LMFDB - Embedded newform 4005.2.a.x.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:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 5 x^{16} - 15 x^{15} + 105 x^{14} + 45 x^{13} - 849 x^{12} + 320 x^{11} + 3371 x^{10} + \cdots + 24 $ x^17 - 5x^16 - 15x^15 + 105x^14 + 45x^13 - 849x^12 + 320x^11 + 3371x^10 - 2456x^9 - 7002x^8 + 6279x^7 + 7299x^6 - 7119x^5 - 3066x^4 + 3184x^3 + 99x^2 - 231x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.56306$ 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)

$f(q)$	$=$	$q-2.56306 q^{2} +4.56927 q^{4} +1.00000 q^{5} -0.860224 q^{7} -6.58519 q^{8} +O(q^{10})$	\(q-2.56306 q^{2} +4.56927 q^{4} +1.00000 q^{5} -0.860224 q^{7} -6.58519 q^{8} -2.56306 q^{10} -4.21170 q^{11} +1.77782 q^{13} +2.20480 q^{14} +7.73970 q^{16} -3.70299 q^{17} +0.190893 q^{19} +4.56927 q^{20} +10.7948 q^{22} +5.54195 q^{23} +1.00000 q^{25} -4.55665 q^{26} -3.93059 q^{28} -5.89248 q^{29} -1.66367 q^{31} -6.66692 q^{32} +9.49097 q^{34} -0.860224 q^{35} -2.99225 q^{37} -0.489269 q^{38} -6.58519 q^{40} -5.52532 q^{41} +9.56745 q^{43} -19.2444 q^{44} -14.2043 q^{46} -0.668780 q^{47} -6.26002 q^{49} -2.56306 q^{50} +8.12332 q^{52} +8.18227 q^{53} -4.21170 q^{55} +5.66474 q^{56} +15.1028 q^{58} -13.5225 q^{59} +7.19117 q^{61} +4.26409 q^{62} +1.60830 q^{64} +1.77782 q^{65} -6.33381 q^{67} -16.9200 q^{68} +2.20480 q^{70} +4.00911 q^{71} +5.79716 q^{73} +7.66932 q^{74} +0.872241 q^{76} +3.62300 q^{77} +14.7351 q^{79} +7.73970 q^{80} +14.1617 q^{82} +6.87780 q^{83} -3.70299 q^{85} -24.5219 q^{86} +27.7348 q^{88} -1.00000 q^{89} -1.52932 q^{91} +25.3227 q^{92} +1.71412 q^{94} +0.190893 q^{95} +8.89776 q^{97} +16.0448 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 5 q^{2} + 21 q^{4} + 17 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) $ 17 * q + 5 * q^2 + 21 * q^4 + 17 * q^5 + 12 * q^7 + 15 * q^8	$ 17 q + 5 q^{2} + 21 q^{4} + 17 q^{5} + 12 q^{7} + 15 q^{8} + 5 q^{10} + 2 q^{11} + 8 q^{13} + 4 q^{14} + 33 q^{16} + 10 q^{17} + 32 q^{19} + 21 q^{20} + 8 q^{22} + 15 q^{23} + 17 q^{25} - 15 q^{26} + 24 q^{28} + q^{29} + 18 q^{31} + 25 q^{32} + 14 q^{34} + 12 q^{35} + 12 q^{37} + 22 q^{38} + 15 q^{40} - 7 q^{41} + 28 q^{43} - 14 q^{44} + 4 q^{46} + 26 q^{47} + 41 q^{49} + 5 q^{50} + 10 q^{52} + 12 q^{53} + 2 q^{55} + 13 q^{56} + 16 q^{58} - 23 q^{59} + 26 q^{61} + 10 q^{62} + 59 q^{64} + 8 q^{65} + 31 q^{67} - q^{68} + 4 q^{70} - 2 q^{71} + 33 q^{73} - 10 q^{74} + 66 q^{76} + 12 q^{77} + 33 q^{79} + 33 q^{80} + 30 q^{82} + 13 q^{83} + 10 q^{85} - 20 q^{86} + 12 q^{88} - 17 q^{89} + 40 q^{91} + 16 q^{92} + 38 q^{94} + 32 q^{95} + 45 q^{97} + 2 q^{98}+O(q^{100}) $ 17 * q + 5 * q^2 + 21 * q^4 + 17 * q^5 + 12 * q^7 + 15 * q^8 + 5 * q^10 + 2 * q^11 + 8 * q^13 + 4 * q^14 + 33 * q^16 + 10 * q^17 + 32 * q^19 + 21 * q^20 + 8 * q^22 + 15 * q^23 + 17 * q^25 - 15 * q^26 + 24 * q^28 + q^29 + 18 * q^31 + 25 * q^32 + 14 * q^34 + 12 * q^35 + 12 * q^37 + 22 * q^38 + 15 * q^40 - 7 * q^41 + 28 * q^43 - 14 * q^44 + 4 * q^46 + 26 * q^47 + 41 * q^49 + 5 * q^50 + 10 * q^52 + 12 * q^53 + 2 * q^55 + 13 * q^56 + 16 * q^58 - 23 * q^59 + 26 * q^61 + 10 * q^62 + 59 * q^64 + 8 * q^65 + 31 * q^67 - q^68 + 4 * q^70 - 2 * q^71 + 33 * q^73 - 10 * q^74 + 66 * q^76 + 12 * q^77 + 33 * q^79 + 33 * q^80 + 30 * q^82 + 13 * q^83 + 10 * q^85 - 20 * q^86 + 12 * q^88 - 17 * q^89 + 40 * q^91 + 16 * q^92 + 38 * q^94 + 32 * q^95 + 45 * 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 $

$2$	−2.56306	−1.81236	−0.906178	−	0.422896i	$-0.861014\pi$
$2$	−2.56306	−1.81236	−0.906178	+	0.422896i	$0.861014\pi$
$3$	0	0
$3$	0	0
$4$	4.56927	2.28464
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.860224	−0.325134	−0.162567	−	0.986698i	$-0.551977\pi$
$7$	−0.860224	−0.325134	−0.162567	+	0.986698i	$0.551977\pi$
$8$	−6.58519	−2.32822
$9$	0	0
$10$	−2.56306	−0.810510
$11$	−4.21170	−1.26987	−0.634937	−	0.772564i	$-0.718974\pi$
$11$	−4.21170	−1.26987	−0.634937	+	0.772564i	$0.718974\pi$
$12$	0	0
$13$	1.77782	0.493077	0.246539	−	0.969133i	$-0.420707\pi$
$13$	1.77782	0.493077	0.246539	+	0.969133i	$0.420707\pi$
$14$	2.20480	0.589259
$15$	0	0
$16$	7.73970	1.93492
$17$	−3.70299	−0.898106	−0.449053	−	0.893505i	$-0.648239\pi$
$17$	−3.70299	−0.898106	−0.449053	+	0.893505i	$0.648239\pi$
$18$	0	0
$19$	0.190893	0.0437938	0.0218969	−	0.999760i	$-0.493029\pi$
$19$	0.190893	0.0437938	0.0218969	+	0.999760i	$0.493029\pi$
$20$	4.56927	1.02172
$21$	0	0
$22$	10.7948	2.30147
$23$	5.54195	1.15558	0.577788	−	0.816187i	$-0.303916\pi$
$23$	5.54195	1.15558	0.577788	+	0.816187i	$0.303916\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.55665	−0.893632
$27$	0	0
$28$	−3.93059	−0.742813
$29$	−5.89248	−1.09421	−0.547103	−	0.837066i	$-0.684269\pi$
$29$	−5.89248	−1.09421	−0.547103	+	0.837066i	$0.684269\pi$
$30$	0	0
$31$	−1.66367	−0.298804	−0.149402	−	0.988777i	$-0.547735\pi$
$31$	−1.66367	−0.298804	−0.149402	+	0.988777i	$0.547735\pi$
$32$	−6.66692	−1.17856
$33$	0	0
$34$	9.49097	1.62769
$35$	−0.860224	−0.145404
$36$	0	0
$37$	−2.99225	−0.491923	−0.245962	−	0.969280i	$-0.579104\pi$
$37$	−2.99225	−0.491923	−0.245962	+	0.969280i	$0.579104\pi$
$38$	−0.489269	−0.0793700
$39$	0	0
$40$	−6.58519	−1.04121
$41$	−5.52532	−0.862910	−0.431455	−	0.902134i	$-0.642000\pi$
