7.12311 q^{31} +6.56155 q^{32} +2.87689 q^{34} +10.0000 q^{37} +8.00000 q^{38} -6.56155 q^{40} -2.68466 q^{41} +10.2462 q^{43} -18.2462 q^{44} +14.2462 q^{46} +8.00000 q^{47} -7.00000 q^{49} +2.56155 q^{50} +9.12311 q^{52} +8.24621 q^{53} +4.00000 q^{55} +6.87689 q^{58} -2.43845 q^{59} +2.87689 q^{61} +18.2462 q^{62} +1.43845 q^{64} -2.00000 q^{65} -15.8078 q^{67} +5.12311 q^{68} -6.24621 q^{71} +2.68466 q^{73} +25.6155 q^{74} +14.2462 q^{76} -2.43845 q^{79} -7.68466 q^{80} -6.87689 q^{82} -4.68466 q^{83} -1.12311 q^{85} +26.2462 q^{86} -26.2462 q^{88} -1.00000 q^{89} +25.3693 q^{92} +20.4924 q^{94} -3.12311 q^{95} -8.43845 q^{97} -17.9309 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 5 q^{4} - 2 q^{5} + 9 q^{8}+O(q^{10}) $ 2 * q + q^2 + 5 * q^4 - 2 * q^5 + 9 * q^8	$ 2 q + q^{2} + 5 q^{4} - 2 q^{5} + 9 q^{8} - q^{10} - 8 q^{11} + 4 q^{13} + 3 q^{16} - 6 q^{17} - 2 q^{19} - 5 q^{20} - 4 q^{22} + 7 q^{23} + 2 q^{25} + 2 q^{26} - 7 q^{29} + 6 q^{31} + 9 q^{32} + 14 q^{34} + 20 q^{37} + 16 q^{38} - 9 q^{40} + 7 q^{41} + 4 q^{43} - 20 q^{44} + 12 q^{46} + 16 q^{47} - 14 q^{49} + q^{50} + 10 q^{52} + 8 q^{55} + 22 q^{58} - 9 q^{59} + 14 q^{61} + 20 q^{62} + 7 q^{64} - 4 q^{65} - 11 q^{67} + 2 q^{68} + 4 q^{71} - 7 q^{73} + 10 q^{74} + 12 q^{76} - 9 q^{79} - 3 q^{80} - 22 q^{82} + 3 q^{83} + 6 q^{85} + 36 q^{86} - 36 q^{88} - 2 q^{89} + 26 q^{92} + 8 q^{94} + 2 q^{95} - 21 q^{97} - 7 q^{98}+O(q^{100}) $ 2 * q + q^2 + 5 * q^4 - 2 * q^5 + 9 * q^8 - q^10 - 8 * q^11 + 4 * q^13 + 3 * q^16 - 6 * q^17 - 2 * q^19 - 5 * q^20 - 4 * q^22 + 7 * q^23 + 2 * q^25 + 2 * q^26 - 7 * q^29 + 6 * q^31 + 9 * q^32 + 14 * q^34 + 20 * q^37 + 16 * q^38 - 9 * q^40 + 7 * q^41 + 4 * q^43 - 20 * q^44 + 12 * q^46 + 16 * q^47 - 14 * q^49 + q^50 + 10 * q^52 + 8 * q^55 + 22 * q^58 - 9 * q^59 + 14 * q^61 + 20 * q^62 + 7 * q^64 - 4 * q^65 - 11 * q^67 + 2 * q^68 + 4 * q^71 - 7 * q^73 + 10 * q^74 + 12 * q^76 - 9 * q^79 - 3 * q^80 - 22 * q^82 + 3 * q^83 + 6 * q^85 + 36 * q^86 - 36 * q^88 - 2 * q^89 + 26 * q^92 + 8 * q^94 + 2 * q^95 - 21 * q^97 - 7 * 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 $

$2$	2.56155	1.81129	0.905646	−	0.424035i	$-0.139387\pi$
$2$	2.56155	1.81129	0.905646	+	0.424035i	$0.139387\pi$
$3$	0	0
$3$	0	0
$4$	4.56155	2.28078
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	6.56155	2.31986
$9$	0	0
$10$	−2.56155	−0.810034
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	7.68466	1.92116
$17$	1.12311	0.272393	0.136197	−	0.990682i	$-0.456512\pi$
$17$	1.12311	0.272393	0.136197	+	0.990682i	$0.456512\pi$
$18$	0	0
$19$	3.12311	0.716490	0.358245	−	0.933628i	$-0.383375\pi$
$19$	3.12311	0.716490	0.358245	+	0.933628i	$0.383375\pi$
$20$	−4.56155	−1.01999
$21$	0	0
$22$	−10.2462	−2.18450
$23$	5.56155	1.15966	0.579832	−	0.814736i	$-0.303118\pi$
$23$	5.56155	1.15966	0.579832	+	0.814736i	$0.303118\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	5.12311	1.00472
$27$	0	0
$28$	0	0
$29$	2.68466	0.498529	0.249264	−	0.968436i	$-0.419811\pi$
$29$	2.68466	0.498529	0.249264	+	0.968436i	$0.419811\pi$
$30$	0	0
$31$	7.12311	1.27935	0.639674	−	0.768647i	$-0.279069\pi$
$31$	7.12311	1.27935	0.639674	+	0.768647i	$0.279069\pi$
$32$	6.56155	1.15993
$33$	0	0
$34$	2.87689	0.493383
$35$	0	0
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	8.00000	1.29777
$39$	0	0
$40$	−6.56155	−1.03747
$41$	−2.68466	−0.419273	−0.209637	−	0.977779i	$-0.567228\pi$
$41$	−2.68466	−0.419273	−0.209637	+	0.977779i	$0.567228\pi$
$42$	0	0
$43$	10.2462	1.56253	0.781266	−	0.624198i	$-0.214574\pi$
$43$	10.2462	1.56253	0.781266	+	0.624198i	$0.214574\pi$
$44$	−18.2462	−2.75072
$45$	0	0
$46$	14.2462	2.10049
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	2.56155	0.362258
$51$	0	0
$52$	9.12311	1.26515
$53$	8.24621	1.13270	0.566352	−	0.824163i	$-0.308354\pi$
$53$	8.24621	1.13270	0.566352	+	0.824163i	$0.308354\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	0	0
$58$	6.87689	0.902980
$59$	−2.43845	−0.317459	−0.158729	−	0.987322i	$-0.550740\pi$
$59$	−2.43845	−0.317459	−0.158729	+	0.987322i	$0.550740\pi$
$60$	0	0
$61$	2.87689	0.368349	0.184174	−	0.982894i	$-0.441039\pi$
$61$	2.87689	0.368349	0.184174	+	0.982894i	$0.441039\pi$
$62$	18.2462	2.31727
$63$	0	0
$64$	1.43845	0.179806
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−15.8078	−1.93123	−0.965613	−	0.259984i	$-0.916283\pi$
$67$	−15.8078	−1.93123	−0.965613	+	0.259984i	$0.916283\pi$
$68$	5.12311	0.621268
$69$	0	0
$70$	0	0
$71$	−6.24621	−0.741289	−0.370644	−	0.928775i	$-0.620863\pi$
$71$	−6.24621	−0.741289	−0.370644	+	0.928775i	$0.620863\pi$
$72$	0	0
$73$	2.68466	0.314216	0.157108	−	0.987581i	$-0.449783\pi$
