LMFDB - Embedded newform 4004.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(1,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.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:	$31.9721009693$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.82709$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.95630 q^{3} -0.209057 q^{5} -1.00000 q^{7} +0.827091 q^{9} +O(q^{10})$	$q-1.95630 q^{3} -0.209057 q^{5} -1.00000 q^{7} +0.827091 q^{9} +1.00000 q^{11} -1.00000 q^{13} +0.408977 q^{15} +2.88558 q^{17} -1.89781 q^{19} +1.95630 q^{21} -1.33826 q^{23} -4.95630 q^{25} +4.25085 q^{27} +5.48127 q^{29} +4.65418 q^{31} -1.95630 q^{33} +0.209057 q^{35} +1.56966 q^{37} +1.95630 q^{39} -7.25999 q^{41} +4.32157 q^{43} -0.172909 q^{45} +3.41056 q^{47} +1.00000 q^{49} -5.64505 q^{51} -10.2499 q^{53} -0.209057 q^{55} +3.71267 q^{57} +8.91068 q^{59} -0.657068 q^{61} -0.827091 q^{63} +0.209057 q^{65} +2.30836 q^{67} +2.61803 q^{69} -8.47214 q^{71} -4.90036 q^{73} +9.69598 q^{75} -1.00000 q^{77} +7.52149 q^{79} -10.7972 q^{81} +1.60134 q^{83} -0.603250 q^{85} -10.7230 q^{87} -11.6507 q^{89} +1.00000 q^{91} -9.10495 q^{93} +0.396750 q^{95} +17.3457 q^{97} +0.827091 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + q^{5} - 4 q^{7} - 3 q^{9}+O(q^{10}) $ 4 * q + q^3 + q^5 - 4 * q^7 - 3 * q^9	$ 4 q + q^{3} + q^{5} - 4 q^{7} - 3 q^{9} + 4 q^{11} - 4 q^{13} - q^{15} + q^{17} - 3 q^{19} - q^{21} - q^{23} - 11 q^{25} - 5 q^{27} + 3 q^{29} + 6 q^{31} + q^{33} - q^{35} + 4 q^{37} - q^{39} - 6 q^{41} - 3 q^{43} - 7 q^{45} - 7 q^{47} + 4 q^{49} - 11 q^{51} - 20 q^{53} + q^{55} - 2 q^{57} - 11 q^{59} - 18 q^{61} + 3 q^{63} - q^{65} - 16 q^{67} + 6 q^{69} - 16 q^{71} + 4 q^{73} + 6 q^{75} - 4 q^{77} + 9 q^{79} - 16 q^{81} - 14 q^{83} - 11 q^{85} - 3 q^{87} - 23 q^{89} + 4 q^{91} - q^{93} - 7 q^{95} + 14 q^{97} - 3 q^{99}+O(q^{100}) $ 4 * q + q^3 + q^5 - 4 * q^7 - 3 * q^9 + 4 * q^11 - 4 * q^13 - q^15 + q^17 - 3 * q^19 - q^21 - q^23 - 11 * q^25 - 5 * q^27 + 3 * q^29 + 6 * q^31 + q^33 - q^35 + 4 * q^37 - q^39 - 6 * q^41 - 3 * q^43 - 7 * q^45 - 7 * q^47 + 4 * q^49 - 11 * q^51 - 20 * q^53 + q^55 - 2 * q^57 - 11 * q^59 - 18 * q^61 + 3 * q^63 - q^65 - 16 * q^67 + 6 * q^69 - 16 * q^71 + 4 * q^73 + 6 * q^75 - 4 * q^77 + 9 * q^79 - 16 * q^81 - 14 * q^83 - 11 * q^85 - 3 * q^87 - 23 * q^89 + 4 * q^91 - q^93 - 7 * q^95 + 14 * q^97 - 3 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.95630	−1.12947	−0.564734	−	0.825273i	$-0.691021\pi$
$3$	−1.95630	−1.12947	−0.564734	+	0.825273i	$0.691021\pi$
$4$	0	0
$5$	−0.209057	−0.0934931	−0.0467465	−	0.998907i	$-0.514885\pi$
$5$	−0.209057	−0.0934931	−0.0467465	+	0.998907i	$0.514885\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.827091	0.275697
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0.408977	0.105597
$16$	0	0
$17$	2.88558	0.699856	0.349928	−	0.936777i	$-0.386206\pi$
$17$	2.88558	0.699856	0.349928	+	0.936777i	$0.386206\pi$
$18$	0	0
$19$	−1.89781	−0.435387	−0.217693	−	0.976017i	$-0.569853\pi$
$19$	−1.89781	−0.435387	−0.217693	+	0.976017i	$0.569853\pi$
$20$	0	0
$21$	1.95630	0.426899
$22$	0	0
$23$	−1.33826	−0.279047	−0.139523	−	0.990219i	$-0.544557\pi$
$23$	−1.33826	−0.279047	−0.139523	+	0.990219i	$0.544557\pi$
$24$	0	0
$25$	−4.95630	−0.991259
$26$	0	0
$27$	4.25085	0.818077
$28$	0	0
$29$	5.48127	1.01785	0.508923	−	0.860812i	$-0.330044\pi$
$29$	5.48127	1.01785	0.508923	+	0.860812i	$0.330044\pi$
$30$	0	0
$31$	4.65418	0.835916	0.417958	−	0.908466i	$-0.362746\pi$
$31$	4.65418	0.835916	0.417958	+	0.908466i	$0.362746\pi$
$32$	0	0
$33$	−1.95630	−0.340547
$34$	0	0
$35$	0.209057	0.0353371
$36$	0	0
$37$	1.56966	0.258050	0.129025	−	0.991641i	$-0.458815\pi$
$37$	1.56966	0.258050	0.129025	+	0.991641i	$0.458815\pi$
$38$	0	0
$39$	1.95630	0.313258
$40$	0	0
$41$	−7.25999	−1.13382	−0.566910	−	0.823780i	$-0.691861\pi$
$41$	−7.25999	−1.13382	−0.566910	+	0.823780i	$0.691861\pi$
$42$	0	0
$43$	4.32157	0.659033	0.329516	−	0.944150i	$-0.393114\pi$
$43$	4.32157	0.659033	0.329516	+	0.944150i	$0.393114\pi$
$44$	0	0
$45$	−0.172909	−0.0257758
$46$	0	0
$47$	3.41056	0.497481	0.248740	−	0.968570i	$-0.419983\pi$
$47$	3.41056	0.497481	0.248740	+	0.968570i	$0.419983\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.64505	−0.790464
$52$	0	0
$53$	−10.2499	−1.40793	−0.703964	−	0.710235i	$-0.748589\pi$
$53$	−10.2499	−1.40793	−0.703964	+	0.710235i	$0.748589\pi$
$54$	0	0
$55$	−0.209057	−0.0281892
$56$	0	0
$57$	3.71267	0.491755
$58$	0	0
$59$	8.91068	1.16007	0.580036	−	0.814591i	$-0.303039\pi$
$59$	8.91068	1.16007	0.580036	+	0.814591i	$0.303039\pi$
$60$	0	0
$61$	−0.657068	−0.0841290	−0.0420645	−	0.999115i	$-0.513393\pi$
$61$	−0.657068	−0.0841290	−0.0420645	+	0.999115i	$0.513393\pi$
$62$	0	0