$41$	−5.52532	−0.862910	−0.431455	+	0.902134i	$0.642000\pi$
$42$	0	0
$43$	9.56745	1.45902	0.729511	−	0.683969i	$-0.239748\pi$
$43$	9.56745	1.45902	0.729511	+	0.683969i	$0.239748\pi$
$44$	−19.2444	−2.90120
$45$	0	0
$46$	−14.2043	−2.09432
$47$	−0.668780	−0.0975515	−0.0487758	−	0.998810i	$-0.515532\pi$
$47$	−0.668780	−0.0975515	−0.0487758	+	0.998810i	$0.515532\pi$
$48$	0	0
$49$	−6.26002	−0.894288
$50$	−2.56306	−0.362471
$51$	0	0
$52$	8.12332	1.12650
$53$	8.18227	1.12392	0.561961	−	0.827164i	$-0.310047\pi$
$53$	8.18227	1.12392	0.561961	+	0.827164i	$0.310047\pi$
$54$	0	0
$55$	−4.21170	−0.567905
$56$	5.66474	0.756983
$57$	0	0
$58$	15.1028	1.98309
$59$	−13.5225	−1.76048	−0.880241	−	0.474528i	$-0.842619\pi$
$59$	−13.5225	−1.76048	−0.880241	+	0.474528i	$0.842619\pi$
$60$	0	0
$61$	7.19117	0.920735	0.460367	−	0.887729i	$-0.347718\pi$
$61$	7.19117	0.920735	0.460367	+	0.887729i	$0.347718\pi$
$62$	4.26409	0.541540
$63$	0	0
$64$	1.60830	0.201038
$65$	1.77782	0.220511
$66$	0	0
$67$	−6.33381	−0.773798	−0.386899	−	0.922122i	$-0.626454\pi$
$67$	−6.33381	−0.773798	−0.386899	+	0.922122i	$0.626454\pi$
$68$	−16.9200	−2.05185
$69$	0	0
$70$	2.20480	0.263524
$71$	4.00911	0.475794	0.237897	−	0.971290i	$-0.423542\pi$
$71$	4.00911	0.475794	0.237897	+	0.971290i	$0.423542\pi$
$72$	0	0
$73$	5.79716	0.678506	0.339253	−	0.940695i	$-0.389826\pi$
$73$	5.79716	0.678506	0.339253	+	0.940695i	$0.389826\pi$
$74$	7.66932	0.891541
$75$	0	0
$76$	0.872241	0.100053
$77$	3.62300	0.412879
$78$	0	0
$79$	14.7351	1.65783	0.828913	−	0.559377i	$-0.188960\pi$
$79$	14.7351	1.65783	0.828913	+	0.559377i	$0.188960\pi$
$80$	7.73970	0.865325
$81$	0	0
$82$	14.1617	1.56390
$83$	6.87780	0.754937	0.377468	−	0.926022i	$-0.376795\pi$
$83$	6.87780	0.754937	0.377468	+	0.926022i	$0.376795\pi$
$84$	0	0
$85$	−3.70299	−0.401645
$86$	−24.5219	−2.64427
$87$	0	0
$88$	27.7348	2.95654
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−1.52932	−0.160316
$92$	25.3227	2.64007
$93$	0	0
$94$	1.71412	0.176798
$95$	0.190893	0.0195852
$96$	0	0
$97$	8.89776	0.903431	0.451715	−	0.892162i	$-0.350812\pi$
$97$	8.89776	0.903431	0.451715	+	0.892162i	$0.350812\pi$
$98$	16.0448	1.62077
$99$	0	0
$100$	4.56927	0.456927
$101$	5.32986	0.530341	0.265170	−	0.964202i	$-0.414572\pi$
$101$	5.32986	0.530341	0.265170	+	0.964202i	$0.414572\pi$
$102$	0	0
$103$	4.09597	0.403588	0.201794	−	0.979428i	$-0.435323\pi$
$103$	4.09597	0.403588	0.201794	+	0.979428i	$0.435323\pi$
$104$	−11.7073	−1.14799
$105$	0	0
$106$	−20.9716	−2.03695
$107$	−1.28993	−0.124702	−0.0623510	−	0.998054i	$-0.519860\pi$
$107$	−1.28993	−0.124702	−0.0623510	+	0.998054i	$0.519860\pi$
$108$	0	0
$109$	−3.61403	−0.346161	−0.173081	−	0.984908i	$-0.555372\pi$
$109$	−3.61403	−0.346161	−0.173081	+	0.984908i	$0.555372\pi$
$110$	10.7948	1.02925
$111$	0	0
$112$	−6.65787	−0.629110
$113$	8.69271	0.817741	0.408871	−	0.912592i	$-0.365923\pi$
$113$	8.69271	0.817741	0.408871	+	0.912592i	$0.365923\pi$
$114$	0	0
$115$	5.54195	0.516789
$116$	−26.9243	−2.49986
$117$	0	0
$118$	34.6590	3.19062
$119$	3.18540	0.292005
$120$	0	0
$121$	6.73840	0.612582
$122$	−18.4314	−1.66870
$123$	0	0
$124$	−7.60177	−0.682659
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.33463	−0.384636	−0.192318	−	0.981333i	$-0.561601\pi$
$127$	−4.33463	−0.384636	−0.192318	+	0.981333i	$0.561601\pi$
$128$	9.21166	0.814203
$129$	0	0
$130$	−4.55665	−0.399644
$131$	10.3636	0.905474	0.452737	−	0.891644i	$-0.350448\pi$
$131$	10.3636	0.905474	0.452737	+	0.891644i	$0.350448\pi$
$132$	0	0
$133$	−0.164210	−0.0142388
$134$	16.2339	1.40240
$135$	0	0
$136$	24.3849	2.09099
$137$	−2.70626	−0.231211	−0.115606	−	0.993295i	$-0.536881\pi$
$137$	−2.70626	−0.231211	−0.115606	+	0.993295i	$0.536881\pi$
$138$	0	0
$139$	1.58671	0.134583	0.0672915	−	0.997733i	$-0.478564\pi$
$139$	1.58671	0.134583	0.0672915	+	0.997733i	$0.478564\pi$
$140$	−3.93059	−0.332196
$141$	0	0
$142$	−10.2756	−0.862309
$143$	−7.48762	−0.626146
$144$	0	0
$145$	−5.89248	−0.489344
$146$	−14.8585	−1.22969
$147$	0	0
$148$	−13.6724	−1.12387
$149$	17.9402	1.46972	0.734859	−	0.678220i	$-0.237248\pi$
$149$	17.9402	1.46972	0.734859	+	0.678220i	$0.237248\pi$
$150$	0	0
$151$	17.7641	1.44562	0.722811	−	0.691046i	$-0.242850\pi$
$151$	17.7641	1.44562	0.722811	+	0.691046i	$0.242850\pi$
$152$	−1.25707	−0.101961
$153$	0	0
$154$	−9.28597	−0.748284
$155$	−1.66367	−0.133629
$156$	0	0
$157$	−2.23448	−0.178331	−0.0891655	−	0.996017i	$-0.528420\pi$
$157$	−2.23448	−0.178331	−0.0891655	+	0.996017i	$0.528420\pi$
$158$	−37.7669	−3.00457
$159$	0	0
$160$	−6.66692	−0.527066
$161$	−4.76731	−0.375717
$162$	0	0
$163$	−9.07070	−0.710472	−0.355236	−	0.934777i	$-0.615599\pi$
$163$	−9.07070	−0.710472	−0.355236	+	0.934777i	$0.615599\pi$
$164$	−25.2467	−1.97143
$165$	0	0
$166$	−17.6282	−1.36821
$167$	−9.06131	−0.701186	−0.350593	−	0.936528i	$-0.614020\pi$
$167$	−9.06131	−0.701186	−0.350593	+	0.936528i	$0.614020\pi$
$168$	0	0
$169$	−9.83937	−0.756875
$170$	9.49097	0.727924
$171$	0	0
$172$	43.7163	3.33333
$173$	23.5716	1.79211	0.896057	−	0.443938i	$-0.146419\pi$
$173$	23.5716	1.79211	0.896057	+	0.443938i	$0.146419\pi$
$174$	0	0
$175$	−0.860224	−0.0650268
$176$	−32.5973	−2.45711
$177$	0	0
$178$	2.56306	0.192109
$179$	−11.7780	−0.880329	−0.440164	−	0.897917i	$-0.645080\pi$