$73$	2.68466	0.314216	0.157108	+	0.987581i	$0.449783\pi$
$74$	25.6155	2.97774
$75$	0	0
$76$	14.2462	1.63415
$77$	0	0
$78$	0	0
$79$	−2.43845	−0.274347	−0.137173	−	0.990547i	$-0.543802\pi$
$79$	−2.43845	−0.274347	−0.137173	+	0.990547i	$0.543802\pi$
$80$	−7.68466	−0.859171
$81$	0	0
$82$	−6.87689	−0.759426
$83$	−4.68466	−0.514208	−0.257104	−	0.966384i	$-0.582768\pi$
$83$	−4.68466	−0.514208	−0.257104	+	0.966384i	$0.582768\pi$
$84$	0	0
$85$	−1.12311	−0.121818
$86$	26.2462	2.83020
$87$	0	0
$88$	−26.2462	−2.79786
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	0	0
$92$	25.3693	2.64493
$93$	0	0
$94$	20.4924	2.11363
$95$	−3.12311	−0.320424
$96$	0	0
$97$	−8.43845	−0.856794	−0.428397	−	0.903591i	$-0.640922\pi$
$97$	−8.43845	−0.856794	−0.428397	+	0.903591i	$0.640922\pi$
$98$	−17.9309	−1.81129
$99$	0	0
$100$	4.56155	0.456155
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−4.87689	−0.480535	−0.240267	−	0.970707i	$-0.577235\pi$
$103$	−4.87689	−0.480535	−0.240267	+	0.970707i	$0.577235\pi$
$104$	13.1231	1.28683
$105$	0	0
$106$	21.1231	2.05166
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−4.43845	−0.425126	−0.212563	−	0.977147i	$-0.568181\pi$
$109$	−4.43845	−0.425126	−0.212563	+	0.977147i	$0.568181\pi$
$110$	10.2462	0.976938
$111$	0	0
$112$	0	0
$113$	−17.8078	−1.67521	−0.837607	−	0.546274i	$-0.816046\pi$
$113$	−17.8078	−1.67521	−0.837607	+	0.546274i	$0.816046\pi$
$114$	0	0
$115$	−5.56155	−0.518617
$116$	12.2462	1.13703
$117$	0	0
$118$	−6.24621	−0.575010
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	7.36932	0.667187
$123$	0	0
$124$	32.4924	2.91791
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.24621	0.554262	0.277131	−	0.960832i	$-0.410616\pi$
$127$	6.24621	0.554262	0.277131	+	0.960832i	$0.410616\pi$
$128$	−9.43845	−0.834249
$129$	0	0
$130$	−5.12311	−0.449326
$131$	−13.3693	−1.16808	−0.584041	−	0.811724i	$-0.698529\pi$
$131$	−13.3693	−1.16808	−0.584041	+	0.811724i	$0.698529\pi$
$132$	0	0
$133$	0	0
$134$	−40.4924	−3.49801
$135$	0	0
$136$	7.36932	0.631914
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	6.43845	0.546102	0.273051	−	0.962000i	$-0.411967\pi$
$139$	6.43845	0.546102	0.273051	+	0.962000i	$0.411967\pi$
$140$	0	0
$141$	0	0
$142$	−16.0000	−1.34269
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−2.68466	−0.222949
$146$	6.87689	0.569136
$147$	0	0
$148$	45.6155	3.74957
$149$	−16.0540	−1.31519	−0.657596	−	0.753370i	$-0.728427\pi$
$149$	−16.0540	−1.31519	−0.657596	+	0.753370i	$0.728427\pi$
$150$	0	0
$151$	−10.2462	−0.833825	−0.416912	−	0.908947i	$-0.636888\pi$
$151$	−10.2462	−0.833825	−0.416912	+	0.908947i	$0.636888\pi$
$152$	20.4924	1.66215
$153$	0	0
$154$	0	0
$155$	−7.12311	−0.572142
$156$	0	0
$157$	5.31534	0.424210	0.212105	−	0.977247i	$-0.431968\pi$
$157$	5.31534	0.424210	0.212105	+	0.977247i	$0.431968\pi$
$158$	−6.24621	−0.496922
$159$	0	0
$160$	−6.56155	−0.518736
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−12.2462	−0.956268
$165$	0	0
$166$	−12.0000	−0.931381
$167$	17.3693	1.34408	0.672039	−	0.740516i	$-0.265419\pi$
$167$	17.3693	1.34408	0.672039	+	0.740516i	$0.265419\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−2.87689	−0.220648
$171$	0	0
$172$	46.7386	3.56379
$173$	6.87689	0.522841	0.261420	−	0.965225i	$-0.415809\pi$
$173$	6.87689	0.522841	0.261420	+	0.965225i	$0.415809\pi$
$174$	0	0
$175$	0	0
$176$	−30.7386	−2.31701
$177$	0	0
$178$	−2.56155	−0.191997
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	15.3693	1.14239	0.571196	−	0.820814i	$-0.306480\pi$
$181$	15.3693	1.14239	0.571196	+	0.820814i	$0.306480\pi$
$182$	0	0
$183$	0	0
$184$	36.4924	2.69026
$185$	−10.0000	−0.735215
$186$	0	0
$187$	−4.49242	−0.328518
$188$	36.4924	2.66148
$189$	0	0
$190$	−8.00000	−0.580381
$191$	11.3153	0.818749	0.409375	−	0.912366i	$-0.365747\pi$
$191$	11.3153	0.818749	0.409375	+	0.912366i	$0.365747\pi$
$192$	0	0
$193$	9.12311	0.656696	0.328348	−	0.944557i	$-0.393508\pi$
$193$	9.12311	0.656696	0.328348	+	0.944557i	$0.393508\pi$
$194$	−21.6155	−1.55190
$195$	0	0
$196$	−31.9309	−2.28078
$197$	−16.2462	−1.15749	−0.578747	−	0.815507i	$-0.696458\pi$
$197$	−16.2462	−1.15749	−0.578747	+	0.815507i	$0.696458\pi$
$198$	0	0
$199$	4.19224	0.297180	0.148590	−	0.988899i	$-0.452527\pi$
$199$	4.19224	0.297180	0.148590	+	0.988899i	$0.452527\pi$
$200$	6.56155	0.463972
$201$	0	0
$202$	−15.3693	−1.08138
$203$	0	0
$204$	0	0
$205$	2.68466	0.187505
$206$	−12.4924	−0.870388
$207$	0	0
$208$	15.3693	1.06567
$209$	−12.4924	−0.864119
$210$	0	0
$211$	22.2462	1.53149	0.765746	−	0.643143i	$-0.222370\pi$
$211$	22.2462	1.53149	0.765746	+	0.643143i	$0.222370\pi$
$212$	37.6155	2.58345
$213$	0	0