$63$	−0.827091	−0.104204
$64$	0	0
$65$	0.209057	0.0259303
$66$	0	0
$67$	2.30836	0.282012	0.141006	−	0.990009i	$-0.454966\pi$
$67$	2.30836	0.282012	0.141006	+	0.990009i	$0.454966\pi$
$68$	0	0
$69$	2.61803	0.315174
$70$	0	0
$71$	−8.47214	−1.00546	−0.502729	−	0.864444i	$-0.667671\pi$
$71$	−8.47214	−1.00546	−0.502729	+	0.864444i	$0.667671\pi$
$72$	0	0
$73$	−4.90036	−0.573544	−0.286772	−	0.957999i	$-0.592582\pi$
$73$	−4.90036	−0.573544	−0.286772	+	0.957999i	$0.592582\pi$
$74$	0	0
$75$	9.69598	1.11959
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	7.52149	0.846233	0.423117	−	0.906075i	$-0.360936\pi$
$79$	7.52149	0.846233	0.423117	+	0.906075i	$0.360936\pi$
$80$	0	0
$81$	−10.7972	−1.19969
$82$	0	0
$83$	1.60134	0.175770	0.0878850	−	0.996131i	$-0.471989\pi$
$83$	1.60134	0.175770	0.0878850	+	0.996131i	$0.471989\pi$
$84$	0	0
$85$	−0.603250	−0.0654317
$86$	0	0
$87$	−10.7230	−1.14962
$88$	0	0
$89$	−11.6507	−1.23497	−0.617485	−	0.786582i	$-0.711849\pi$
$89$	−11.6507	−1.23497	−0.617485	+	0.786582i	$0.711849\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−9.10495	−0.944140
$94$	0	0
$95$	0.396750	0.0407057
$96$	0	0
$97$	17.3457	1.76119	0.880594	−	0.473871i	$-0.157144\pi$
$97$	17.3457	1.76119	0.880594	+	0.473871i	$0.157144\pi$
$98$	0	0
$99$	0.827091	0.0831258
$100$	0	0
$101$	6.00118	0.597140	0.298570	−	0.954388i	$-0.403490\pi$
$101$	6.00118	0.597140	0.298570	+	0.954388i	$0.403490\pi$
$102$	0	0
$103$	6.51742	0.642181	0.321090	−	0.947049i	$-0.395951\pi$
$103$	6.51742	0.642181	0.321090	+	0.947049i	$0.395951\pi$
$104$	0	0
$105$	−0.408977	−0.0399121
$106$	0	0
$107$	1.47056	0.142164	0.0710820	−	0.997470i	$-0.477355\pi$
$107$	1.47056	0.142164	0.0710820	+	0.997470i	$0.477355\pi$
$108$	0	0
$109$	−8.63591	−0.827170	−0.413585	−	0.910466i	$-0.635724\pi$
$109$	−8.63591	−0.827170	−0.413585	+	0.910466i	$0.635724\pi$
$110$	0	0
$111$	−3.07072	−0.291459
$112$	0	0
$113$	−14.0342	−1.32023	−0.660115	−	0.751165i	$-0.729492\pi$
$113$	−14.0342	−1.32023	−0.660115	+	0.751165i	$0.729492\pi$
$114$	0	0
$115$	0.279773	0.0260889
$116$	0	0
$117$	−0.827091	−0.0764646
$118$	0	0
$119$	−2.88558	−0.264521
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	14.2027	1.28061
$124$	0	0
$125$	2.08143	0.186169
$126$	0	0
$127$	−3.39675	−0.301413	−0.150706	−	0.988579i	$-0.548155\pi$
$127$	−3.39675	−0.301413	−0.150706	+	0.988579i	$0.548155\pi$
$128$	0	0
$129$	−8.45426	−0.744356
$130$	0	0
$131$	−11.4310	−0.998730	−0.499365	−	0.866392i	$-0.666433\pi$
$131$	−11.4310	−0.998730	−0.499365	+	0.866392i	$0.666433\pi$
$132$	0	0
$133$	1.89781	0.164561
$134$	0	0
$135$	−0.888670	−0.0764845
$136$	0	0
$137$	18.4161	1.57339	0.786696	−	0.617341i	$-0.211790\pi$
$137$	18.4161	1.57339	0.786696	+	0.617341i	$0.211790\pi$
$138$	0	0
$139$	0.0864759	0.00733479	0.00366739	−	0.999993i	$-0.498833\pi$
$139$	0.0864759	0.00733479	0.00366739	+	0.999993i	$0.498833\pi$
$140$	0	0
$141$	−6.67206	−0.561888
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−1.14590	−0.0951617
$146$	0	0
$147$	−1.95630	−0.161353
$148$	0	0
$149$	−14.4240	−1.18166	−0.590829	−	0.806797i	$-0.701199\pi$
$149$	−14.4240	−1.18166	−0.590829	+	0.806797i	$0.701199\pi$
$150$	0	0
$151$	−4.83720	−0.393646	−0.196823	−	0.980439i	$-0.563062\pi$
$151$	−4.83720	−0.393646	−0.196823	+	0.980439i	$0.563062\pi$
$152$	0	0
$153$	2.38664	0.192948
$154$	0	0
$155$	−0.972989	−0.0781524
$156$	0	0
$157$	−9.01032	−0.719102	−0.359551	−	0.933126i	$-0.617070\pi$
$157$	−9.01032	−0.719102	−0.359551	+	0.933126i	$0.617070\pi$
$158$	0	0
$159$	20.0518	1.59021
$160$	0	0
$161$	1.33826	0.105470
$162$	0	0
$163$	−23.8163	−1.86544	−0.932719	−	0.360604i	$-0.882571\pi$
$163$	−23.8163	−1.86544	−0.932719	+	0.360604i	$0.882571\pi$
$164$	0	0
$165$	0.408977	0.0318388
$166$	0	0
$167$	3.99691	0.309290	0.154645	−	0.987970i	$-0.450577\pi$
$167$	3.99691	0.309290	0.154645	+	0.987970i	$0.450577\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.56966	−0.120035
$172$	0	0
$173$	−19.6142	−1.49124	−0.745621	−	0.666371i	$-0.767847\pi$
$173$	−19.6142	−1.49124	−0.745621	+	0.666371i	$0.767847\pi$
$174$	0	0
$175$	4.95630	0.374661
$176$	0	0
$177$	−17.4319	−1.31026
$178$	0	0
$179$	−7.55508	−0.564693	−0.282347	−	0.959312i	$-0.591113\pi$
$179$	−7.55508	−0.564693	−0.282347	+	0.959312i	$0.591113\pi$
$180$	0	0
$181$	−9.94776	−0.739411	−0.369706	−	0.929149i	$-0.620541\pi$
$181$	−9.94776	−0.739411	−0.369706	+	0.929149i	$0.620541\pi$
$182$	0	0
$183$	1.28542	0.0950210
$184$	0	0
$185$	−0.328148	−0.0241259
$186$	0	0
$187$	2.88558	0.211014
$188$	0	0
$189$	−4.25085	−0.309204
$190$	0	0
$191$	−13.7793	−0.997037	−0.498518	−	0.866879i	$-0.666122\pi$
$191$	−13.7793	−0.997037	−0.498518	+	0.866879i	$0.666122\pi$
$192$	0	0
$193$	−5.58891	−0.402298	−0.201149	−	0.979561i	$-0.564468\pi$