$179$	−11.7780	−0.880329	−0.440164	+	0.897917i	$0.645080\pi$
$180$	0	0
$181$	4.52315	0.336203	0.168101	−	0.985770i	$-0.446236\pi$
$181$	4.52315	0.336203	0.168101	+	0.985770i	$0.446236\pi$
$182$	3.91973	0.290550
$183$	0	0
$184$	−36.4948	−2.69043
$185$	−2.99225	−0.219995
$186$	0	0
$187$	15.5959	1.14048
$188$	−3.05584	−0.222870
$189$	0	0
$190$	−0.489269	−0.0354953
$191$	10.7099	0.774943	0.387472	−	0.921882i	$-0.373349\pi$
$191$	10.7099	0.774943	0.387472	+	0.921882i	$0.373349\pi$
$192$	0	0
$193$	−7.32803	−0.527483	−0.263741	−	0.964593i	$-0.584957\pi$
$193$	−7.32803	−0.527483	−0.263741	+	0.964593i	$0.584957\pi$
$194$	−22.8055	−1.63734
$195$	0	0
$196$	−28.6037	−2.04312
$197$	−10.9883	−0.782883	−0.391442	−	0.920203i	$-0.628024\pi$
$197$	−10.9883	−0.782883	−0.391442	+	0.920203i	$0.628024\pi$
$198$	0	0
$199$	−28.0600	−1.98912	−0.994561	−	0.104159i	$-0.966785\pi$
$199$	−28.0600	−1.98912	−0.994561	+	0.104159i	$0.966785\pi$
$200$	−6.58519	−0.465644
$201$	0	0
$202$	−13.6607	−0.961166
$203$	5.06885	0.355763
$204$	0	0
$205$	−5.52532	−0.385905
$206$	−10.4982	−0.731445
$207$	0	0
$208$	13.7598	0.954067
$209$	−0.803982	−0.0556126
$210$	0	0
$211$	2.40553	0.165603	0.0828017	−	0.996566i	$-0.473613\pi$
$211$	2.40553	0.165603	0.0828017	+	0.996566i	$0.473613\pi$
$212$	37.3870	2.56775
$213$	0	0
$214$	3.30616	0.226005
$215$	9.56745	0.652495
$216$	0	0
$217$	1.43113	0.0971514
$218$	9.26297	0.627368
$219$	0	0
$220$	−19.2444	−1.29746
$221$	−6.58323	−0.442836
$222$	0	0
$223$	5.68142	0.380456	0.190228	−	0.981740i	$-0.439077\pi$
$223$	5.68142	0.380456	0.190228	+	0.981740i	$0.439077\pi$
$224$	5.73504	0.383188
$225$	0	0
$226$	−22.2799	−1.48204
$227$	5.00230	0.332014	0.166007	−	0.986125i	$-0.446912\pi$
$227$	5.00230	0.332014	0.166007	+	0.986125i	$0.446912\pi$
$228$	0	0
$229$	18.5748	1.22746	0.613728	−	0.789518i	$-0.289669\pi$
$229$	18.5748	1.22746	0.613728	+	0.789518i	$0.289669\pi$
$230$	−14.2043	−0.936606
$231$	0	0
$232$	38.8031	2.54755
$233$	12.7591	0.835879	0.417939	−	0.908475i	$-0.362752\pi$
$233$	12.7591	0.835879	0.417939	+	0.908475i	$0.362752\pi$
$234$	0	0
$235$	−0.668780	−0.0436264
$236$	−61.7880	−4.02206
$237$	0	0
$238$	−8.16436	−0.529217
$239$	19.0658	1.23327	0.616634	−	0.787250i	$-0.288496\pi$
$239$	19.0658	1.23327	0.616634	+	0.787250i	$0.288496\pi$
$240$	0	0
$241$	−15.1174	−0.973798	−0.486899	−	0.873458i	$-0.661872\pi$
$241$	−15.1174	−0.973798	−0.486899	+	0.873458i	$0.661872\pi$
$242$	−17.2709	−1.11022
$243$	0	0
$244$	32.8584	2.10354
$245$	−6.26002	−0.399938
$246$	0	0
$247$	0.339372	0.0215937
$248$	10.9556	0.695682
$249$	0	0
$250$	−2.56306	−0.162102
$251$	−3.85765	−0.243493	−0.121746	−	0.992561i	$-0.538849\pi$
$251$	−3.85765	−0.243493	−0.121746	+	0.992561i	$0.538849\pi$
$252$	0	0
$253$	−23.3410	−1.46744
$254$	11.1099	0.697098
$255$	0	0
$256$	−26.8266	−1.67666
$257$	−12.9105	−0.805334	−0.402667	−	0.915347i	$-0.631917\pi$
$257$	−12.9105	−0.805334	−0.402667	+	0.915347i	$0.631917\pi$
$258$	0	0
$259$	2.57401	0.159941
$260$	8.12332	0.503787
$261$	0	0
$262$	−26.5626	−1.64104
$263$	20.6207	1.27153	0.635764	−	0.771884i	$-0.280685\pi$
$263$	20.6207	1.27153	0.635764	+	0.771884i	$0.280685\pi$
$264$	0	0
$265$	8.18227	0.502633
$266$	0.420881	0.0258059
$267$	0	0
$268$	−28.9409	−1.76785
$269$	12.5419	0.764693	0.382346	−	0.924019i	$-0.375116\pi$
$269$	12.5419	0.764693	0.382346	+	0.924019i	$0.375116\pi$
$270$	0	0
$271$	−3.25633	−0.197808	−0.0989040	−	0.995097i	$-0.531534\pi$
$271$	−3.25633	−0.197808	−0.0989040	+	0.995097i	$0.531534\pi$
$272$	−28.6600	−1.73777
$273$	0	0
$274$	6.93630	0.419037
$275$	−4.21170	−0.253975
$276$	0	0
$277$	−19.4188	−1.16677	−0.583383	−	0.812198i	$-0.698271\pi$
$277$	−19.4188	−1.16677	−0.583383	+	0.812198i	$0.698271\pi$
$278$	−4.06683	−0.243913
$279$	0	0
$280$	5.66474	0.338533
$281$	−14.6740	−0.875377	−0.437688	−	0.899127i	$-0.644203\pi$
$281$	−14.6740	−0.875377	−0.437688	+	0.899127i	$0.644203\pi$
$282$	0	0
$283$	−6.75792	−0.401717	−0.200858	−	0.979620i	$-0.564373\pi$
$283$	−6.75792	−0.401717	−0.200858	+	0.979620i	$0.564373\pi$
$284$	18.3187	1.08702
$285$	0	0
$286$	19.1912	1.13480
$287$	4.75301	0.280561
$288$	0	0
$289$	−3.28789	−0.193405
$290$	15.1028	0.886865
$291$	0	0
$292$	26.4888	1.55014
$293$	8.47376	0.495043	0.247521	−	0.968882i	$-0.420384\pi$
$293$	8.47376	0.495043	0.247521	+	0.968882i	$0.420384\pi$
$294$	0	0
$295$	−13.5225	−0.787311
$296$	19.7046	1.14530
$297$	0	0
$298$	−45.9818	−2.66365
$299$	9.85256	0.569788
$300$	0	0
$301$	−8.23015	−0.474378
$302$	−45.5304	−2.61998
$303$	0	0
$304$	1.47745	0.0847377
$305$	7.19117	0.411765
$306$	0	0
$307$	13.3411	0.761417	0.380709	−	0.924695i	$-0.375680\pi$
$307$	13.3411	0.761417	0.380709	+	0.924695i	$0.375680\pi$
$308$	16.5545	0.943279
$309$	0	0
$310$	4.26409	0.242184
$311$	23.1415	1.31223	0.656116	−	0.754660i	$-0.272198\pi$
$311$	23.1415	1.31223	0.656116	+	0.754660i	$0.272198\pi$
$312$	0	0
$313$	−4.78885	−0.270682	−0.135341	−	0.990799i	$-0.543213\pi$
$313$	−4.78885	−0.270682	−0.135341	+	0.990799i	$0.543213\pi$
$314$	5.72711	0.323199
$315$	0	0
$316$	67.3286	3.78753
$317$	11.7786	0.661553	0.330777	−	0.943709i	$-0.392689\pi$
$317$	11.7786	0.661553	0.330777	+	0.943709i	$0.392689\pi$
$318$	0	0
$319$	24.8173	1.38950
$320$	1.60830	0.0899068
$321$	0	0
$322$	12.2189	0.680933