$214$	−10.2462	−0.700417
$215$	−10.2462	−0.698786
$216$	0	0
$217$	0	0
$218$	−11.3693	−0.770027
$219$	0	0
$220$	18.2462	1.23016
$221$	2.24621	0.151097
$222$	0	0
$223$	−5.56155	−0.372429	−0.186215	−	0.982509i	$-0.559622\pi$
$223$	−5.56155	−0.372429	−0.186215	+	0.982509i	$0.559622\pi$
$224$	0	0
$225$	0	0
$226$	−45.6155	−3.03430
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	7.36932	0.486978	0.243489	−	0.969904i	$-0.421708\pi$
$229$	7.36932	0.486978	0.243489	+	0.969904i	$0.421708\pi$
$230$	−14.2462	−0.939367
$231$	0	0
$232$	17.6155	1.15652
$233$	−14.8769	−0.974618	−0.487309	−	0.873230i	$-0.662021\pi$
$233$	−14.8769	−0.974618	−0.487309	+	0.873230i	$0.662021\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	−11.1231	−0.724053
$237$	0	0
$238$	0	0
$239$	−7.80776	−0.505042	−0.252521	−	0.967591i	$-0.581260\pi$
$239$	−7.80776	−0.505042	−0.252521	+	0.967591i	$0.581260\pi$
$240$	0	0
$241$	−1.12311	−0.0723456	−0.0361728	−	0.999346i	$-0.511517\pi$
$241$	−1.12311	−0.0723456	−0.0361728	+	0.999346i	$0.511517\pi$
$242$	12.8078	0.823314
$243$	0	0
$244$	13.1231	0.840121
$245$	7.00000	0.447214
$246$	0	0
$247$	6.24621	0.397437
$248$	46.7386	2.96791
$249$	0	0
$250$	−2.56155	−0.162007
$251$	−21.3693	−1.34882	−0.674410	−	0.738357i	$-0.735602\pi$
$251$	−21.3693	−1.34882	−0.674410	+	0.738357i	$0.735602\pi$
$252$	0	0
$253$	−22.2462	−1.39861
$254$	16.0000	1.00393
$255$	0	0
$256$	−27.0540	−1.69087
$257$	−6.87689	−0.428969	−0.214484	−	0.976727i	$-0.568807\pi$
$257$	−6.87689	−0.428969	−0.214484	+	0.976727i	$0.568807\pi$
$258$	0	0
$259$	0	0
$260$	−9.12311	−0.565791
$261$	0	0
$262$	−34.2462	−2.11574
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−8.24621	−0.506561
$266$	0	0
$267$	0	0
$268$	−72.1080	−4.40469
$269$	30.4924	1.85916	0.929578	−	0.368626i	$-0.120172\pi$
$269$	30.4924	1.85916	0.929578	+	0.368626i	$0.120172\pi$
$270$	0	0
$271$	−26.7386	−1.62426	−0.812128	−	0.583479i	$-0.801691\pi$
$271$	−26.7386	−1.62426	−0.812128	+	0.583479i	$0.801691\pi$
$272$	8.63068	0.523312
$273$	0	0
$274$	25.6155	1.54749
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−0.930870	−0.0559305	−0.0279653	−	0.999609i	$-0.508903\pi$
$277$	−0.930870	−0.0559305	−0.0279653	+	0.999609i	$0.508903\pi$
$278$	16.4924	0.989150
$279$	0	0
$280$	0	0
$281$	12.2462	0.730548	0.365274	−	0.930900i	$-0.380975\pi$
$281$	12.2462	0.730548	0.365274	+	0.930900i	$0.380975\pi$
$282$	0	0
$283$	−2.24621	−0.133523	−0.0667617	−	0.997769i	$-0.521267\pi$
$283$	−2.24621	−0.133523	−0.0667617	+	0.997769i	$0.521267\pi$
$284$	−28.4924	−1.69071
$285$	0	0
$286$	−20.4924	−1.21174
$287$	0	0
$288$	0	0
$289$	−15.7386	−0.925802
$290$	−6.87689	−0.403825
$291$	0	0
$292$	12.2462	0.716655
$293$	−29.4233	−1.71893	−0.859464	−	0.511197i	$-0.829202\pi$
$293$	−29.4233	−1.71893	−0.859464	+	0.511197i	$0.829202\pi$
$294$	0	0
$295$	2.43845	0.141972
$296$	65.6155	3.81383
$297$	0	0
$298$	−41.1231	−2.38220
$299$	11.1231	0.643266
$300$	0	0
$301$	0	0
$302$	−26.2462	−1.51030
$303$	0	0
$304$	24.0000	1.37649
$305$	−2.87689	−0.164730
$306$	0	0
$307$	10.9309	0.623858	0.311929	−	0.950105i	$-0.399025\pi$
$307$	10.9309	0.623858	0.311929	+	0.950105i	$0.399025\pi$
$308$	0	0
$309$	0	0
$310$	−18.2462	−1.03632
$311$	17.3693	0.984924	0.492462	−	0.870334i	$-0.336097\pi$
$311$	17.3693	0.984924	0.492462	+	0.870334i	$0.336097\pi$
$312$	0	0
$313$	13.6155	0.769595	0.384798	−	0.923001i	$-0.374271\pi$
$313$	13.6155	0.769595	0.384798	+	0.923001i	$0.374271\pi$
$314$	13.6155	0.768369
$315$	0	0
$316$	−11.1231	−0.625724
$317$	−2.49242	−0.139988	−0.0699942	−	0.997547i	$-0.522298\pi$
$317$	−2.49242	−0.139988	−0.0699942	+	0.997547i	$0.522298\pi$
$318$	0	0
$319$	−10.7386	−0.601248
$320$	−1.43845	−0.0804116
$321$	0	0
$322$	0	0
$323$	3.50758	0.195167
$324$	0	0
$325$	2.00000	0.110940
$326$	10.2462	0.567485
$327$	0	0
$328$	−17.6155	−0.972655
$329$	0	0
$330$	0	0
$331$	−32.4924	−1.78595	−0.892973	−	0.450111i	$-0.851384\pi$
$331$	−32.4924	−1.78595	−0.892973	+	0.450111i	$0.851384\pi$
$332$	−21.3693	−1.17279
$333$	0	0
$334$	44.4924	2.43452
$335$	15.8078	0.863670
$336$	0	0
$337$	−11.3693	−0.619326	−0.309663	−	0.950846i	$-0.600216\pi$
$337$	−11.3693	−0.619326	−0.309663	+	0.950846i	$0.600216\pi$
$338$	−23.0540	−1.25397
$339$	0	0
$340$	−5.12311	−0.277839
$341$	−28.4924	−1.54295
$342$	0	0
$343$	0	0
$344$	67.2311	3.62486
$345$	0	0
$346$	17.6155	0.947017
$347$	19.6155	1.05302	0.526508	−	0.850170i	$-0.323501\pi$
$347$	19.6155	1.05302	0.526508	+	0.850170i	$0.323501\pi$
$348$	0	0
$349$	18.8769	1.01046	0.505228	−	0.862986i	$-0.331408\pi$
$349$	18.8769	1.01046	0.505228	+	0.862986i	$0.331408\pi$
$350$	0	0
$351$	0	0
$352$	−26.2462	−1.39893