$193$	−5.58891	−0.402298	−0.201149	+	0.979561i	$0.564468\pi$
$194$	0	0
$195$	−0.408977	−0.0292875
$196$	0	0
$197$	−5.49703	−0.391647	−0.195824	−	0.980639i	$-0.562738\pi$
$197$	−5.49703	−0.391647	−0.195824	+	0.980639i	$0.562738\pi$
$198$	0	0
$199$	3.72141	0.263804	0.131902	−	0.991263i	$-0.457892\pi$
$199$	3.72141	0.263804	0.131902	+	0.991263i	$0.457892\pi$
$200$	0	0
$201$	−4.51584	−0.318523
$202$	0	0
$203$	−5.48127	−0.384710
$204$	0	0
$205$	1.51775	0.106004
$206$	0	0
$207$	−1.10686	−0.0769323
$208$	0	0
$209$	−1.89781	−0.131274
$210$	0	0
$211$	−21.8493	−1.50417	−0.752084	−	0.659067i	$-0.770951\pi$
$211$	−21.8493	−1.50417	−0.752084	+	0.659067i	$0.770951\pi$
$212$	0	0
$213$	16.5740	1.13563
$214$	0	0
$215$	−0.903454	−0.0616150
$216$	0	0
$217$	−4.65418	−0.315946
$218$	0	0
$219$	9.58656	0.647799
$220$	0	0
$221$	−2.88558	−0.194105
$222$	0	0
$223$	6.24460	0.418169	0.209085	−	0.977898i	$-0.432952\pi$
$223$	6.24460	0.418169	0.209085	+	0.977898i	$0.432952\pi$
$224$	0	0
$225$	−4.09931	−0.273287
$226$	0	0
$227$	−11.0506	−0.733454	−0.366727	−	0.930329i	$-0.619522\pi$
$227$	−11.0506	−0.733454	−0.366727	+	0.930329i	$0.619522\pi$
$228$	0	0
$229$	−28.5742	−1.88824	−0.944118	−	0.329607i	$-0.893084\pi$
$229$	−28.5742	−1.88824	−0.944118	+	0.329607i	$0.893084\pi$
$230$	0	0
$231$	1.95630	0.128715
$232$	0	0
$233$	23.1433	1.51617	0.758083	−	0.652158i	$-0.226136\pi$
$233$	23.1433	1.51617	0.758083	+	0.652158i	$0.226136\pi$
$234$	0	0
$235$	−0.713001	−0.0465110
$236$	0	0
$237$	−14.7143	−0.955793
$238$	0	0
$239$	−12.8025	−0.828126	−0.414063	−	0.910248i	$-0.635891\pi$
$239$	−12.8025	−0.828126	−0.414063	+	0.910248i	$0.635891\pi$
$240$	0	0
$241$	23.8896	1.53886	0.769432	−	0.638729i	$-0.220539\pi$
$241$	23.8896	1.53886	0.769432	+	0.638729i	$0.220539\pi$
$242$	0	0
$243$	8.36994	0.536932
$244$	0	0
$245$	−0.209057	−0.0133562
$246$	0	0
$247$	1.89781	0.120755
$248$	0	0
$249$	−3.13269	−0.198526
$250$	0	0
$251$	21.2303	1.34004	0.670022	−	0.742341i	$-0.266285\pi$
$251$	21.2303	1.34004	0.670022	+	0.742341i	$0.266285\pi$
$252$	0	0
$253$	−1.33826	−0.0841358
$254$	0	0
$255$	1.18014	0.0739030
$256$	0	0
$257$	−26.1829	−1.63324	−0.816622	−	0.577173i	$-0.804156\pi$
$257$	−26.1829	−1.63324	−0.816622	+	0.577173i	$0.804156\pi$
$258$	0	0
$259$	−1.56966	−0.0975338
$260$	0	0
$261$	4.53351	0.280617
$262$	0	0
$263$	20.2822	1.25065	0.625327	−	0.780363i	$-0.284966\pi$
$263$	20.2822	1.25065	0.625327	+	0.780363i	$0.284966\pi$
$264$	0	0
$265$	2.14281	0.131632
$266$	0	0
$267$	22.7922	1.39486
$268$	0	0
$269$	1.27786	0.0779127	0.0389563	−	0.999241i	$-0.487597\pi$
$269$	1.27786	0.0779127	0.0389563	+	0.999241i	$0.487597\pi$
$270$	0	0
$271$	−23.7060	−1.44003	−0.720017	−	0.693956i	$-0.755866\pi$
$271$	−23.7060	−1.44003	−0.720017	+	0.693956i	$0.755866\pi$
$272$	0	0
$273$	−1.95630	−0.118400
$274$	0	0
$275$	−4.95630	−0.298876
$276$	0	0
$277$	−14.7201	−0.884445	−0.442223	−	0.896905i	$-0.645810\pi$
$277$	−14.7201	−0.884445	−0.442223	+	0.896905i	$0.645810\pi$
$278$	0	0
$279$	3.84943	0.230459
$280$	0	0
$281$	−18.9767	−1.13206	−0.566028	−	0.824386i	$-0.691521\pi$
$281$	−18.9767	−1.13206	−0.566028	+	0.824386i	$0.691521\pi$
$282$	0	0
$283$	−28.9696	−1.72207	−0.861033	−	0.508548i	$-0.830182\pi$
$283$	−28.9696	−1.72207	−0.861033	+	0.508548i	$0.830182\pi$
$284$	0	0
$285$	−0.776159	−0.0459757
$286$	0	0
$287$	7.25999	0.428544
$288$	0	0
$289$	−8.67343	−0.510202
$290$	0	0
$291$	−33.9333	−1.98920
$292$	0	0
$293$	28.7265	1.67822	0.839109	−	0.543963i	$-0.183077\pi$
$293$	28.7265	1.67822	0.839109	+	0.543963i	$0.183077\pi$
$294$	0	0
$295$	−1.86284	−0.108459
$296$	0	0
$297$	4.25085	0.246659
$298$	0	0
$299$	1.33826	0.0773936
$300$	0	0
$301$	−4.32157	−0.249091
$302$	0	0
$303$	−11.7401	−0.674450
$304$	0	0
$305$	0.137365	0.00786548
$306$	0	0
$307$	20.2216	1.15411	0.577054	−	0.816706i	$-0.304202\pi$
$307$	20.2216	1.15411	0.577054	+	0.816706i	$0.304202\pi$
$308$	0	0
$309$	−12.7500	−0.725322
$310$	0	0
$311$	9.74687	0.552694	0.276347	−	0.961058i	$-0.410876\pi$
$311$	9.74687	0.552694	0.276347	+	0.961058i	$0.410876\pi$
$312$	0	0
$313$	11.5319	0.651820	0.325910	−	0.945401i	$-0.394329\pi$
$313$	11.5319	0.651820	0.325910	+	0.945401i	$0.394329\pi$
$314$	0	0
$315$	0.172909	0.00974232
$316$	0	0
$317$	−12.3016	−0.690926	−0.345463	−	0.938432i	$-0.612278\pi$
$317$	−12.3016	−0.690926	−0.345463	+	0.938432i	$0.612278\pi$
$318$	0	0
$319$	5.48127	0.306892
$320$	0	0
$321$	−2.87684	−0.160570
$322$	0	0
$323$	−5.47627	−0.304708
$324$	0	0
$325$	4.95630	0.274926
$326$	0	0
$327$	16.8944	0.934262
$328$	0	0
$329$	−3.41056	−0.188030
$330$	0	0
$331$	−13.4848	−0.741192	−0.370596	−	0.928794i	$-0.620847\pi$
$331$	−13.4848	−0.741192	−0.370596	+	0.928794i	$0.620847\pi$
$332$	0	0
$333$	1.29825	0.0711437
$334$	0	0