$323$	−0.706873	−0.0393315
$324$	0	0
$325$	1.77782	0.0986155
$326$	23.2487	1.28763
$327$	0	0
$328$	36.3853	2.00904
$329$	0.575300	0.0317173
$330$	0	0
$331$	−4.77849	−0.262649	−0.131325	−	0.991339i	$-0.541923\pi$
$331$	−4.77849	−0.262649	−0.131325	+	0.991339i	$0.541923\pi$
$332$	31.4265	1.72476
$333$	0	0
$334$	23.2247	1.27080
$335$	−6.33381	−0.346053
$336$	0	0
$337$	16.6581	0.907424	0.453712	−	0.891148i	$-0.350100\pi$
$337$	16.6581	0.907424	0.453712	+	0.891148i	$0.350100\pi$
$338$	25.2189	1.37173
$339$	0	0
$340$	−16.9200	−0.917613
$341$	7.00689	0.379444
$342$	0	0
$343$	11.4066	0.615897
$344$	−63.0035	−3.39692
$345$	0	0
$346$	−60.4154	−3.24795
$347$	32.2014	1.72866	0.864331	−	0.502923i	$-0.167742\pi$
$347$	32.2014	1.72866	0.864331	+	0.502923i	$0.167742\pi$
$348$	0	0
$349$	24.1379	1.29207	0.646036	−	0.763307i	$-0.276425\pi$
$349$	24.1379	1.29207	0.646036	+	0.763307i	$0.276425\pi$
$350$	2.20480	0.117852
$351$	0	0
$352$	28.0790	1.49662
$353$	28.7843	1.53203	0.766016	−	0.642822i	$-0.222237\pi$
$353$	28.7843	1.53203	0.766016	+	0.642822i	$0.222237\pi$
$354$	0	0
$355$	4.00911	0.212782
$356$	−4.56927	−0.242171
$357$	0	0
$358$	30.1877	1.59547
$359$	2.59764	0.137098	0.0685491	−	0.997648i	$-0.478163\pi$
$359$	2.59764	0.137098	0.0685491	+	0.997648i	$0.478163\pi$
$360$	0	0
$361$	−18.9636	−0.998082
$362$	−11.5931	−0.609319
$363$	0	0
$364$	−6.98787	−0.366264
$365$	5.79716	0.303437
$366$	0	0
$367$	25.2994	1.32062	0.660309	−	0.750994i	$-0.270425\pi$
$367$	25.2994	1.32062	0.660309	+	0.750994i	$0.270425\pi$
$368$	42.8930	2.23595
$369$	0	0
$370$	7.66932	0.398709
$371$	−7.03858	−0.365425
$372$	0	0
$373$	−27.3505	−1.41616	−0.708079	−	0.706133i	$-0.750438\pi$
$373$	−27.3505	−1.41616	−0.708079	+	0.706133i	$0.750438\pi$
$374$	−39.9731	−2.06696
$375$	0	0
$376$	4.40404	0.227121
$377$	−10.4757	−0.539528
$378$	0	0
$379$	1.45828	0.0749067	0.0374533	−	0.999298i	$-0.488075\pi$
$379$	1.45828	0.0749067	0.0374533	+	0.999298i	$0.488075\pi$
$380$	0.872241	0.0447450
$381$	0	0
$382$	−27.4502	−1.40447
$383$	−26.9178	−1.37544	−0.687718	−	0.725978i	$-0.741388\pi$
$383$	−26.9178	−1.37544	−0.687718	+	0.725978i	$0.741388\pi$
$384$	0	0
$385$	3.62300	0.184645
$386$	18.7822	0.955987
$387$	0	0
$388$	40.6563	2.06401
$389$	12.0636	0.611647	0.305823	−	0.952088i	$-0.401068\pi$
$389$	12.0636	0.611647	0.305823	+	0.952088i	$0.401068\pi$
$390$	0	0
$391$	−20.5218	−1.03783
$392$	41.2234	2.08210
$393$	0	0
$394$	28.1636	1.41886
$395$	14.7351	0.741403
$396$	0	0
$397$	−15.9222	−0.799111	−0.399556	−	0.916709i	$-0.630836\pi$
$397$	−15.9222	−0.799111	−0.399556	+	0.916709i	$0.630836\pi$
$398$	71.9195	3.60500
$399$	0	0
$400$	7.73970	0.386985
$401$	2.22221	0.110972	0.0554860	−	0.998459i	$-0.482329\pi$
$401$	2.22221	0.110972	0.0554860	+	0.998459i	$0.482329\pi$
$402$	0	0
$403$	−2.95770	−0.147334
$404$	24.3536	1.21164
$405$	0	0
$406$	−12.9918	−0.644770
$407$	12.6025	0.624681
$408$	0	0
$409$	19.3621	0.957396	0.478698	−	0.877980i	$-0.341109\pi$
$409$	19.3621	0.957396	0.478698	+	0.877980i	$0.341109\pi$
$410$	14.1617	0.699398
$411$	0	0
$412$	18.7156	0.922051
$413$	11.6324	0.572392
$414$	0	0
$415$	6.87780	0.337618
$416$	−11.8525	−0.581119
$417$	0	0
$418$	2.06065	0.100790
$419$	−23.6038	−1.15312	−0.576560	−	0.817055i	$-0.695605\pi$
$419$	−23.6038	−1.15312	−0.576560	+	0.817055i	$0.695605\pi$
$420$	0	0
$421$	−27.7387	−1.35190	−0.675950	−	0.736947i	$-0.736267\pi$
$421$	−27.7387	−1.35190	−0.675950	+	0.736947i	$0.736267\pi$
$422$	−6.16551	−0.300132
$423$	0	0
$424$	−53.8819	−2.61673
$425$	−3.70299	−0.179621
$426$	0	0
$427$	−6.18601	−0.299362
$428$	−5.89403	−0.284899
$429$	0	0
$430$	−24.5219	−1.18255
$431$	12.5891	0.606394	0.303197	−	0.952928i	$-0.401946\pi$
$431$	12.5891	0.606394	0.303197	+	0.952928i	$0.401946\pi$
$432$	0	0
$433$	19.5451	0.939277	0.469639	−	0.882859i	$-0.344384\pi$
$433$	19.5451	0.939277	0.469639	+	0.882859i	$0.344384\pi$
$434$	−3.66807	−0.176073
$435$	0	0
$436$	−16.5135	−0.790853
$437$	1.05792	0.0506070
$438$	0	0
$439$	18.2107	0.869151	0.434575	−	0.900635i	$-0.356898\pi$
$439$	18.2107	0.869151	0.434575	+	0.900635i	$0.356898\pi$
$440$	27.7348	1.32221
$441$	0	0
$442$	16.8732	0.802576
$443$	31.4813	1.49572	0.747862	−	0.663855i	$-0.231081\pi$
$443$	31.4813	1.49572	0.747862	+	0.663855i	$0.231081\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−14.5618	−0.689521
$447$	0	0
$448$	−1.38350	−0.0653642
$449$	−22.1058	−1.04324	−0.521619	−	0.853179i	$-0.674672\pi$
$449$	−22.1058	−1.04324	−0.521619	+	0.853179i	$0.674672\pi$
$450$	0	0
$451$	23.2710	1.09579
$452$	39.7193	1.86824
$453$	0	0
$454$	−12.8212	−0.601728
$455$	−1.52932	−0.0716956
$456$	0	0
$457$	−6.34062	−0.296602	−0.148301	−	0.988942i	$-0.547380\pi$
$457$	−6.34062	−0.296602	−0.148301	+	0.988942i	$0.547380\pi$
$458$	−47.6082	−2.22459
$459$	0	0
$460$	25.3227	1.18068
$461$	−6.78304	−0.315918	−0.157959	−	0.987446i	$-0.550491\pi$
$461$	−6.78304	−0.315918	−0.157959	+	0.987446i	$0.550491\pi$
$462$	0	0
$463$	−5.19182	−0.241284	−0.120642	−	0.992696i	$-0.538495\pi$
$463$	−5.19182	−0.241284	−0.120642	+	0.992696i	$0.538495\pi$
$464$	−45.6060	−2.11721
$465$	0	0
$466$	−32.7024	−1.51491
$467$	26.1645	1.21075	0.605375	−	0.795941i	$-0.293023\pi$
$467$	26.1645	1.21075	0.605375	+	0.795941i	$0.293023\pi$
$468$	0	0
$469$	5.44849	0.251588