$353$	2.68466	0.142890	0.0714450	−	0.997445i	$-0.477239\pi$
$353$	2.68466	0.142890	0.0714450	+	0.997445i	$0.477239\pi$
$354$	0	0
$355$	6.24621	0.331514
$356$	−4.56155	−0.241762
$357$	0	0
$358$	−30.7386	−1.62459
$359$	34.2462	1.80745	0.903723	−	0.428118i	$-0.140823\pi$
$359$	34.2462	1.80745	0.903723	+	0.428118i	$0.140823\pi$
$360$	0	0
$361$	−9.24621	−0.486643
$362$	39.3693	2.06921
$363$	0	0
$364$	0	0
$365$	−2.68466	−0.140521
$366$	0	0
$367$	32.9848	1.72179	0.860897	−	0.508779i	$-0.169903\pi$
$367$	32.9848	1.72179	0.860897	+	0.508779i	$0.169903\pi$
$368$	42.7386	2.22791
$369$	0	0
$370$	−25.6155	−1.33169
$371$	0	0
$372$	0	0
$373$	6.68466	0.346118	0.173059	−	0.984911i	$-0.444635\pi$
$373$	6.68466	0.346118	0.173059	+	0.984911i	$0.444635\pi$
$374$	−11.5076	−0.595043
$375$	0	0
$376$	52.4924	2.70709
$377$	5.36932	0.276534
$378$	0	0
$379$	−9.36932	−0.481269	−0.240635	−	0.970616i	$-0.577356\pi$
$379$	−9.36932	−0.481269	−0.240635	+	0.970616i	$0.577356\pi$
$380$	−14.2462	−0.730815
$381$	0	0
$382$	28.9848	1.48299
$383$	9.75379	0.498395	0.249198	−	0.968453i	$-0.419833\pi$
$383$	9.75379	0.498395	0.249198	+	0.968453i	$0.419833\pi$
$384$	0	0
$385$	0	0
$386$	23.3693	1.18947
$387$	0	0
$388$	−38.4924	−1.95416
$389$	22.4924	1.14041	0.570206	−	0.821502i	$-0.306864\pi$
$389$	22.4924	1.14041	0.570206	+	0.821502i	$0.306864\pi$
$390$	0	0
$391$	6.24621	0.315884
$392$	−45.9309	−2.31986
$393$	0	0
$394$	−41.6155	−2.09656
$395$	2.43845	0.122692
$396$	0	0
$397$	−6.00000	−0.301131	−0.150566	−	0.988600i	$-0.548110\pi$
$397$	−6.00000	−0.301131	−0.150566	+	0.988600i	$0.548110\pi$
$398$	10.7386	0.538279
$399$	0	0
$400$	7.68466	0.384233
$401$	−0.630683	−0.0314948	−0.0157474	−	0.999876i	$-0.505013\pi$
$401$	−0.630683	−0.0314948	−0.0157474	+	0.999876i	$0.505013\pi$
$402$	0	0
$403$	14.2462	0.709654
$404$	−27.3693	−1.36167
$405$	0	0
$406$	0	0
$407$	−40.0000	−1.98273
$408$	0	0
$409$	−20.9309	−1.03496	−0.517482	−	0.855694i	$-0.673131\pi$
$409$	−20.9309	−1.03496	−0.517482	+	0.855694i	$0.673131\pi$
$410$	6.87689	0.339626
$411$	0	0
$412$	−22.2462	−1.09599
$413$	0	0
$414$	0	0
$415$	4.68466	0.229961
$416$	13.1231	0.643413
$417$	0	0
$418$	−32.0000	−1.56517
$419$	26.0540	1.27282	0.636410	−	0.771351i	$-0.280419\pi$
$419$	26.0540	1.27282	0.636410	+	0.771351i	$0.280419\pi$
$420$	0	0
$421$	−0.246211	−0.0119996	−0.00599980	−	0.999982i	$-0.501910\pi$
$421$	−0.246211	−0.0119996	−0.00599980	+	0.999982i	$0.501910\pi$
$422$	56.9848	2.77398
$423$	0	0
$424$	54.1080	2.62771
$425$	1.12311	0.0544786
$426$	0	0
$427$	0	0
$428$	−18.2462	−0.881964
$429$	0	0
$430$	−26.2462	−1.26570
$431$	26.2462	1.26424	0.632118	−	0.774872i	$-0.282186\pi$
$431$	26.2462	1.26424	0.632118	+	0.774872i	$0.282186\pi$
$432$	0	0
$433$	−6.87689	−0.330482	−0.165241	−	0.986253i	$-0.552840\pi$
$433$	−6.87689	−0.330482	−0.165241	+	0.986253i	$0.552840\pi$
$434$	0	0
$435$	0	0
$436$	−20.2462	−0.969618
$437$	17.3693	0.830887
$438$	0	0
$439$	−5.36932	−0.256264	−0.128132	−	0.991757i	$-0.540898\pi$
$439$	−5.36932	−0.256264	−0.128132	+	0.991757i	$0.540898\pi$
$440$	26.2462	1.25124
$441$	0	0
$442$	5.75379	0.273680
$443$	−19.6155	−0.931962	−0.465981	−	0.884795i	$-0.654298\pi$
$443$	−19.6155	−0.931962	−0.465981	+	0.884795i	$0.654298\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−14.2462	−0.674578
$447$	0	0
$448$	0	0
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	10.7386	0.505663
$452$	−81.2311	−3.82079
$453$	0	0
$454$	−10.2462	−0.480879
$455$	0	0
$456$	0	0
$457$	−9.61553	−0.449795	−0.224898	−	0.974382i	$-0.572205\pi$
$457$	−9.61553	−0.449795	−0.224898	+	0.974382i	$0.572205\pi$
$458$	18.8769	0.882059
$459$	0	0
$460$	−25.3693	−1.18285
$461$	−16.7386	−0.779596	−0.389798	−	0.920900i	$-0.627455\pi$
$461$	−16.7386	−0.779596	−0.389798	+	0.920900i	$0.627455\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	20.6307	0.957755
$465$	0	0
$466$	−38.1080	−1.76532
$467$	16.4924	0.763178	0.381589	−	0.924332i	$-0.375377\pi$
$467$	16.4924	0.763178	0.381589	+	0.924332i	$0.375377\pi$
$468$	0	0
$469$	0	0
$470$	−20.4924	−0.945245
$471$	0	0
$472$	−16.0000	−0.736460
$473$	−40.9848	−1.88449
$474$	0	0
$475$	3.12311	0.143298
$476$	0	0
$477$	0	0
$478$	−20.0000	−0.914779
$479$	27.1231	1.23929	0.619643	−	0.784884i	$-0.287277\pi$
$479$	27.1231	1.23929	0.619643	+	0.784884i	$0.287277\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	−2.87689	−0.131039
$483$	0	0
$484$	22.8078	1.03672
$485$	8.43845	0.383170
$486$	0	0
$487$	2.43845	0.110497	0.0552483	−	0.998473i	$-0.482405\pi$
$487$	2.43845	0.110497	0.0552483	+	0.998473i	$0.482405\pi$
$488$	18.8769	0.854517
$489$	0	0
$490$	17.9309	0.810034
$491$	−27.8078	−1.25495	−0.627473	−	0.778638i	$-0.715911\pi$