$335$	−0.482579	−0.0263661
$336$	0	0
$337$	−30.7424	−1.67465	−0.837324	−	0.546707i	$-0.815881\pi$
$337$	−30.7424	−1.67465	−0.837324	+	0.546707i	$0.815881\pi$
$338$	0	0
$339$	27.4551	1.49116
$340$	0	0
$341$	4.65418	0.252038
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.547318	−0.0294666
$346$	0	0
$347$	−14.3391	−0.769762	−0.384881	−	0.922966i	$-0.625757\pi$
$347$	−14.3391	−0.769762	−0.384881	+	0.922966i	$0.625757\pi$
$348$	0	0
$349$	−4.07327	−0.218037	−0.109019	−	0.994040i	$-0.534771\pi$
$349$	−4.07327	−0.218037	−0.109019	+	0.994040i	$0.534771\pi$
$350$	0	0
$351$	−4.25085	−0.226894
$352$	0	0
$353$	−22.6914	−1.20774	−0.603870	−	0.797083i	$-0.706375\pi$
$353$	−22.6914	−1.20774	−0.603870	+	0.797083i	$0.706375\pi$
$354$	0	0
$355$	1.77116	0.0940033
$356$	0	0
$357$	5.64505	0.298767
$358$	0	0
$359$	1.18672	0.0626325	0.0313162	−	0.999510i	$-0.490030\pi$
$359$	1.18672	0.0626325	0.0313162	+	0.999510i	$0.490030\pi$
$360$	0	0
$361$	−15.3983	−0.810438
$362$	0	0
$363$	−1.95630	−0.102679
$364$	0	0
$365$	1.02445	0.0536224
$366$	0	0
$367$	23.3916	1.22103	0.610515	−	0.792004i	$-0.290962\pi$
$367$	23.3916	1.22103	0.610515	+	0.792004i	$0.290962\pi$
$368$	0	0
$369$	−6.00467	−0.312591
$370$	0	0
$371$	10.2499	0.532147
$372$	0	0
$373$	−2.64814	−0.137115	−0.0685576	−	0.997647i	$-0.521840\pi$
$373$	−2.64814	−0.137115	−0.0685576	+	0.997647i	$0.521840\pi$
$374$	0	0
$375$	−4.07190	−0.210272
$376$	0	0
$377$	−5.48127	−0.282300
$378$	0	0
$379$	17.5188	0.899881	0.449940	−	0.893059i	$-0.351445\pi$
$379$	17.5188	0.899881	0.449940	+	0.893059i	$0.351445\pi$
$380$	0	0
$381$	6.64505	0.340436
$382$	0	0
$383$	24.8242	1.26846	0.634229	−	0.773145i	$-0.281318\pi$
$383$	24.8242	1.26846	0.634229	+	0.773145i	$0.281318\pi$
$384$	0	0
$385$	0.209057	0.0106545
$386$	0	0
$387$	3.57433	0.181693
$388$	0	0
$389$	−1.70833	−0.0866158	−0.0433079	−	0.999062i	$-0.513790\pi$
$389$	−1.70833	−0.0866158	−0.0433079	+	0.999062i	$0.513790\pi$
$390$	0	0
$391$	−3.86166	−0.195292
$392$	0	0
$393$	22.3624	1.12803
$394$	0	0
$395$	−1.57242	−0.0791170
$396$	0	0
$397$	−17.1485	−0.860656	−0.430328	−	0.902673i	$-0.641602\pi$
$397$	−17.1485	−0.860656	−0.430328	+	0.902673i	$0.641602\pi$
$398$	0	0
$399$	−3.71267	−0.185866
$400$	0	0
$401$	−27.0057	−1.34860	−0.674301	−	0.738457i	$-0.735555\pi$
$401$	−27.0057	−1.34860	−0.674301	+	0.738457i	$0.735555\pi$
$402$	0	0
$403$	−4.65418	−0.231841
$404$	0	0
$405$	2.25723	0.112163
$406$	0	0
$407$	1.56966	0.0778051
$408$	0	0
$409$	−5.59065	−0.276440	−0.138220	−	0.990402i	$-0.544138\pi$
$409$	−5.59065	−0.276440	−0.138220	+	0.990402i	$0.544138\pi$
$410$	0	0
$411$	−36.0273	−1.77709
$412$	0	0
$413$	−8.91068	−0.438466
$414$	0	0
$415$	−0.334771	−0.0164333
$416$	0	0
$417$	−0.169172	−0.00828441
$418$	0	0
$419$	7.25150	0.354259	0.177129	−	0.984188i	$-0.443319\pi$
$419$	7.25150	0.354259	0.177129	+	0.984188i	$0.443319\pi$
$420$	0	0
$421$	−24.8961	−1.21336	−0.606682	−	0.794945i	$-0.707500\pi$
$421$	−24.8961	−1.21336	−0.606682	+	0.794945i	$0.707500\pi$
$422$	0	0
$423$	2.82084	0.137154
$424$	0	0
$425$	−14.3018	−0.693738
$426$	0	0
$427$	0.657068	0.0317978
$428$	0	0
$429$	1.95630	0.0944508
$430$	0	0
$431$	19.5851	0.943381	0.471690	−	0.881764i	$-0.343644\pi$
$431$	19.5851	0.943381	0.471690	+	0.881764i	$0.343644\pi$
$432$	0	0
$433$	−2.56462	−0.123248	−0.0616238	−	0.998099i	$-0.519628\pi$
$433$	−2.56462	−0.123248	−0.0616238	+	0.998099i	$0.519628\pi$
$434$	0	0
$435$	2.24171	0.107482
$436$	0	0
$437$	2.53976	0.121493
$438$	0	0
$439$	−18.2681	−0.871887	−0.435944	−	0.899974i	$-0.643585\pi$
$439$	−18.2681	−0.871887	−0.435944	+	0.899974i	$0.643585\pi$
$440$	0	0
$441$	0.827091	0.0393853
$442$	0	0
$443$	−6.82169	−0.324108	−0.162054	−	0.986782i	$-0.551812\pi$
$443$	−6.82169	−0.324108	−0.162054	+	0.986782i	$0.551812\pi$
$444$	0	0
$445$	2.43566	0.115461
$446$	0	0
$447$	28.2175	1.33464
$448$	0	0
$449$	27.9057	1.31695	0.658475	−	0.752602i	$-0.271202\pi$
$449$	27.9057	1.31695	0.658475	+	0.752602i	$0.271202\pi$
$450$	0	0
$451$	−7.25999	−0.341860
$452$	0	0
$453$	9.46300	0.444610
$454$	0	0
$455$	−0.209057	−0.00980074
$456$	0	0
$457$	−12.1920	−0.570316	−0.285158	−	0.958481i	$-0.592046\pi$
$457$	−12.1920	−0.570316	−0.285158	+	0.958481i	$0.592046\pi$
$458$	0	0
$459$	12.2662	0.572536
$460$	0	0
$461$	11.3774	0.529897	0.264948	−	0.964263i	$-0.414645\pi$
$461$	11.3774	0.529897	0.264948	+	0.964263i	$0.414645\pi$
$462$	0	0
$463$	12.1689	0.565536	0.282768	−	0.959188i	$-0.408747\pi$
$463$	12.1689	0.565536	0.282768	+	0.959188i	$0.408747\pi$
$464$	0	0
$465$	1.90345	0.0882705
$466$	0	0
$467$	−18.3884	−0.850915	−0.425457	−	0.904978i	$-0.639887\pi$
$467$	−18.3884	−0.850915	−0.425457	+	0.904978i	$0.639887\pi$
$468$	0	0
$469$	−2.30836	−0.106590
$470$	0	0
$471$	17.6268	0.812202
$472$	0	0