$470$	1.71412	0.0790665
$471$	0	0
$472$	89.0484	4.09878
$473$	−40.2952	−1.85278
$474$	0	0
$475$	0.190893	0.00875876
$476$	14.5549	0.667125
$477$	0	0
$478$	−48.8669	−2.23512
$479$	7.79448	0.356139	0.178069	−	0.984018i	$-0.443015\pi$
$479$	7.79448	0.356139	0.178069	+	0.984018i	$0.443015\pi$
$480$	0	0
$481$	−5.31967	−0.242556
$482$	38.7468	1.76487
$483$	0	0
$484$	30.7896	1.39953
$485$	8.89776	0.404027
$486$	0	0
$487$	31.0535	1.40717	0.703585	−	0.710611i	$-0.251581\pi$
$487$	31.0535	1.40717	0.703585	+	0.710611i	$0.251581\pi$
$488$	−47.3552	−2.14367
$489$	0	0
$490$	16.0448	0.724830
$491$	27.9471	1.26124	0.630618	−	0.776093i	$-0.282802\pi$
$491$	27.9471	1.26124	0.630618	+	0.776093i	$0.282802\pi$
$492$	0	0
$493$	21.8198	0.982713
$494$	−0.869830	−0.0391355
$495$	0	0
$496$	−12.8763	−0.578164
$497$	−3.44873	−0.154697
$498$	0	0
$499$	39.3746	1.76265	0.881326	−	0.472510i	$-0.156652\pi$
$499$	39.3746	1.76265	0.881326	+	0.472510i	$0.156652\pi$
$500$	4.56927	0.204344
$501$	0	0
$502$	9.88738	0.441295
$503$	31.7858	1.41726	0.708630	−	0.705580i	$-0.249313\pi$
$503$	31.7858	1.41726	0.708630	+	0.705580i	$0.249313\pi$
$504$	0	0
$505$	5.32986	0.237176
$506$	59.8244	2.65952
$507$	0	0
$508$	−19.8061	−0.878754
$509$	9.33281	0.413669	0.206835	−	0.978376i	$-0.433684\pi$
$509$	9.33281	0.413669	0.206835	+	0.978376i	$0.433684\pi$
$510$	0	0
$511$	−4.98685	−0.220605
$512$	50.3349	2.22451
$513$	0	0
$514$	33.0903	1.45955
$515$	4.09597	0.180490
$516$	0	0
$517$	2.81670	0.123878
$518$	−6.59733	−0.289870
$519$	0	0
$520$	−11.7073	−0.513397
$521$	14.0197	0.614216	0.307108	−	0.951675i	$-0.400639\pi$
$521$	14.0197	0.614216	0.307108	+	0.951675i	$0.400639\pi$
$522$	0	0
$523$	15.7379	0.688171	0.344086	−	0.938938i	$-0.388189\pi$
$523$	15.7379	0.688171	0.344086	+	0.938938i	$0.388189\pi$
$524$	47.3542	2.06868
$525$	0	0
$526$	−52.8521	−2.30446
$527$	6.16056	0.268358
$528$	0	0
$529$	7.71317	0.335355
$530$	−20.9716	−0.910950
$531$	0	0
$532$	−0.750322	−0.0325306
$533$	−9.82300	−0.425481
$534$	0	0
$535$	−1.28993	−0.0557684
$536$	41.7094	1.80157
$537$	0	0
$538$	−32.1456	−1.38590
$539$	26.3653	1.13563
$540$	0	0
$541$	20.5401	0.883088	0.441544	−	0.897240i	$-0.354431\pi$
$541$	20.5401	0.883088	0.441544	+	0.897240i	$0.354431\pi$
$542$	8.34617	0.358499
$543$	0	0
$544$	24.6875	1.05847
$545$	−3.61403	−0.154808
$546$	0	0
$547$	28.0903	1.20105	0.600527	−	0.799604i	$-0.294957\pi$
$547$	28.0903	1.20105	0.600527	+	0.799604i	$0.294957\pi$
$548$	−12.3656	−0.528234
$549$	0	0
$550$	10.7948	0.460293
$551$	−1.12483	−0.0479194
$552$	0	0
$553$	−12.6755	−0.539016
$554$	49.7716	2.11459
$555$	0	0
$556$	7.25011	0.307473
$557$	26.9430	1.14161	0.570807	−	0.821085i	$-0.306631\pi$
$557$	26.9430	1.14161	0.570807	+	0.821085i	$0.306631\pi$
$558$	0	0
$559$	17.0092	0.719411
$560$	−6.65787	−0.281346
$561$	0	0
$562$	37.6103	1.58649
$563$	8.65594	0.364804	0.182402	−	0.983224i	$-0.441613\pi$
$563$	8.65594	0.364804	0.182402	+	0.983224i	$0.441613\pi$
$564$	0	0
$565$	8.69271	0.365705
$566$	17.3209	0.728054
$567$	0	0
$568$	−26.4008	−1.10775
$569$	−43.8785	−1.83948	−0.919742	−	0.392523i	$-0.871602\pi$
$569$	−43.8785	−1.83948	−0.919742	+	0.392523i	$0.871602\pi$
$570$	0	0
$571$	−12.6616	−0.529873	−0.264937	−	0.964266i	$-0.585351\pi$
$571$	−12.6616	−0.529873	−0.264937	+	0.964266i	$0.585351\pi$
$572$	−34.2130	−1.43052
$573$	0	0
$574$	−12.1822	−0.508477
$575$	5.54195	0.231115
$576$	0	0
$577$	−8.67563	−0.361171	−0.180586	−	0.983559i	$-0.557799\pi$
$577$	−8.67563	−0.361171	−0.180586	+	0.983559i	$0.557799\pi$
$578$	8.42705	0.350519
$579$	0	0
$580$	−26.9243	−1.11797
$581$	−5.91645	−0.245456
$582$	0	0
$583$	−34.4613	−1.42724
$584$	−38.1754	−1.57971
$585$	0	0
$586$	−21.7188	−0.897194
$587$	−17.5075	−0.722610	−0.361305	−	0.932448i	$-0.617669\pi$
$587$	−17.5075	−0.722610	−0.361305	+	0.932448i	$0.617669\pi$
$588$	0	0
$589$	−0.317583	−0.0130858
$590$	34.6590	1.42689
$591$	0	0
$592$	−23.1591	−0.951835
$593$	−33.3611	−1.36998	−0.684988	−	0.728554i	$-0.740193\pi$
$593$	−33.3611	−1.36998	−0.684988	+	0.728554i	$0.740193\pi$
$594$	0	0
$595$	3.18540	0.130589
$596$	81.9736	3.35777
$597$	0	0
$598$	−25.2527	−1.03266
$599$	30.1645	1.23249	0.616244	−	0.787555i	$-0.288653\pi$
$599$	30.1645	1.23249	0.616244	+	0.787555i	$0.288653\pi$
$600$	0	0
$601$	38.5326	1.57178	0.785888	−	0.618368i	$-0.212206\pi$
$601$	38.5326	1.57178	0.785888	+	0.618368i	$0.212206\pi$
$602$	21.0943	0.859741
$603$	0	0
$604$	81.1690	3.30272
$605$	6.73840	0.273955
$606$	0	0
$607$	33.6770	1.36691	0.683454	−	0.729994i	$-0.260477\pi$
$607$	33.6770	1.36691	0.683454	+	0.729994i	$0.260477\pi$
$608$	−1.27267	−0.0516134
$609$	0	0
$610$	−18.4314	−0.746265
$611$	−1.18897	−0.0481004
$612$	0	0
$613$	−8.61530	−0.347969	−0.173984	−	0.984748i	$-0.555664\pi$
$613$	−8.61530	−0.347969	−0.173984	+	0.984748i	$0.555664\pi$
$614$	−34.1941	−1.37996
$615$	0	0
$616$	−23.8582	−0.961273
$617$	−28.9396	−1.16506	−0.582531	−	0.812808i	$-0.697938\pi$
$617$	−28.9396	−1.16506	−0.582531	+	0.812808i	$0.697938\pi$
$618$	0	0
$619$	−2.35278	−0.0945664	−0.0472832	−	0.998882i	$-0.515056\pi$
$619$	−2.35278	−0.0945664	−0.0472832	+	0.998882i	$0.515056\pi$
$620$	−7.60177	−0.305294
$621$	0	0
$622$	−59.3129	−2.37823
$623$	0.860224	0.0344641
$624$	0	0
$625$	1.00000	0.0400000