$491$	−27.8078	−1.25495	−0.627473	+	0.778638i	$0.715911\pi$
$492$	0	0
$493$	3.01515	0.135796
$494$	16.0000	0.719874
$495$	0	0
$496$	54.7386	2.45784
$497$	0	0
$498$	0	0
$499$	−1.36932	−0.0612990	−0.0306495	−	0.999530i	$-0.509758\pi$
$499$	−1.36932	−0.0612990	−0.0306495	+	0.999530i	$0.509758\pi$
$500$	−4.56155	−0.203999
$501$	0	0
$502$	−54.7386	−2.44310
$503$	−11.5076	−0.513098	−0.256549	−	0.966531i	$-0.582585\pi$
$503$	−11.5076	−0.513098	−0.256549	+	0.966531i	$0.582585\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	−56.9848	−2.53329
$507$	0	0
$508$	28.4924	1.26415
$509$	−5.61553	−0.248904	−0.124452	−	0.992226i	$-0.539717\pi$
$509$	−5.61553	−0.248904	−0.124452	+	0.992226i	$0.539717\pi$
$510$	0	0
$511$	0	0
$512$	−50.4233	−2.22842
$513$	0	0
$514$	−17.6155	−0.776988
$515$	4.87689	0.214902
$516$	0	0
$517$	−32.0000	−1.40736
$518$	0	0
$519$	0	0
$520$	−13.1231	−0.575486
$521$	10.4924	0.459681	0.229841	−	0.973228i	$-0.426179\pi$
$521$	10.4924	0.459681	0.229841	+	0.973228i	$0.426179\pi$
$522$	0	0
$523$	−44.9848	−1.96705	−0.983525	−	0.180772i	$-0.942140\pi$
$523$	−44.9848	−1.96705	−0.983525	+	0.180772i	$0.942140\pi$
$524$	−60.9848	−2.66414
$525$	0	0
$526$	61.4773	2.68054
$527$	8.00000	0.348485
$528$	0	0
$529$	7.93087	0.344820
$530$	−21.1231	−0.917529
$531$	0	0
$532$	0	0
$533$	−5.36932	−0.232571
$534$	0	0
$535$	4.00000	0.172935
$536$	−103.723	−4.48017
$537$	0	0
$538$	78.1080	3.36747
$539$	28.0000	1.20605
$540$	0	0
$541$	−28.7386	−1.23557	−0.617785	−	0.786347i	$-0.711970\pi$
$541$	−28.7386	−1.23557	−0.617785	+	0.786347i	$0.711970\pi$
$542$	−68.4924	−2.94200
$543$	0	0
$544$	7.36932	0.315957
$545$	4.43845	0.190122
$546$	0	0
$547$	−11.6155	−0.496644	−0.248322	−	0.968678i	$-0.579879\pi$
$547$	−11.6155	−0.496644	−0.248322	+	0.968678i	$0.579879\pi$
$548$	45.6155	1.94860
$549$	0	0
$550$	−10.2462	−0.436900
$551$	8.38447	0.357191
$552$	0	0
$553$	0	0
$554$	−2.38447	−0.101307
$555$	0	0
$556$	29.3693	1.24554
$557$	43.5616	1.84576	0.922881	−	0.385085i	$-0.125828\pi$
$557$	43.5616	1.84576	0.922881	+	0.385085i	$0.125828\pi$
$558$	0	0
$559$	20.4924	0.866737
$560$	0	0
$561$	0	0
$562$	31.3693	1.32323
$563$	−25.1771	−1.06109	−0.530544	−	0.847658i	$-0.678012\pi$
$563$	−25.1771	−1.06109	−0.530544	+	0.847658i	$0.678012\pi$
$564$	0	0
$565$	17.8078	0.749178
$566$	−5.75379	−0.241850
$567$	0	0
$568$	−40.9848	−1.71969
$569$	19.1771	0.803945	0.401973	−	0.915652i	$-0.368325\pi$
$569$	19.1771	0.803945	0.401973	+	0.915652i	$0.368325\pi$
$570$	0	0
$571$	3.50758	0.146788	0.0733938	−	0.997303i	$-0.476617\pi$
$571$	3.50758	0.146788	0.0733938	+	0.997303i	$0.476617\pi$
$572$	−36.4924	−1.52582
$573$	0	0
$574$	0	0
$575$	5.56155	0.231933
$576$	0	0
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	−40.3153	−1.67690
$579$	0	0
$580$	−12.2462	−0.508496
$581$	0	0
$582$	0	0
$583$	−32.9848	−1.36609
$584$	17.6155	0.728936
$585$	0	0
$586$	−75.3693	−3.11348
$587$	46.3542	1.91324	0.956621	−	0.291337i	$-0.0941001\pi$
$587$	46.3542	1.91324	0.956621	+	0.291337i	$0.0941001\pi$
$588$	0	0
$589$	22.2462	0.916639
$590$	6.24621	0.257152
$591$	0	0
$592$	76.8466	3.15838
$593$	4.43845	0.182265	0.0911326	−	0.995839i	$-0.470951\pi$
$593$	4.43845	0.182265	0.0911326	+	0.995839i	$0.470951\pi$
$594$	0	0
$595$	0	0
$596$	−73.2311	−2.99966
$597$	0	0
$598$	28.4924	1.16514
$599$	13.7538	0.561965	0.280982	−	0.959713i	$-0.409340\pi$
$599$	13.7538	0.561965	0.280982	+	0.959713i	$0.409340\pi$
$600$	0	0
$601$	−3.56155	−0.145279	−0.0726394	−	0.997358i	$-0.523142\pi$
$601$	−3.56155	−0.145279	−0.0726394	+	0.997358i	$0.523142\pi$
$602$	0	0
$603$	0	0
$604$	−46.7386	−1.90177
$605$	−5.00000	−0.203279
$606$	0	0
$607$	10.0540	0.408078	0.204039	−	0.978963i	$-0.434593\pi$
$607$	10.0540	0.408078	0.204039	+	0.978963i	$0.434593\pi$
$608$	20.4924	0.831077
$609$	0	0
$610$	−7.36932	−0.298375
$611$	16.0000	0.647291
$612$	0	0
$613$	19.5616	0.790084	0.395042	−	0.918663i	$-0.370730\pi$
$613$	19.5616	0.790084	0.395042	+	0.918663i	$0.370730\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	0	0
$617$	1.31534	0.0529537	0.0264768	−	0.999649i	$-0.491571\pi$
$617$	1.31534	0.0529537	0.0264768	+	0.999649i	$0.491571\pi$
$618$	0	0
$619$	47.8078	1.92156	0.960778	−	0.277318i	$-0.0894456\pi$
$619$	47.8078	1.92156	0.960778	+	0.277318i	$0.0894456\pi$
$620$	−32.4924	−1.30493
$621$	0	0
$622$	44.4924	1.78398
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	34.8769	1.39396
$627$	0	0
$628$	24.2462	0.967529
$629$	11.2311	0.447812
$630$	0	0
$631$	−43.4233	−1.72865	−0.864327	−	0.502930i	$-0.832255\pi$
$631$	−43.4233	−1.72865	−0.864327	+	0.502930i	$0.832255\pi$
$632$	−16.0000	−0.636446
$633$	0	0