$473$	4.32157	0.198706
$474$	0	0
$475$	9.40609	0.431581
$476$	0	0
$477$	−8.47758	−0.388162
$478$	0	0
$479$	−2.53681	−0.115910	−0.0579548	−	0.998319i	$-0.518458\pi$
$479$	−2.53681	−0.115910	−0.0579548	+	0.998319i	$0.518458\pi$
$480$	0	0
$481$	−1.56966	−0.0715703
$482$	0	0
$483$	−2.61803	−0.119125
$484$	0	0
$485$	−3.62624	−0.164659
$486$	0	0
$487$	−23.8874	−1.08244	−0.541221	−	0.840880i	$-0.682038\pi$
$487$	−23.8874	−1.08244	−0.541221	+	0.840880i	$0.682038\pi$
$488$	0	0
$489$	46.5917	2.10695
$490$	0	0
$491$	34.8015	1.57057	0.785286	−	0.619134i	$-0.212516\pi$
$491$	34.8015	1.57057	0.785286	+	0.619134i	$0.212516\pi$
$492$	0	0
$493$	15.8166	0.712346
$494$	0	0
$495$	−0.172909	−0.00777169
$496$	0	0
$497$	8.47214	0.380027
$498$	0	0
$499$	−15.3688	−0.688002	−0.344001	−	0.938969i	$-0.611782\pi$
$499$	−15.3688	−0.688002	−0.344001	+	0.938969i	$0.611782\pi$
$500$	0	0
$501$	−7.81913	−0.349333
$502$	0	0
$503$	38.1964	1.70309	0.851546	−	0.524280i	$-0.175665\pi$
$503$	38.1964	1.70309	0.851546	+	0.524280i	$0.175665\pi$
$504$	0	0
$505$	−1.25459	−0.0558284
$506$	0	0
$507$	−1.95630	−0.0868821
$508$	0	0
$509$	4.71458	0.208970	0.104485	−	0.994526i	$-0.466681\pi$
$509$	4.71458	0.208970	0.104485	+	0.994526i	$0.466681\pi$
$510$	0	0
$511$	4.90036	0.216779
$512$	0	0
$513$	−8.06729	−0.356180
$514$	0	0
$515$	−1.36251	−0.0600394
$516$	0	0
$517$	3.41056	0.149996
$518$	0	0
$519$	38.3712	1.68431
$520$	0	0
$521$	0.00309062	0.000135403 0	6.77013e−5	−	1.00000i	$-0.499978\pi$
$521$	0.00309062	0.000135403 0	6.77013e−5		1.00000i	$0.499978\pi$
$522$	0	0
$523$	−26.9621	−1.17897	−0.589486	−	0.807778i	$-0.700670\pi$
$523$	−26.9621	−1.17897	−0.589486	+	0.807778i	$0.700670\pi$
$524$	0	0
$525$	−9.69598	−0.423167
$526$	0	0
$527$	13.4300	0.585020
$528$	0	0
$529$	−21.2091	−0.922133
$530$	0	0
$531$	7.36994	0.319828
$532$	0	0
$533$	7.25999	0.314465
$534$	0	0
$535$	−0.307430	−0.0132914
$536$	0	0
$537$	14.7800	0.637803
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−12.0859	−0.519613	−0.259807	−	0.965661i	$-0.583659\pi$
$541$	−12.0859	−0.519613	−0.259807	+	0.965661i	$0.583659\pi$
$542$	0	0
$543$	19.4608	0.835141
$544$	0	0
$545$	1.80540	0.0773347
$546$	0	0
$547$	−22.0071	−0.940956	−0.470478	−	0.882412i	$-0.655918\pi$
$547$	−22.0071	−0.940956	−0.470478	+	0.882412i	$0.655918\pi$
$548$	0	0
$549$	−0.543455	−0.0231941
$550$	0	0
$551$	−10.4024	−0.443157
$552$	0	0
$553$	−7.52149	−0.319846
$554$	0	0
$555$	0.641954	0.0272494
$556$	0	0
$557$	−15.8015	−0.669532	−0.334766	−	0.942301i	$-0.608657\pi$
$557$	−15.8015	−0.669532	−0.334766	+	0.942301i	$0.608657\pi$
$558$	0	0
$559$	−4.32157	−0.182783
$560$	0	0
$561$	−5.64505	−0.238334
$562$	0	0
$563$	25.2334	1.06346	0.531731	−	0.846913i	$-0.321542\pi$
$563$	25.2334	1.06346	0.531731	+	0.846913i	$0.321542\pi$
$564$	0	0
$565$	2.93395	0.123432
$566$	0	0
$567$	10.7972	0.453439
$568$	0	0
$569$	−16.4888	−0.691248	−0.345624	−	0.938373i	$-0.612333\pi$
$569$	−16.4888	−0.691248	−0.345624	+	0.938373i	$0.612333\pi$
$570$	0	0
$571$	41.1774	1.72322	0.861611	−	0.507569i	$-0.169456\pi$
$571$	41.1774	1.72322	0.861611	+	0.507569i	$0.169456\pi$
$572$	0	0
$573$	26.9564	1.12612
$574$	0	0
$575$	6.63282	0.276608
$576$	0	0
$577$	15.8423	0.659523	0.329761	−	0.944064i	$-0.393032\pi$
$577$	15.8423	0.659523	0.329761	+	0.944064i	$0.393032\pi$
$578$	0	0
$579$	10.9336	0.454383
$580$	0	0
$581$	−1.60134	−0.0664348
$582$	0	0
$583$	−10.2499	−0.424506
$584$	0	0
$585$	0.172909	0.00714891
$586$	0	0
$587$	−9.65267	−0.398408	−0.199204	−	0.979958i	$-0.563836\pi$
$587$	−9.65267	−0.398408	−0.199204	+	0.979958i	$0.563836\pi$
$588$	0	0
$589$	−8.83274	−0.363947
$590$	0	0
$591$	10.7538	0.442353
$592$	0	0
$593$	17.6830	0.726153	0.363076	−	0.931759i	$-0.381726\pi$
$593$	17.6830	0.726153	0.363076	+	0.931759i	$0.381726\pi$
$594$	0	0
$595$	0.603250	0.0247309
$596$	0	0
$597$	−7.28017	−0.297958
$598$	0	0
$599$	0.919748	0.0375799	0.0187899	−	0.999823i	$-0.494019\pi$
$599$	0.919748	0.0375799	0.0187899	+	0.999823i	$0.494019\pi$
$600$	0	0
$601$	−16.9946	−0.693225	−0.346613	−	0.938008i	$-0.612668\pi$
$601$	−16.9946	−0.693225	−0.346613	+	0.938008i	$0.612668\pi$
$602$	0	0
$603$	1.90923	0.0777497
$604$	0	0
$605$	−0.209057	−0.00849937
$606$	0	0
$607$	9.15353	0.371530	0.185765	−	0.982594i	$-0.440524\pi$
$607$	9.15353	0.371530	0.185765	+	0.982594i	$0.440524\pi$
$608$	0	0
$609$	10.7230	0.434517
$610$	0	0
$611$	−3.41056	−0.137976
$612$	0	0
$613$	−4.23956	−0.171234	−0.0856171	−	0.996328i	$-0.527286\pi$
$613$	−4.23956	−0.171234	−0.0856171	+	0.996328i	$0.527286\pi$
$614$	0	0
$615$	−2.96917	−0.119728
$616$	0	0
$617$	20.0216	0.806040	0.403020	−	0.915191i	$-0.367961\pi$
$617$	20.0216	0.806040	0.403020	+	0.915191i	$0.367961\pi$
$618$	0	0
$619$	−15.5916	−0.626680	−0.313340	−	0.949641i	$-0.601448\pi$