$626$	12.2741	0.490572
$627$	0	0
$628$	−10.2100	−0.407421
$629$	11.0803	0.441800
$630$	0	0
$631$	23.0574	0.917902	0.458951	−	0.888462i	$-0.348225\pi$
$631$	23.0574	0.917902	0.458951	+	0.888462i	$0.348225\pi$
$632$	−97.0334	−3.85978
$633$	0	0
$634$	−30.1893	−1.19897
$635$	−4.33463	−0.172015
$636$	0	0
$637$	−11.1292	−0.440953
$638$	−63.6083	−2.51828
$639$	0	0
$640$	9.21166	0.364123
$641$	−42.7957	−1.69033	−0.845164	−	0.534507i	$-0.820497\pi$
$641$	−42.7957	−1.69033	−0.845164	+	0.534507i	$0.820497\pi$
$642$	0	0
$643$	−0.392776	−0.0154896	−0.00774478	−	0.999970i	$-0.502465\pi$
$643$	−0.392776	−0.0154896	−0.00774478	+	0.999970i	$0.502465\pi$
$644$	−21.7831	−0.858376
$645$	0	0
$646$	1.81176	0.0712827
$647$	37.5369	1.47573	0.737863	−	0.674950i	$-0.235835\pi$
$647$	37.5369	1.47573	0.737863	+	0.674950i	$0.235835\pi$
$648$	0	0
$649$	56.9527	2.23559
$650$	−4.55665	−0.178726
$651$	0	0
$652$	−41.4465	−1.62317
$653$	−25.9450	−1.01531	−0.507653	−	0.861562i	$-0.669487\pi$
$653$	−25.9450	−1.01531	−0.507653	+	0.861562i	$0.669487\pi$
$654$	0	0
$655$	10.3636	0.404940
$656$	−42.7643	−1.66967
$657$	0	0
$658$	−1.47453	−0.0574831
$659$	−34.8175	−1.35630	−0.678148	−	0.734925i	$-0.737217\pi$
$659$	−34.8175	−1.35630	−0.678148	+	0.734925i	$0.737217\pi$
$660$	0	0
$661$	−4.52020	−0.175816	−0.0879078	−	0.996129i	$-0.528018\pi$
$661$	−4.52020	−0.175816	−0.0879078	+	0.996129i	$0.528018\pi$
$662$	12.2475	0.476014
$663$	0	0
$664$	−45.2917	−1.75766
$665$	−0.164210	−0.00636781
$666$	0	0
$667$	−32.6558	−1.26444
$668$	−41.4036	−1.60195
$669$	0	0
$670$	16.2339	0.627171
$671$	−30.2870	−1.16922
$672$	0	0
$673$	−6.99025	−0.269454	−0.134727	−	0.990883i	$-0.543016\pi$
$673$	−6.99025	−0.269454	−0.134727	+	0.990883i	$0.543016\pi$
$674$	−42.6957	−1.64458
$675$	0	0
$676$	−44.9588	−1.72918
$677$	−34.9173	−1.34198	−0.670991	−	0.741465i	$-0.734131\pi$
$677$	−34.9173	−1.34198	−0.670991	+	0.741465i	$0.734131\pi$
$678$	0	0
$679$	−7.65407	−0.293736
$680$	24.3849	0.935118
$681$	0	0
$682$	−17.9591	−0.687688
$683$	−0.0378289	−0.00144748	−0.000723741	−	1.00000i	$-0.500230\pi$
$683$	−0.0378289	−0.00144748	−0.000723741		1.00000i	$0.500230\pi$
$684$	0	0
$685$	−2.70626	−0.103401
$686$	−29.2357	−1.11623
$687$	0	0
$688$	74.0492	2.82310
$689$	14.5466	0.554180
$690$	0	0
$691$	−32.6694	−1.24280	−0.621401	−	0.783492i	$-0.713436\pi$
$691$	−32.6694	−1.24280	−0.621401	+	0.783492i	$0.713436\pi$
$692$	107.705	4.09433
$693$	0	0
$694$	−82.5341	−3.13295
$695$	1.58671	0.0601874
$696$	0	0
$697$	20.4602	0.774985
$698$	−61.8669	−2.34170
$699$	0	0
$700$	−3.93059	−0.148563
$701$	14.5708	0.550330	0.275165	−	0.961397i	$-0.411267\pi$
$701$	14.5708	0.550330	0.275165	+	0.961397i	$0.411267\pi$
$702$	0	0
$703$	−0.571199	−0.0215432
$704$	−6.77368	−0.255293
$705$	0	0
$706$	−73.7757	−2.77659
$707$	−4.58487	−0.172432
$708$	0	0
$709$	18.3721	0.689978	0.344989	−	0.938607i	$-0.387883\pi$
$709$	18.3721	0.689978	0.344989	+	0.938607i	$0.387883\pi$
$710$	−10.2756	−0.385636
$711$	0	0
$712$	6.58519	0.246791
$713$	−9.21998	−0.345291
$714$	0	0
$715$	−7.48762	−0.280021
$716$	−53.8169	−2.01123
$717$	0	0
$718$	−6.65791	−0.248471
$719$	−27.3152	−1.01868	−0.509342	−	0.860564i	$-0.670111\pi$
$719$	−27.3152	−1.01868	−0.509342	+	0.860564i	$0.670111\pi$
$720$	0	0
$721$	−3.52345	−0.131220
$722$	48.6047	1.80888
$723$	0	0
$724$	20.6675	0.768101
$725$	−5.89248	−0.218841
$726$	0	0
$727$	32.8175	1.21713	0.608566	−	0.793503i	$-0.291745\pi$
$727$	32.8175	1.21713	0.608566	+	0.793503i	$0.291745\pi$
$728$	10.0709	0.373251
$729$	0	0
$730$	−14.8585	−0.549936
$731$	−35.4281	−1.31036
$732$	0	0
$733$	−38.6320	−1.42690	−0.713452	−	0.700704i	$-0.752869\pi$
$733$	−38.6320	−1.42690	−0.713452	+	0.700704i	$0.752869\pi$
$734$	−64.8439	−2.39343
$735$	0	0
$736$	−36.9477	−1.36191
$737$	26.6761	0.982626
$738$	0	0
$739$	14.9120	0.548547	0.274274	−	0.961652i	$-0.411563\pi$
$739$	14.9120	0.548547	0.274274	+	0.961652i	$0.411563\pi$
$740$	−13.6724	−0.502608
$741$	0	0
$742$	18.0403	0.662280
$743$	19.8206	0.727148	0.363574	−	0.931565i	$-0.381556\pi$
$743$	19.8206	0.727148	0.363574	+	0.931565i	$0.381556\pi$
$744$	0	0
$745$	17.9402	0.657278
$746$	70.1011	2.56658
$747$	0	0
$748$	71.2617	2.60559
$749$	1.10963	0.0405449
$750$	0	0
$751$	47.2991	1.72597	0.862985	−	0.505230i	$-0.168592\pi$
$751$	47.2991	1.72597	0.862985	+	0.505230i	$0.168592\pi$
$752$	−5.17615	−0.188755
$753$	0	0
$754$	26.8499	0.977817
$755$	17.7641	0.646502
$756$	0	0
$757$	−17.4793	−0.635295	−0.317647	−	0.948209i	$-0.602893\pi$
$757$	−17.4793	−0.635295	−0.317647	+	0.948209i	$0.602893\pi$
$758$	−3.73765	−0.135758
$759$	0	0
$760$	−1.25707	−0.0455986
$761$	−14.3928	−0.521740	−0.260870	−	0.965374i	$-0.584009\pi$
$761$	−14.3928	−0.521740	−0.260870	+	0.965374i	$0.584009\pi$
$762$	0	0
$763$	3.10887	0.112549
$764$	48.9366	1.77046
$765$	0	0
$766$	68.9920	2.49278
$767$	−24.0405	−0.868053
$768$	0	0
$769$	−4.16171	−0.150075	−0.0750375	−	0.997181i	$-0.523908\pi$
$769$	−4.16171	−0.150075	−0.0750375	+	0.997181i	$0.523908\pi$
$770$	−9.28597	−0.334643
$771$	0	0
$772$	−33.4837	−1.20511
$773$	−26.2623	−0.944591	−0.472296	−	0.881440i	$-0.656574\pi$
$773$	−26.2623	−0.944591	−0.472296	+	0.881440i	$0.656574\pi$
$774$	0	0
$775$	−1.66367	−0.0597609
$776$	−58.5935	−2.10338
$777$	0	0
$778$	−30.9196	−1.10852