$634$	−6.38447	−0.253560
$635$	−6.24621	−0.247873
$636$	0	0
$637$	−14.0000	−0.554700
$638$	−27.5076	−1.08904
$639$	0	0
$640$	9.43845	0.373087
$641$	−35.3693	−1.39700	−0.698502	−	0.715608i	$-0.746150\pi$
$641$	−35.3693	−1.39700	−0.698502	+	0.715608i	$0.746150\pi$
$642$	0	0
$643$	25.5616	1.00805	0.504025	−	0.863689i	$-0.331852\pi$
$643$	25.5616	1.00805	0.504025	+	0.863689i	$0.331852\pi$
$644$	0	0
$645$	0	0
$646$	8.98485	0.353504
$647$	−18.0540	−0.709775	−0.354888	−	0.934909i	$-0.615481\pi$
$647$	−18.0540	−0.709775	−0.354888	+	0.934909i	$0.615481\pi$
$648$	0	0
$649$	9.75379	0.382870
$650$	5.12311	0.200945
$651$	0	0
$652$	18.2462	0.714577
$653$	−48.9309	−1.91481	−0.957406	−	0.288744i	$-0.906762\pi$
$653$	−48.9309	−1.91481	−0.957406	+	0.288744i	$0.906762\pi$
$654$	0	0
$655$	13.3693	0.522382
$656$	−20.6307	−0.805493
$657$	0	0
$658$	0	0
$659$	5.36932	0.209159	0.104579	−	0.994517i	$-0.466650\pi$
$659$	5.36932	0.209159	0.104579	+	0.994517i	$0.466650\pi$
$660$	0	0
$661$	−12.7386	−0.495475	−0.247738	−	0.968827i	$-0.579687\pi$
$661$	−12.7386	−0.495475	−0.247738	+	0.968827i	$0.579687\pi$
$662$	−83.2311	−3.23487
$663$	0	0
$664$	−30.7386	−1.19289
$665$	0	0
$666$	0	0
$667$	14.9309	0.578126
$668$	79.2311	3.06554
$669$	0	0
$670$	40.4924	1.56436
$671$	−11.5076	−0.444245
$672$	0	0
$673$	−13.3153	−0.513269	−0.256634	−	0.966509i	$-0.582614\pi$
$673$	−13.3153	−0.513269	−0.256634	+	0.966509i	$0.582614\pi$
$674$	−29.1231	−1.12178
$675$	0	0
$676$	−41.0540	−1.57900
$677$	−46.4924	−1.78685	−0.893424	−	0.449213i	$-0.851704\pi$
$677$	−46.4924	−1.78685	−0.893424	+	0.449213i	$0.851704\pi$
$678$	0	0
$679$	0	0
$680$	−7.36932	−0.282600
$681$	0	0
$682$	−72.9848	−2.79473
$683$	−0.192236	−0.00735570	−0.00367785	−	0.999993i	$-0.501171\pi$
$683$	−0.192236	−0.00735570	−0.00367785	+	0.999993i	$0.501171\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	0	0
$687$	0	0
$688$	78.7386	3.00188
$689$	16.4924	0.628311
$690$	0	0
$691$	−43.9157	−1.67063	−0.835316	−	0.549770i	$-0.814715\pi$
$691$	−43.9157	−1.67063	−0.835316	+	0.549770i	$0.814715\pi$
$692$	31.3693	1.19248
$693$	0	0
$694$	50.2462	1.90732
$695$	−6.43845	−0.244224
$696$	0	0
$697$	−3.01515	−0.114207
$698$	48.3542	1.83023
$699$	0	0
$700$	0	0
$701$	25.6155	0.967485	0.483743	−	0.875210i	$-0.339277\pi$
$701$	25.6155	0.967485	0.483743	+	0.875210i	$0.339277\pi$
$702$	0	0
$703$	31.2311	1.17790
$704$	−5.75379	−0.216854
$705$	0	0
$706$	6.87689	0.258815
$707$	0	0
$708$	0	0
$709$	1.12311	0.0421791	0.0210896	−	0.999778i	$-0.493286\pi$
$709$	1.12311	0.0421791	0.0210896	+	0.999778i	$0.493286\pi$
$710$	16.0000	0.600469
$711$	0	0
$712$	−6.56155	−0.245905
$713$	39.6155	1.48361
$714$	0	0
$715$	8.00000	0.299183
$716$	−54.7386	−2.04568
$717$	0	0
$718$	87.7235	3.27381
$719$	42.2462	1.57552	0.787759	−	0.615984i	$-0.211241\pi$
$719$	42.2462	1.57552	0.787759	+	0.615984i	$0.211241\pi$
$720$	0	0
$721$	0	0
$722$	−23.6847	−0.881452
$723$	0	0
$724$	70.1080	2.60554
$725$	2.68466	0.0997057
$726$	0	0
$727$	−45.8617	−1.70092	−0.850459	−	0.526042i	$-0.823676\pi$
$727$	−45.8617	−1.70092	−0.850459	+	0.526042i	$0.823676\pi$
$728$	0	0
$729$	0	0
$730$	−6.87689	−0.254525
$731$	11.5076	0.425623
$732$	0	0
$733$	−9.31534	−0.344070	−0.172035	−	0.985091i	$-0.555034\pi$
$733$	−9.31534	−0.344070	−0.172035	+	0.985091i	$0.555034\pi$
$734$	84.4924	3.11867
$735$	0	0
$736$	36.4924	1.34513
$737$	63.2311	2.32915
$738$	0	0
$739$	−19.5076	−0.717598	−0.358799	−	0.933415i	$-0.616814\pi$
$739$	−19.5076	−0.717598	−0.358799	+	0.933415i	$0.616814\pi$
$740$	−45.6155	−1.67686
$741$	0	0
$742$	0	0
$743$	−30.5464	−1.12064	−0.560319	−	0.828277i	$-0.689322\pi$
$743$	−30.5464	−1.12064	−0.560319	+	0.828277i	$0.689322\pi$
$744$	0	0
$745$	16.0540	0.588172
$746$	17.1231	0.626921
$747$	0	0
$748$	−20.4924	−0.749277
$749$	0	0
$750$	0	0
$751$	−4.49242	−0.163931	−0.0819654	−	0.996635i	$-0.526120\pi$
$751$	−4.49242	−0.163931	−0.0819654	+	0.996635i	$0.526120\pi$
$752$	61.4773	2.24185
$753$	0	0
$754$	13.7538	0.500883
$755$	10.2462	0.372898
$756$	0	0
$757$	24.0540	0.874257	0.437128	−	0.899399i	$-0.355996\pi$
$757$	24.0540	0.874257	0.437128	+	0.899399i	$0.355996\pi$
$758$	−24.0000	−0.871719
$759$	0	0
$760$	−20.4924	−0.743338
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	0	0
$764$	51.6155	1.86738
$765$	0	0
$766$	24.9848	0.902739
$767$	−4.87689	−0.176094
$768$	0	0
$769$	−47.6695	−1.71901	−0.859503	−	0.511130i	$-0.829227\pi$
$769$	−47.6695	−1.71901	−0.859503	+	0.511130i	$0.829227\pi$
$770$	0	0
$771$	0	0
$772$	41.6155	1.49778
$773$	22.3002	0.802082	0.401041	−	0.916060i	$-0.368648\pi$
$773$	22.3002	0.802082	0.401041	+	0.916060i	$0.368648\pi$
$774$	0	0
$775$	7.12311	0.255870