$619$	−15.5916	−0.626680	−0.313340	+	0.949641i	$0.601448\pi$
$620$	0	0
$621$	−5.68875	−0.228282
$622$	0	0
$623$	11.6507	0.466775
$624$	0	0
$625$	24.3463	0.973854
$626$	0	0
$627$	3.71267	0.148270
$628$	0	0
$629$	4.52937	0.180598
$630$	0	0
$631$	0.208123	0.00828525	0.00414263	−	0.999991i	$-0.498681\pi$
$631$	0.208123	0.00828525	0.00414263	+	0.999991i	$0.498681\pi$
$632$	0	0
$633$	42.7437	1.69891
$634$	0	0
$635$	0.710114	0.0281800
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−7.00723	−0.277202
$640$	0	0
$641$	15.8374	0.625540	0.312770	−	0.949829i	$-0.398743\pi$
$641$	15.8374	0.625540	0.312770	+	0.949829i	$0.398743\pi$
$642$	0	0
$643$	22.0356	0.869000	0.434500	−	0.900672i	$-0.356925\pi$
$643$	22.0356	0.869000	0.434500	+	0.900672i	$0.356925\pi$
$644$	0	0
$645$	1.76742	0.0695922
$646$	0	0
$647$	−35.1180	−1.38063	−0.690315	−	0.723509i	$-0.742528\pi$
$647$	−35.1180	−1.38063	−0.690315	+	0.723509i	$0.742528\pi$
$648$	0	0
$649$	8.91068	0.349775
$650$	0	0
$651$	9.10495	0.356851
$652$	0	0
$653$	12.7941	0.500671	0.250335	−	0.968159i	$-0.419459\pi$
$653$	12.7941	0.500671	0.250335	+	0.968159i	$0.419459\pi$
$654$	0	0
$655$	2.38973	0.0933744
$656$	0	0
$657$	−4.05305	−0.158124
$658$	0	0
$659$	−22.0707	−0.859751	−0.429875	−	0.902888i	$-0.641443\pi$
$659$	−22.0707	−0.859751	−0.429875	+	0.902888i	$0.641443\pi$
$660$	0	0
$661$	−24.9203	−0.969286	−0.484643	−	0.874712i	$-0.661050\pi$
$661$	−24.9203	−0.969286	−0.484643	+	0.874712i	$0.661050\pi$
$662$	0	0
$663$	5.64505	0.219235
$664$	0	0
$665$	−0.396750	−0.0153853
$666$	0	0
$667$	−7.33537	−0.284027
$668$	0	0
$669$	−12.2163	−0.472309
$670$	0	0
$671$	−0.657068	−0.0253658
$672$	0	0
$673$	2.77193	0.106850	0.0534250	−	0.998572i	$-0.482986\pi$
$673$	2.77193	0.106850	0.0534250	+	0.998572i	$0.482986\pi$
$674$	0	0
$675$	−21.0685	−0.810926
$676$	0	0
$677$	6.70471	0.257683	0.128842	−	0.991665i	$-0.458874\pi$
$677$	6.70471	0.257683	0.128842	+	0.991665i	$0.458874\pi$
$678$	0	0
$679$	−17.3457	−0.665667
$680$	0	0
$681$	21.6182	0.828413
$682$	0	0
$683$	9.86270	0.377386	0.188693	−	0.982036i	$-0.439575\pi$
$683$	9.86270	0.377386	0.188693	+	0.982036i	$0.439575\pi$
$684$	0	0
$685$	−3.85001	−0.147101
$686$	0	0
$687$	55.8996	2.13270
$688$	0	0
$689$	10.2499	0.390489
$690$	0	0
$691$	−43.4556	−1.65313	−0.826565	−	0.562841i	$-0.809708\pi$
$691$	−43.4556	−1.65313	−0.826565	+	0.562841i	$0.809708\pi$
$692$	0	0
$693$	−0.827091	−0.0314186
$694$	0	0
$695$	−0.0180784	−0.000685752 0
$696$	0	0
$697$	−20.9493	−0.793510
$698$	0	0
$699$	−45.2751	−1.71246
$700$	0	0
$701$	−48.2091	−1.82083	−0.910417	−	0.413692i	$-0.864239\pi$
$701$	−48.2091	−1.82083	−0.910417	+	0.413692i	$0.864239\pi$
$702$	0	0
$703$	−2.97891	−0.112352
$704$	0	0
$705$	1.39484	0.0525327
$706$	0	0
$707$	−6.00118	−0.225698
$708$	0	0
$709$	24.8563	0.933499	0.466749	−	0.884390i	$-0.345425\pi$
$709$	24.8563	0.933499	0.466749	+	0.884390i	$0.345425\pi$
$710$	0	0
$711$	6.22095	0.233304
$712$	0	0
$713$	−6.22851	−0.233260
$714$	0	0
$715$	0.209057	0.00781829
$716$	0	0
$717$	25.0455	0.935341
$718$	0	0
$719$	14.6682	0.547031	0.273516	−	0.961868i	$-0.411813\pi$
$719$	14.6682	0.547031	0.273516	+	0.961868i	$0.411813\pi$
$720$	0	0
$721$	−6.51742	−0.242721
$722$	0	0
$723$	−46.7351	−1.73810
$724$	0	0
$725$	−27.1668	−1.00895
$726$	0	0
$727$	49.2095	1.82508	0.912540	−	0.408987i	$-0.134118\pi$
$727$	49.2095	1.82508	0.912540	+	0.408987i	$0.134118\pi$
$728$	0	0
$729$	16.0175	0.593241
$730$	0	0
$731$	12.4702	0.461228
$732$	0	0
$733$	−15.0506	−0.555905	−0.277953	−	0.960595i	$-0.589656\pi$
$733$	−15.0506	−0.555905	−0.277953	+	0.960595i	$0.589656\pi$
$734$	0	0
$735$	0.408977	0.0150853
$736$	0	0
$737$	2.30836	0.0850297
$738$	0	0
$739$	24.0329	0.884066	0.442033	−	0.896999i	$-0.354257\pi$
$739$	24.0329	0.884066	0.442033	+	0.896999i	$0.354257\pi$
$740$	0	0
$741$	−3.71267	−0.136388
$742$	0	0
$743$	8.18802	0.300389	0.150195	−	0.988656i	$-0.452010\pi$
$743$	8.18802	0.300389	0.150195	+	0.988656i	$0.452010\pi$
$744$	0	0
$745$	3.01543	0.110477
$746$	0	0
$747$	1.32445	0.0484592
$748$	0	0
$749$	−1.47056	−0.0537330
$750$	0	0
$751$	31.6082	1.15340	0.576700	−	0.816956i	$-0.304340\pi$
$751$	31.6082	1.15340	0.576700	+	0.816956i	$0.304340\pi$
$752$	0	0
$753$	−41.5327	−1.51354
$754$	0	0
$755$	1.01125	0.0368032
$756$	0	0
$757$	54.5743	1.98354	0.991769	−	0.128042i	$-0.0408692\pi$
$757$	54.5743	1.98354	0.991769	+	0.128042i	$0.0408692\pi$
$758$	0	0
$759$	2.61803	0.0950286
$760$	0	0
$761$	28.5128	1.03359	0.516794	−	0.856110i	$-0.327125\pi$
$761$	28.5128	1.03359	0.516794	+	0.856110i	$0.327125\pi$
$762$	0	0
$763$	8.63591	0.312641
$764$	0	0
$765$	−0.498943	−0.0180393
$766$	0	0
$767$	−8.91068	−0.321746
$768$	0	0
$769$	24.7838	0.893726	0.446863	−	0.894602i	$-0.352541\pi$