$779$	−1.05474	−0.0377901
$780$	0	0
$781$	−16.8852	−0.604199
$782$	52.5985	1.88092
$783$	0	0
$784$	−48.4506	−1.73038
$785$	−2.23448	−0.0797521
$786$	0	0
$787$	0.553635	0.0197349	0.00986747	−	0.999951i	$-0.496859\pi$
$787$	0.553635	0.0197349	0.00986747	+	0.999951i	$0.496859\pi$
$788$	−50.2085	−1.78860
$789$	0	0
$790$	−37.7669	−1.34369
$791$	−7.47767	−0.265875
$792$	0	0
$793$	12.7846	0.453993
$794$	40.8095	1.44827
$795$	0	0
$796$	−128.214	−4.54442
$797$	−22.2385	−0.787728	−0.393864	−	0.919169i	$-0.628862\pi$
$797$	−22.2385	−0.787728	−0.393864	+	0.919169i	$0.628862\pi$
$798$	0	0
$799$	2.47648	0.0876116
$800$	−6.66692	−0.235711
$801$	0	0
$802$	−5.69566	−0.201121
$803$	−24.4159	−0.861617
$804$	0	0
$805$	−4.76731	−0.168026
$806$	7.58077	0.267021
$807$	0	0
$808$	−35.0982	−1.23475
$809$	4.61163	0.162136	0.0810681	−	0.996709i	$-0.474167\pi$
$809$	4.61163	0.162136	0.0810681	+	0.996709i	$0.474167\pi$
$810$	0	0
$811$	12.8357	0.450723	0.225362	−	0.974275i	$-0.427644\pi$
$811$	12.8357	0.450723	0.225362	+	0.974275i	$0.427644\pi$
$812$	23.1609	0.812790
$813$	0	0
$814$	−32.3009	−1.13214
$815$	−9.07070	−0.317733
$816$	0	0
$817$	1.82636	0.0638961
$818$	−49.6263	−1.73514
$819$	0	0
$820$	−25.2467	−0.881653
$821$	43.3925	1.51441	0.757205	−	0.653178i	$-0.226565\pi$
$821$	43.3925	1.51441	0.757205	+	0.653178i	$0.226565\pi$
$822$	0	0
$823$	−23.2779	−0.811416	−0.405708	−	0.914003i	$-0.632975\pi$
$823$	−23.2779	−0.811416	−0.405708	+	0.914003i	$0.632975\pi$
$824$	−26.9727	−0.939640
$825$	0	0
$826$	−29.8145	−1.03738
$827$	−17.0207	−0.591869	−0.295935	−	0.955208i	$-0.595631\pi$
$827$	−17.0207	−0.591869	−0.295935	+	0.955208i	$0.595631\pi$
$828$	0	0
$829$	24.6429	0.855884	0.427942	−	0.903806i	$-0.359239\pi$
$829$	24.6429	0.855884	0.427942	+	0.903806i	$0.359239\pi$
$830$	−17.6282	−0.611884
$831$	0	0
$832$	2.85926	0.0991271
$833$	23.1808	0.803166
$834$	0	0
$835$	−9.06131	−0.313580
$836$	−3.67361	−0.127055
$837$	0	0
$838$	60.4978	2.08986
$839$	24.9914	0.862797	0.431399	−	0.902161i	$-0.358020\pi$
$839$	24.9914	0.862797	0.431399	+	0.902161i	$0.358020\pi$
$840$	0	0
$841$	5.72128	0.197285
$842$	71.0959	2.45013
$843$	0	0
$844$	10.9915	0.378343
$845$	−9.83937	−0.338485
$846$	0	0
$847$	−5.79653	−0.199171
$848$	63.3283	2.17470
$849$	0	0
$850$	9.49097	0.325538
$851$	−16.5829	−0.568455
$852$	0	0
$853$	24.7310	0.846774	0.423387	−	0.905949i	$-0.360841\pi$
$853$	24.7310	0.846774	0.423387	+	0.905949i	$0.360841\pi$
$854$	15.8551	0.542551
$855$	0	0
$856$	8.49443	0.290334
$857$	5.54047	0.189259	0.0946294	−	0.995513i	$-0.469833\pi$
$857$	5.54047	0.189259	0.0946294	+	0.995513i	$0.469833\pi$
$858$	0	0
$859$	8.95441	0.305521	0.152760	−	0.988263i	$-0.451184\pi$
$859$	8.95441	0.305521	0.152760	+	0.988263i	$0.451184\pi$
$860$	43.7163	1.49071
$861$	0	0
$862$	−32.2665	−1.09900
$863$	−6.08527	−0.207145	−0.103573	−	0.994622i	$-0.533027\pi$
$863$	−6.08527	−0.207145	−0.103573	+	0.994622i	$0.533027\pi$
$864$	0	0
$865$	23.5716	0.801458
$866$	−50.0952	−1.70231
$867$	0	0
$868$	6.53922	0.221956
$869$	−62.0597	−2.10523
$870$	0	0
$871$	−11.2603	−0.381542
$872$	23.7991	0.805939
$873$	0	0
$874$	−2.71150	−0.0917180
$875$	−0.860224	−0.0290809
$876$	0	0
$877$	−14.1540	−0.477947	−0.238974	−	0.971026i	$-0.576811\pi$
$877$	−14.1540	−0.477947	−0.238974	+	0.971026i	$0.576811\pi$
$878$	−46.6752	−1.57521
$879$	0	0
$880$	−32.5973	−1.09885
$881$	−25.1223	−0.846392	−0.423196	−	0.906038i	$-0.639092\pi$
$881$	−25.1223	−0.846392	−0.423196	+	0.906038i	$0.639092\pi$
$882$	0	0
$883$	−20.8908	−0.703031	−0.351516	−	0.936182i	$-0.614334\pi$
$883$	−20.8908	−0.703031	−0.351516	+	0.936182i	$0.614334\pi$
$884$	−30.0806	−1.01172
$885$	0	0
$886$	−80.6885	−2.71078
$887$	1.26866	0.0425973	0.0212987	−	0.999773i	$-0.493220\pi$
$887$	1.26866	0.0425973	0.0212987	+	0.999773i	$0.493220\pi$
$888$	0	0
$889$	3.72875	0.125058
$890$	2.56306	0.0859139
$891$	0	0
$892$	25.9599	0.869203
$893$	−0.127665	−0.00427215
$894$	0	0
$895$	−11.7780	−0.393695
$896$	−7.92409	−0.264725
$897$	0	0
$898$	56.6585	1.89072
$899$	9.80315	0.326953
$900$	0	0
$901$	−30.2989	−1.00940
$902$	−59.6449	−1.98596
$903$	0	0
$904$	−57.2432	−1.90388
$905$	4.52315	0.150354
$906$	0	0
$907$	22.9355	0.761562	0.380781	−	0.924665i	$-0.375655\pi$
$907$	22.9355	0.761562	0.380781	+	0.924665i	$0.375655\pi$
$908$	22.8569	0.758532
$909$	0	0
$910$	3.91973	0.129938
$911$	−31.5500	−1.04530	−0.522649	−	0.852548i	$-0.675056\pi$
$911$	−31.5500	−1.04530	−0.522649	+	0.852548i	$0.675056\pi$
$912$	0	0
$913$	−28.9672	−0.958675
$914$	16.2514	0.537548
$915$	0	0
$916$	84.8732	2.80429
$917$	−8.91503	−0.294400
$918$	0	0
$919$	−2.13081	−0.0702889	−0.0351445	−	0.999382i	$-0.511189\pi$
$919$	−2.13081	−0.0702889	−0.0351445	+	0.999382i	$0.511189\pi$
$920$	−36.4948	−1.20320
$921$	0	0
$922$	17.3853	0.572555
$923$	7.12746	0.234603
$924$	0	0
$925$	−2.99225	−0.0983847
$926$	13.3069	0.437293
$927$	0	0
$928$	39.2846	1.28958
$929$	43.5650	1.42932	0.714660	−	0.699472i	$-0.246581\pi$
$929$	43.5650	1.42932	0.714660	+	0.699472i	$0.246581\pi$
$930$	0	0
$931$	−1.19499	−0.0391643
$932$	58.2999	1.90968
$933$	0	0
$934$	−67.0612	−2.19431
$935$	15.5959	0.510039
$936$	0	0
$937$	−20.3131	−0.663601	−0.331801	−	0.943350i	$-0.607656\pi$
$937$	−20.3131	−0.663601	−0.331801	+	0.943350i	$0.607656\pi$