$776$	−55.3693	−1.98764
$777$	0	0
$778$	57.6155	2.06562
$779$	−8.38447	−0.300405
$780$	0	0
$781$	24.9848	0.894028
$782$	16.0000	0.572159
$783$	0	0
$784$	−53.7926	−1.92116
$785$	−5.31534	−0.189713
$786$	0	0
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	−74.1080	−2.63999
$789$	0	0
$790$	6.24621	0.222230
$791$	0	0
$792$	0	0
$793$	5.75379	0.204323
$794$	−15.3693	−0.545437
$795$	0	0
$796$	19.1231	0.677801
$797$	32.2462	1.14222	0.571110	−	0.820874i	$-0.306513\pi$
$797$	32.2462	1.14222	0.571110	+	0.820874i	$0.306513\pi$
$798$	0	0
$799$	8.98485	0.317861
$800$	6.56155	0.231986
$801$	0	0
$802$	−1.61553	−0.0570463
$803$	−10.7386	−0.378958
$804$	0	0
$805$	0	0
$806$	36.4924	1.28539
$807$	0	0
$808$	−39.3693	−1.38501
$809$	−37.1231	−1.30518	−0.652589	−	0.757712i	$-0.726317\pi$
$809$	−37.1231	−1.30518	−0.652589	+	0.757712i	$0.726317\pi$
$810$	0	0
$811$	38.0540	1.33626	0.668128	−	0.744046i	$-0.267096\pi$
$811$	38.0540	1.33626	0.668128	+	0.744046i	$0.267096\pi$
$812$	0	0
$813$	0	0
$814$	−102.462	−3.59130
$815$	−4.00000	−0.140114
$816$	0	0
$817$	32.0000	1.11954
$818$	−53.6155	−1.87462
$819$	0	0
$820$	12.2462	0.427656
$821$	−10.8769	−0.379606	−0.189803	−	0.981822i	$-0.560785\pi$
$821$	−10.8769	−0.379606	−0.189803	+	0.981822i	$0.560785\pi$
$822$	0	0
$823$	25.0691	0.873855	0.436927	−	0.899497i	$-0.356067\pi$
$823$	25.0691	0.873855	0.436927	+	0.899497i	$0.356067\pi$
$824$	−32.0000	−1.11477
$825$	0	0
$826$	0	0
$827$	−25.1771	−0.875493	−0.437746	−	0.899098i	$-0.644223\pi$
$827$	−25.1771	−0.875493	−0.437746	+	0.899098i	$0.644223\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	12.0000	0.416526
$831$	0	0
$832$	2.87689	0.0997384
$833$	−7.86174	−0.272393
$834$	0	0
$835$	−17.3693	−0.601090
$836$	−56.9848	−1.97086
$837$	0	0
$838$	66.7386	2.30545
$839$	23.8078	0.821935	0.410968	−	0.911650i	$-0.365191\pi$
$839$	23.8078	0.821935	0.410968	+	0.911650i	$0.365191\pi$
$840$	0	0
$841$	−21.7926	−0.751469
$842$	−0.630683	−0.0217348
$843$	0	0
$844$	101.477	3.49299
$845$	9.00000	0.309609
$846$	0	0
$847$	0	0
$848$	63.3693	2.17611
$849$	0	0
$850$	2.87689	0.0986767
$851$	55.6155	1.90648
$852$	0	0
$853$	−49.1231	−1.68194	−0.840972	−	0.541079i	$-0.818016\pi$
$853$	−49.1231	−1.68194	−0.840972	+	0.541079i	$0.818016\pi$
$854$	0	0
$855$	0	0
$856$	−26.2462	−0.897077
$857$	−32.7386	−1.11833	−0.559165	−	0.829056i	$-0.688878\pi$
$857$	−32.7386	−1.11833	−0.559165	+	0.829056i	$0.688878\pi$
$858$	0	0
$859$	−28.8769	−0.985267	−0.492633	−	0.870237i	$-0.663966\pi$
$859$	−28.8769	−0.985267	−0.492633	+	0.870237i	$0.663966\pi$
$860$	−46.7386	−1.59377
$861$	0	0
$862$	67.2311	2.28990
$863$	−11.5076	−0.391722	−0.195861	−	0.980632i	$-0.562750\pi$
$863$	−11.5076	−0.391722	−0.195861	+	0.980632i	$0.562750\pi$
$864$	0	0
$865$	−6.87689	−0.233821
$866$	−17.6155	−0.598600
$867$	0	0
$868$	0	0
$869$	9.75379	0.330875
$870$	0	0
$871$	−31.6155	−1.07125
$872$	−29.1231	−0.986233
$873$	0	0
$874$	44.4924	1.50498
$875$	0	0
$876$	0	0
$877$	40.2462	1.35902	0.679509	−	0.733667i	$-0.262193\pi$
$877$	40.2462	1.35902	0.679509	+	0.733667i	$0.262193\pi$
$878$	−13.7538	−0.464168
$879$	0	0
$880$	30.7386	1.03620
$881$	−0.630683	−0.0212483	−0.0106241	−	0.999944i	$-0.503382\pi$
$881$	−0.630683	−0.0212483	−0.0106241	+	0.999944i	$0.503382\pi$
$882$	0	0
$883$	51.6155	1.73700	0.868500	−	0.495688i	$-0.165084\pi$
$883$	51.6155	1.73700	0.868500	+	0.495688i	$0.165084\pi$
$884$	10.2462	0.344617
$885$	0	0
$886$	−50.2462	−1.68805
$887$	4.19224	0.140762	0.0703808	−	0.997520i	$-0.477579\pi$
$887$	4.19224	0.140762	0.0703808	+	0.997520i	$0.477579\pi$
$888$	0	0
$889$	0	0
$890$	2.56155	0.0858634
$891$	0	0
$892$	−25.3693	−0.849428
$893$	24.9848	0.836086
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	0	0
$898$	−87.0928	−2.90632
$899$	19.1231	0.637791
$900$	0	0
$901$	9.26137	0.308541
$902$	27.5076	0.915902
$903$	0	0
$904$	−116.847	−3.88626
$905$	−15.3693	−0.510893
$906$	0	0
$907$	26.5464	0.881459	0.440729	−	0.897640i	$-0.354720\pi$
$907$	26.5464	0.881459	0.440729	+	0.897640i	$0.354720\pi$
$908$	−18.2462	−0.605522
$909$	0	0
$910$	0	0
$911$	42.7386	1.41599	0.707997	−	0.706215i	$-0.249599\pi$
$911$	42.7386	1.41599	0.707997	+	0.706215i	$0.249599\pi$
$912$	0	0
$913$	18.7386	0.620158
$914$	−24.6307	−0.814711
$915$	0	0
$916$	33.6155	1.11069
$917$	0	0
$918$	0	0
$919$	−47.1231	−1.55445	−0.777224	−	0.629224i	$-0.783373\pi$
$919$	−47.1231	−1.55445	−0.777224	+	0.629224i	$0.783373\pi$
$920$	−36.4924	−1.20312
$921$	0	0
$922$	−42.8769	−1.41208
$923$	−12.4924	−0.411193
$924$	0	0
$925$	10.0000	0.328798
$926$	−81.9697	−2.69369
$927$	0	0
$928$	17.6155	0.578258
$929$	27.8617	0.914114	0.457057	−	0.889437i	$-0.348904\pi$