$769$	24.7838	0.893726	0.446863	+	0.894602i	$0.352541\pi$
$770$	0	0
$771$	51.2215	1.84470
$772$	0	0
$773$	37.3681	1.34404	0.672019	−	0.740534i	$-0.265427\pi$
$773$	37.3681	1.34404	0.672019	+	0.740534i	$0.265427\pi$
$774$	0	0
$775$	−23.0675	−0.828609
$776$	0	0
$777$	3.07072	0.110161
$778$	0	0
$779$	13.7781	0.493650
$780$	0	0
$781$	−8.47214	−0.303157
$782$	0	0
$783$	23.3001	0.832677
$784$	0	0
$785$	1.88367	0.0672310
$786$	0	0
$787$	29.6368	1.05644	0.528218	−	0.849109i	$-0.322860\pi$
$787$	29.6368	1.05644	0.528218	+	0.849109i	$0.322860\pi$
$788$	0	0
$789$	−39.6780	−1.41257
$790$	0	0
$791$	14.0342	0.499000
$792$	0	0
$793$	0.657068	0.0233332
$794$	0	0
$795$	−4.19196	−0.148674
$796$	0	0
$797$	48.1623	1.70600	0.852998	−	0.521914i	$-0.174782\pi$
$797$	48.1623	1.70600	0.852998	+	0.521914i	$0.174782\pi$
$798$	0	0
$799$	9.84143	0.348165
$800$	0	0
$801$	−9.63618	−0.340478
$802$	0	0
$803$	−4.90036	−0.172930
$804$	0	0
$805$	−0.279773	−0.00986069
$806$	0	0
$807$	−2.49988	−0.0879998
$808$	0	0
$809$	−8.89700	−0.312802	−0.156401	−	0.987694i	$-0.549989\pi$
$809$	−8.89700	−0.312802	−0.156401	+	0.987694i	$0.549989\pi$
$810$	0	0
$811$	40.2919	1.41484	0.707420	−	0.706794i	$-0.249859\pi$
$811$	40.2919	1.41484	0.707420	+	0.706794i	$0.249859\pi$
$812$	0	0
$813$	46.3759	1.62647
$814$	0	0
$815$	4.97897	0.174406
$816$	0	0
$817$	−8.20150	−0.286934
$818$	0	0
$819$	0.827091	0.0289009
$820$	0	0
$821$	26.7345	0.933040	0.466520	−	0.884511i	$-0.345508\pi$
$821$	26.7345	0.933040	0.466520	+	0.884511i	$0.345508\pi$
$822$	0	0
$823$	20.3282	0.708597	0.354299	−	0.935132i	$-0.384720\pi$
$823$	20.3282	0.708597	0.354299	+	0.935132i	$0.384720\pi$
$824$	0	0
$825$	9.69598	0.337571
$826$	0	0
$827$	14.5553	0.506137	0.253068	−	0.967448i	$-0.418560\pi$
$827$	14.5553	0.506137	0.253068	+	0.967448i	$0.418560\pi$
$828$	0	0
$829$	−5.10740	−0.177387	−0.0886936	−	0.996059i	$-0.528269\pi$
$829$	−5.10740	−0.177387	−0.0886936	+	0.996059i	$0.528269\pi$
$830$	0	0
$831$	28.7969	0.998952
$832$	0	0
$833$	2.88558	0.0999794
$834$	0	0
$835$	−0.835582	−0.0289165
$836$	0	0
$837$	19.7842	0.683843
$838$	0	0
$839$	−16.4676	−0.568525	−0.284263	−	0.958746i	$-0.591749\pi$
$839$	−16.4676	−0.568525	−0.284263	+	0.958746i	$0.591749\pi$
$840$	0	0
$841$	1.04435	0.0360121
$842$	0	0
$843$	37.1241	1.27862
$844$	0	0
$845$	−0.209057	−0.00719178
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	56.6732	1.94502
$850$	0	0
$851$	−2.10061	−0.0720081
$852$	0	0
$853$	−53.4026	−1.82847	−0.914235	−	0.405184i	$-0.867208\pi$
$853$	−53.4026	−1.82847	−0.914235	+	0.405184i	$0.867208\pi$
$854$	0	0
$855$	0.328148	0.0112224
$856$	0	0
$857$	3.97785	0.135881	0.0679405	−	0.997689i	$-0.478357\pi$
$857$	3.97785	0.135881	0.0679405	+	0.997689i	$0.478357\pi$
$858$	0	0
$859$	15.7488	0.537343	0.268671	−	0.963232i	$-0.413415\pi$
$859$	15.7488	0.537343	0.268671	+	0.963232i	$0.413415\pi$
$860$	0	0
$861$	−14.2027	−0.484026
$862$	0	0
$863$	−0.435385	−0.0148207	−0.00741033	−	0.999973i	$-0.502359\pi$
$863$	−0.435385	−0.0148207	−0.00741033	+	0.999973i	$0.502359\pi$
$864$	0	0
$865$	4.10049	0.139421
$866$	0	0
$867$	16.9678	0.576256
$868$	0	0
$869$	7.52149	0.255149
$870$	0	0
$871$	−2.30836	−0.0782159
$872$	0	0
$873$	14.3465	0.485554
$874$	0	0
$875$	−2.08143	−0.0703653
$876$	0	0
$877$	39.5309	1.33487	0.667433	−	0.744670i	$-0.267393\pi$
$877$	39.5309	1.33487	0.667433	+	0.744670i	$0.267393\pi$
$878$	0	0
$879$	−56.1975	−1.89549
$880$	0	0
$881$	−16.1849	−0.545282	−0.272641	−	0.962116i	$-0.587897\pi$
$881$	−16.1849	−0.545282	−0.272641	+	0.962116i	$0.587897\pi$
$882$	0	0
$883$	−50.4914	−1.69917	−0.849586	−	0.527451i	$-0.823148\pi$
$883$	−50.4914	−1.69917	−0.849586	+	0.527451i	$0.823148\pi$
$884$	0	0
$885$	3.64426	0.122501
$886$	0	0
$887$	−11.6509	−0.391199	−0.195599	−	0.980684i	$-0.562665\pi$
$887$	−11.6509	−0.391199	−0.195599	+	0.980684i	$0.562665\pi$
$888$	0	0
$889$	3.39675	0.113923
$890$	0	0
$891$	−10.7972	−0.361720
$892$	0	0
$893$	−6.47258	−0.216597
$894$	0	0
$895$	1.57944	0.0527949
$896$	0	0
$897$	−2.61803	−0.0874136
$898$	0	0
$899$	25.5108	0.850834
$900$	0	0
$901$	−29.5768	−0.985347
$902$	0	0
$903$	8.45426	0.281340
$904$	0	0
$905$	2.07965	0.0691299
$906$	0	0
$907$	32.2878	1.07210	0.536049	−	0.844187i	$-0.319916\pi$
$907$	32.2878	1.07210	0.536049	+	0.844187i	$0.319916\pi$
$908$	0	0
$909$	4.96352	0.164630
$910$	0	0
$911$	−32.9248	−1.09085	−0.545424	−	0.838160i	$-0.683631\pi$
$911$	−32.9248	−1.09085	−0.545424	+	0.838160i	$0.683631\pi$
$912$	0	0
$913$	1.60134	0.0529966
$914$	0	0
$915$	−0.268726	−0.00888380
$916$	0	0
$917$	11.4310	0.377484
$918$	0	0
$919$	−21.3097	−0.702941	−0.351471	−	0.936199i	$-0.614318\pi$
$919$	−21.3097	−0.702941	−0.351471	+	0.936199i	$0.614318\pi$
$920$	0	0
$921$	−39.5594	−1.30353
$922$	0	0