$938$	−13.9648	−0.455967
$939$	0	0
$940$	−3.05584	−0.0996704
$941$	39.7773	1.29670	0.648351	−	0.761342i	$-0.275459\pi$
$941$	39.7773	1.29670	0.648351	+	0.761342i	$0.275459\pi$
$942$	0	0
$943$	−30.6210	−0.997158
$944$	−104.660	−3.40640
$945$	0	0
$946$	103.279	3.35789
$947$	10.4083	0.338225	0.169113	−	0.985597i	$-0.445910\pi$
$947$	10.4083	0.338225	0.169113	+	0.985597i	$0.445910\pi$
$948$	0	0
$949$	10.3063	0.334556
$950$	−0.489269	−0.0158740
$951$	0	0
$952$	−20.9765	−0.679851
$953$	16.0580	0.520169	0.260084	−	0.965586i	$-0.416250\pi$
$953$	16.0580	0.520169	0.260084	+	0.965586i	$0.416250\pi$
$954$	0	0
$955$	10.7099	0.346565
$956$	87.1170	2.81757
$957$	0	0
$958$	−19.9777	−0.645450
$959$	2.32799	0.0751746
$960$	0	0
$961$	−28.2322	−0.910716
$962$	13.6346	0.439598
$963$	0	0
$964$	−69.0756	−2.22477
$965$	−7.32803	−0.235898
$966$	0	0
$967$	6.28179	0.202009	0.101004	−	0.994886i	$-0.467794\pi$
$967$	6.28179	0.202009	0.101004	+	0.994886i	$0.467794\pi$
$968$	−44.3737	−1.42622
$969$	0	0
$970$	−22.8055	−0.732240
$971$	−3.20201	−0.102757	−0.0513786	−	0.998679i	$-0.516362\pi$
$971$	−3.20201	−0.102757	−0.0513786	+	0.998679i	$0.516362\pi$
$972$	0	0
$973$	−1.36493	−0.0437575
$974$	−79.5921	−2.55029
$975$	0	0
$976$	55.6575	1.78155
$977$	50.3806	1.61182	0.805909	−	0.592039i	$-0.201677\pi$
$977$	50.3806	1.61182	0.805909	+	0.592039i	$0.201677\pi$
$978$	0	0
$979$	4.21170	0.134606
$980$	−28.6037	−0.913712
$981$	0	0
$982$	−71.6301	−2.28581
$983$	14.8459	0.473511	0.236755	−	0.971569i	$-0.423916\pi$
$983$	14.8459	0.473511	0.236755	+	0.971569i	$0.423916\pi$
$984$	0	0
$985$	−10.9883	−0.350116
$986$	−55.9253	−1.78103
$987$	0	0
$988$	1.55068	0.0493338
$989$	53.0223	1.68601
$990$	0	0
$991$	−46.6856	−1.48302	−0.741508	−	0.670944i	$-0.765889\pi$
$991$	−46.6856	−1.48302	−0.741508	+	0.670944i	$0.765889\pi$
$992$	11.0916	0.352158
$993$	0	0
$994$	8.83931	0.280366
$995$	−28.0600	−0.889562
$996$	0	0
$997$	51.2816	1.62410	0.812052	−	0.583585i	$-0.198351\pi$
$997$	51.2816	1.62410	0.812052	+	0.583585i	$0.198351\pi$
$998$	−100.920	−3.19455
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.x.1.1	yes	17
3.2	odd	2		4005.2.a.w.1.17	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.w.1.17	✓	17	3.2	odd	2
4005.2.a.x.1.1	yes	17	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)

\(f(q)\)	\(=\)	\(q-2.56306 q^{2} +4.56927 q^{4} +1.00000 q^{5} -0.860224 q^{7} -6.58519 q^{8} +O(q^{10})\)	\(q-2.56306 q^{2} +4.56927 q^{4} +1.00000 q^{5} -0.860224 q^{7} -6.58519 q^{8} -2.56306 q^{10} -4.21170 q^{11} +1.77782 q^{13} +2.20480 q^{14} +7.73970 q^{16} -3.70299 q^{17} +0.190893 q^{19} +4.56927 q^{20} +10.7948 q^{22} +5.54195 q^{23} +1.00000 q^{25} -4.55665 q^{26} -3.93059 q^{28} -5.89248 q^{29} -1.66367 q^{31} -6.66692 q^{32} +9.49097 q^{34} -0.860224 q^{35} -2.99225 q^{37} -0.489269 q^{38} -6.58519 q^{40} -5.52532 q^{41} +9.56745 q^{43} -19.2444 q^{44} -14.2043 q^{46} -0.668780 q^{47} -6.26002 q^{49} -2.56306 q^{50} +8.12332 q^{52} +8.18227 q^{53} -4.21170 q^{55} +5.66474 q^{56} +15.1028 q^{58} -13.5225 q^{59} +7.19117 q^{61} +4.26409 q^{62} +1.60830 q^{64} +1.77782 q^{65} -6.33381 q^{67} -16.9200 q^{68} +2.20480 q^{70} +4.00911 q^{71} +5.79716 q^{73} +7.66932 q^{74} +0.872241 q^{76} +3.62300 q^{77} +14.7351 q^{79} +7.73970 q^{80} +14.1617 q^{82} +6.87780 q^{83} -3.70299 q^{85} -24.5219 q^{86} +27.7348 q^{88} -1.00000 q^{89} -1.52932 q^{91} +25.3227 q^{92} +1.71412 q^{94} +0.190893 q^{95} +8.89776 q^{97} +16.0448 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 5 q^{2} + 21 q^{4} + 17 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) 17 * q + 5 * q^2 + 21 * q^4 + 17 * q^5 + 12 * q^7 + 15 * q^8	\( 17 q + 5 q^{2} + 21 q^{4} + 17 q^{5} + 12 q^{7} + 15 q^{8} + 5 q^{10} + 2 q^{11} + 8 q^{13} + 4 q^{14} + 33 q^{16} + 10 q^{17} + 32 q^{19} + 21 q^{20} + 8 q^{22} + 15 q^{23} + 17 q^{25} - 15 q^{26} + 24 q^{28} + q^{29} + 18 q^{31} + 25 q^{32} + 14 q^{34} + 12 q^{35} + 12 q^{37} + 22 q^{38} + 15 q^{40} - 7 q^{41} + 28 q^{43} - 14 q^{44} + 4 q^{46} + 26 q^{47} + 41 q^{49} + 5 q^{50} + 10 q^{52} + 12 q^{53} + 2 q^{55} + 13 q^{56} + 16 q^{58} - 23 q^{59} + 26 q^{61} + 10 q^{62} + 59 q^{64} + 8 q^{65} + 31 q^{67} - q^{68} + 4 q^{70} - 2 q^{71} + 33 q^{73} - 10 q^{74} + 66 q^{76} + 12 q^{77} + 33 q^{79} + 33 q^{80} + 30 q^{82} + 13 q^{83} + 10 q^{85} - 20 q^{86} + 12 q^{88} - 17 q^{89} + 40 q^{91} + 16 q^{92} + 38 q^{94} + 32 q^{95} + 45 q^{97} + 2 q^{98}+O(q^{100}) \) 17 * q + 5 * q^2 + 21 * q^4 + 17 * q^5 + 12 * q^7 + 15 * q^8 + 5 * q^10 + 2 * q^11 + 8 * q^13 + 4 * q^14 + 33 * q^16 + 10 * q^17 + 32 * q^19 + 21 * q^20 + 8 * q^22 + 15 * q^23 + 17 * q^25 - 15 * q^26 + 24 * q^28 + q^29 + 18 * q^31 + 25 * q^32 + 14 * q^34 + 12 * q^35 + 12 * q^37 + 22 * q^38 + 15 * q^40 - 7 * q^41 + 28 * q^43 - 14 * q^44 + 4 * q^46 + 26 * q^47 + 41 * q^49 + 5 * q^50 + 10 * q^52 + 12 * q^53 + 2 * q^55 + 13 * q^56 + 16 * q^58 - 23 * q^59 + 26 * q^61 + 10 * q^62 + 59 * q^64 + 8 * q^65 + 31 * q^67 - q^68 + 4 * q^70 - 2 * q^71 + 33 * q^73 - 10 * q^74 + 66 * q^76 + 12 * q^77 + 33 * q^79 + 33 * q^80 + 30 * q^82 + 13 * q^83 + 10 * q^85 - 20 * q^86 + 12 * q^88 - 17 * q^89 + 40 * q^91 + 16 * q^92 + 38 * q^94 + 32 * q^95 + 45 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more