$929$	27.8617	0.914114	0.457057	+	0.889437i	$0.348904\pi$
$930$	0	0
$931$	−21.8617	−0.716490
$932$	−67.8617	−2.22289
$933$	0	0
$934$	42.2462	1.38234
$935$	4.49242	0.146918
$936$	0	0
$937$	45.4233	1.48391	0.741957	−	0.670447i	$-0.233898\pi$
$937$	45.4233	1.48391	0.741957	+	0.670447i	$0.233898\pi$
$938$	0	0
$939$	0	0
$940$	−36.4924	−1.19025
$941$	−30.3002	−0.987758	−0.493879	−	0.869531i	$-0.664421\pi$
$941$	−30.3002	−0.987758	−0.493879	+	0.869531i	$0.664421\pi$
$942$	0	0
$943$	−14.9309	−0.486216
$944$	−18.7386	−0.609891
$945$	0	0
$946$	−104.985	−3.41335
$947$	11.6155	0.377454	0.188727	−	0.982030i	$-0.439564\pi$
$947$	11.6155	0.377454	0.188727	+	0.982030i	$0.439564\pi$
$948$	0	0
$949$	5.36932	0.174295
$950$	8.00000	0.259554
$951$	0	0
$952$	0	0
$953$	6.19224	0.200586	0.100293	−	0.994958i	$-0.468022\pi$
$953$	6.19224	0.200586	0.100293	+	0.994958i	$0.468022\pi$
$954$	0	0
$955$	−11.3153	−0.366156
$956$	−35.6155	−1.15189
$957$	0	0
$958$	69.4773	2.24471
$959$	0	0
$960$	0	0
$961$	19.7386	0.636730
$962$	51.2311	1.65176
$963$	0	0
$964$	−5.12311	−0.165004
$965$	−9.12311	−0.293683
$966$	0	0
$967$	−36.4924	−1.17352	−0.586759	−	0.809762i	$-0.699596\pi$
$967$	−36.4924	−1.17352	−0.586759	+	0.809762i	$0.699596\pi$
$968$	32.8078	1.05448
$969$	0	0
$970$	21.6155	0.694033
$971$	−48.1080	−1.54386	−0.771929	−	0.635709i	$-0.780708\pi$
$971$	−48.1080	−1.54386	−0.771929	+	0.635709i	$0.780708\pi$
$972$	0	0
$973$	0	0
$974$	6.24621	0.200142
$975$	0	0
$976$	22.1080	0.707658
$977$	−57.2311	−1.83098	−0.915492	−	0.402337i	$-0.868198\pi$
$977$	−57.2311	−1.83098	−0.915492	+	0.402337i	$0.868198\pi$
$978$	0	0
$979$	4.00000	0.127841
$980$	31.9309	1.01999
$981$	0	0
$982$	−71.2311	−2.27307
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	16.2462	0.517647
$986$	7.72348	0.245966
$987$	0	0
$988$	28.4924	0.906465
$989$	56.9848	1.81201
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	46.7386	1.48395
$993$	0	0
$994$	0	0
$995$	−4.19224	−0.132903
$996$	0	0
$997$	0.438447	0.0138858	0.00694288	−	0.999976i	$-0.497790\pi$
$997$	0.438447	0.0138858	0.00694288	+	0.999976i	$0.497790\pi$
$998$	−3.50758	−0.111030
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{5} +6.56155 q^{8} +O(q^{10})\)	\(q+2.56155 q^{2} +4.56155 q^{4} -1.00000 q^{5} +6.56155 q^{8} -2.56155 q^{10} -4.00000 q^{11} +2.00000 q^{13} +7.68466 q^{16} +1.12311 q^{17} +3.12311 q^{19} -4.56155 q^{20} -10.2462 q^{22} +5.56155 q^{23} +1.00000 q^{25} +5.12311 q^{26} +2.68466 q^{29} +7.12311 q^{31} +6.56155 q^{32} +2.87689 q^{34} +10.0000 q^{37} +8.00000 q^{38} -6.56155 q^{40} -2.68466 q^{41} +10.2462 q^{43} -18.2462 q^{44} +14.2462 q^{46} +8.00000 q^{47} -7.00000 q^{49} +2.56155 q^{50} +9.12311 q^{52} +8.24621 q^{53} +4.00000 q^{55} +6.87689 q^{58} -2.43845 q^{59} +2.87689 q^{61} +18.2462 q^{62} +1.43845 q^{64} -2.00000 q^{65} -15.8078 q^{67} +5.12311 q^{68} -6.24621 q^{71} +2.68466 q^{73} +25.6155 q^{74} +14.2462 q^{76} -2.43845 q^{79} -7.68466 q^{80} -6.87689 q^{82} -4.68466 q^{83} -1.12311 q^{85} +26.2462 q^{86} -26.2462 q^{88} -1.00000 q^{89} +25.3693 q^{92} +20.4924 q^{94} -3.12311 q^{95} -8.43845 q^{97} -17.9309 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 5 q^{4} - 2 q^{5} + 9 q^{8}+O(q^{10}) \) 2 * q + q^2 + 5 * q^4 - 2 * q^5 + 9 * q^8	\( 2 q + q^{2} + 5 q^{4} - 2 q^{5} + 9 q^{8} - q^{10} - 8 q^{11} + 4 q^{13} + 3 q^{16} - 6 q^{17} - 2 q^{19} - 5 q^{20} - 4 q^{22} + 7 q^{23} + 2 q^{25} + 2 q^{26} - 7 q^{29} + 6 q^{31} + 9 q^{32} + 14 q^{34} + 20 q^{37} + 16 q^{38} - 9 q^{40} + 7 q^{41} + 4 q^{43} - 20 q^{44} + 12 q^{46} + 16 q^{47} - 14 q^{49} + q^{50} + 10 q^{52} + 8 q^{55} + 22 q^{58} - 9 q^{59} + 14 q^{61} + 20 q^{62} + 7 q^{64} - 4 q^{65} - 11 q^{67} + 2 q^{68} + 4 q^{71} - 7 q^{73} + 10 q^{74} + 12 q^{76} - 9 q^{79} - 3 q^{80} - 22 q^{82} + 3 q^{83} + 6 q^{85} + 36 q^{86} - 36 q^{88} - 2 q^{89} + 26 q^{92} + 8 q^{94} + 2 q^{95} - 21 q^{97} - 7 q^{98}+O(q^{100}) \) 2 * q + q^2 + 5 * q^4 - 2 * q^5 + 9 * q^8 - q^10 - 8 * q^11 + 4 * q^13 + 3 * q^16 - 6 * q^17 - 2 * q^19 - 5 * q^20 - 4 * q^22 + 7 * q^23 + 2 * q^25 + 2 * q^26 - 7 * q^29 + 6 * q^31 + 9 * q^32 + 14 * q^34 + 20 * q^37 + 16 * q^38 - 9 * q^40 + 7 * q^41 + 4 * q^43 - 20 * q^44 + 12 * q^46 + 16 * q^47 - 14 * q^49 + q^50 + 10 * q^52 + 8 * q^55 + 22 * q^58 - 9 * q^59 + 14 * q^61 + 20 * q^62 + 7 * q^64 - 4 * q^65 - 11 * q^67 + 2 * q^68 + 4 * q^71 - 7 * q^73 + 10 * q^74 + 12 * q^76 - 9 * q^79 - 3 * q^80 - 22 * q^82 + 3 * q^83 + 6 * q^85 + 36 * q^86 - 36 * q^88 - 2 * q^89 + 26 * q^92 + 8 * q^94 + 2 * q^95 - 21 * q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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