$923$	8.47214	0.278864
$924$	0	0
$925$	−7.77969	−0.255795
$926$	0	0
$927$	5.39050	0.177047
$928$	0	0
$929$	27.6726	0.907908	0.453954	−	0.891025i	$-0.350013\pi$
$929$	27.6726	0.907908	0.453954	+	0.891025i	$0.350013\pi$
$930$	0	0
$931$	−1.89781	−0.0621981
$932$	0	0
$933$	−19.0677	−0.624250
$934$	0	0
$935$	−0.603250	−0.0197284
$936$	0	0
$937$	32.5657	1.06388	0.531938	−	0.846784i	$-0.321464\pi$
$937$	32.5657	1.06388	0.531938	+	0.846784i	$0.321464\pi$
$938$	0	0
$939$	−22.5597	−0.736210
$940$	0	0
$941$	−53.3860	−1.74033	−0.870167	−	0.492757i	$-0.835989\pi$
$941$	−53.3860	−1.74033	−0.870167	+	0.492757i	$0.835989\pi$
$942$	0	0
$943$	9.71576	0.316389
$944$	0	0
$945$	0.888670	0.0289084
$946$	0	0
$947$	−15.1314	−0.491703	−0.245852	−	0.969307i	$-0.579068\pi$
$947$	−15.1314	−0.491703	−0.245852	+	0.969307i	$0.579068\pi$
$948$	0	0
$949$	4.90036	0.159073
$950$	0	0
$951$	24.0655	0.780378
$952$	0	0
$953$	20.7272	0.671419	0.335710	−	0.941966i	$-0.391024\pi$
$953$	20.7272	0.671419	0.335710	+	0.941966i	$0.391024\pi$
$954$	0	0
$955$	2.88066	0.0932160
$956$	0	0
$957$	−10.7230	−0.346625
$958$	0	0
$959$	−18.4161	−0.594686
$960$	0	0
$961$	−9.33859	−0.301245
$962$	0	0
$963$	1.21628	0.0391942
$964$	0	0
$965$	1.16840	0.0376121
$966$	0	0
$967$	33.1103	1.06475	0.532377	−	0.846507i	$-0.321299\pi$
$967$	33.1103	1.06475	0.532377	+	0.846507i	$0.321299\pi$
$968$	0	0
$969$	10.7132	0.344158
$970$	0	0
$971$	−17.4819	−0.561021	−0.280511	−	0.959851i	$-0.590504\pi$
$971$	−17.4819	−0.561021	−0.280511	+	0.959851i	$0.590504\pi$
$972$	0	0
$973$	−0.0864759	−0.00277229
$974$	0	0
$975$	−9.69598	−0.310520
$976$	0	0
$977$	54.2965	1.73710	0.868549	−	0.495603i	$-0.165053\pi$
$977$	54.2965	1.73710	0.868549	+	0.495603i	$0.165053\pi$
$978$	0	0
$979$	−11.6507	−0.372358
$980$	0	0
$981$	−7.14268	−0.228048
$982$	0	0
$983$	15.7233	0.501494	0.250747	−	0.968053i	$-0.419324\pi$
$983$	15.7233	0.501494	0.250747	+	0.968053i	$0.419324\pi$
$984$	0	0
$985$	1.14919	0.0366163
$986$	0	0
$987$	6.67206	0.212374
$988$	0	0
$989$	−5.78339	−0.183901
$990$	0	0
$991$	−51.3993	−1.63275	−0.816377	−	0.577519i	$-0.804021\pi$
$991$	−51.3993	−1.63275	−0.816377	+	0.577519i	$0.804021\pi$
$992$	0	0
$993$	26.3803	0.837152
$994$	0	0
$995$	−0.777986	−0.0246638
$996$	0	0
$997$	−5.14788	−0.163035	−0.0815175	−	0.996672i	$-0.525977\pi$
$997$	−5.14788	−0.163035	−0.0815175	+	0.996672i	$0.525977\pi$
$998$	0	0
$999$	6.67239	0.211105

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.e.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.e.1.1	✓	4	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.95630 q^{3} -0.209057 q^{5} -1.00000 q^{7} +0.827091 q^{9} +O(q^{10})\)	\(q-1.95630 q^{3} -0.209057 q^{5} -1.00000 q^{7} +0.827091 q^{9} +1.00000 q^{11} -1.00000 q^{13} +0.408977 q^{15} +2.88558 q^{17} -1.89781 q^{19} +1.95630 q^{21} -1.33826 q^{23} -4.95630 q^{25} +4.25085 q^{27} +5.48127 q^{29} +4.65418 q^{31} -1.95630 q^{33} +0.209057 q^{35} +1.56966 q^{37} +1.95630 q^{39} -7.25999 q^{41} +4.32157 q^{43} -0.172909 q^{45} +3.41056 q^{47} +1.00000 q^{49} -5.64505 q^{51} -10.2499 q^{53} -0.209057 q^{55} +3.71267 q^{57} +8.91068 q^{59} -0.657068 q^{61} -0.827091 q^{63} +0.209057 q^{65} +2.30836 q^{67} +2.61803 q^{69} -8.47214 q^{71} -4.90036 q^{73} +9.69598 q^{75} -1.00000 q^{77} +7.52149 q^{79} -10.7972 q^{81} +1.60134 q^{83} -0.603250 q^{85} -10.7230 q^{87} -11.6507 q^{89} +1.00000 q^{91} -9.10495 q^{93} +0.396750 q^{95} +17.3457 q^{97} +0.827091 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + q^{5} - 4 q^{7} - 3 q^{9}+O(q^{10}) \) 4 * q + q^3 + q^5 - 4 * q^7 - 3 * q^9	\( 4 q + q^{3} + q^{5} - 4 q^{7} - 3 q^{9} + 4 q^{11} - 4 q^{13} - q^{15} + q^{17} - 3 q^{19} - q^{21} - q^{23} - 11 q^{25} - 5 q^{27} + 3 q^{29} + 6 q^{31} + q^{33} - q^{35} + 4 q^{37} - q^{39} - 6 q^{41} - 3 q^{43} - 7 q^{45} - 7 q^{47} + 4 q^{49} - 11 q^{51} - 20 q^{53} + q^{55} - 2 q^{57} - 11 q^{59} - 18 q^{61} + 3 q^{63} - q^{65} - 16 q^{67} + 6 q^{69} - 16 q^{71} + 4 q^{73} + 6 q^{75} - 4 q^{77} + 9 q^{79} - 16 q^{81} - 14 q^{83} - 11 q^{85} - 3 q^{87} - 23 q^{89} + 4 q^{91} - q^{93} - 7 q^{95} + 14 q^{97} - 3 q^{99}+O(q^{100}) \) 4 * q + q^3 + q^5 - 4 * q^7 - 3 * q^9 + 4 * q^11 - 4 * q^13 - q^15 + q^17 - 3 * q^19 - q^21 - q^23 - 11 * q^25 - 5 * q^27 + 3 * q^29 + 6 * q^31 + q^33 - q^35 + 4 * q^37 - q^39 - 6 * q^41 - 3 * q^43 - 7 * q^45 - 7 * q^47 + 4 * q^49 - 11 * q^51 - 20 * q^53 + q^55 - 2 * q^57 - 11 * q^59 - 18 * q^61 + 3 * q^63 - q^65 - 16 * q^67 + 6 * q^69 - 16 * q^71 + 4 * q^73 + 6 * q^75 - 4 * q^77 + 9 * q^79 - 16 * q^81 - 14 * q^83 - 11 * q^85 - 3 * q^87 - 23 * q^89 + 4 * q^91 - q^93 - 7 * q^95 + 14 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

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