LMFDB - Embedded newform 4004.2.a.e.1.2

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			1.2
Root			$-1.95630$ 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)

$f(q)$	$=$	$q-0.209057 q^{3} +1.33826 q^{5} -1.00000 q^{7} -2.95630 q^{9} +O(q^{10})$	$q-0.209057 q^{3} +1.33826 q^{5} -1.00000 q^{7} -2.95630 q^{9} +1.00000 q^{11} -1.00000 q^{13} -0.279773 q^{15} +2.31592 q^{17} +3.06316 q^{19} +0.209057 q^{21} -1.82709 q^{23} -3.20906 q^{25} +1.24520 q^{27} -5.86889 q^{29} -2.91259 q^{31} -0.209057 q^{33} -1.33826 q^{35} +9.05560 q^{37} +0.209057 q^{39} -1.84846 q^{41} +0.138341 q^{43} -3.95630 q^{45} -6.76202 q^{47} +1.00000 q^{49} -0.484159 q^{51} +8.09740 q^{53} +1.33826 q^{55} -0.640375 q^{57} +2.21064 q^{59} -11.6375 q^{61} +2.95630 q^{63} -1.33826 q^{65} -12.8252 q^{67} +0.381966 q^{69} +0.472136 q^{71} -5.79719 q^{73} +0.670876 q^{75} -1.00000 q^{77} +4.19683 q^{79} +8.60857 q^{81} -5.30678 q^{83} +3.09931 q^{85} +1.22693 q^{87} -10.3622 q^{89} +1.00000 q^{91} +0.608897 q^{93} +4.09931 q^{95} -8.30954 q^{97} -2.95630 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.209057	−0.120699	−0.0603495	−	0.998177i	$-0.519222\pi$
$3$	−0.209057	−0.120699	−0.0603495	+	0.998177i	$0.519222\pi$
$4$	0	0
$5$	1.33826	0.598489	0.299244	−	0.954177i	$-0.403265\pi$
$5$	1.33826	0.598489	0.299244	+	0.954177i	$0.403265\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.95630	−0.985432
$10$	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
$15$	−0.279773	−0.0722370
$16$	0	0
$17$	2.31592	0.561693	0.280847	−	0.959753i	$-0.409385\pi$
$17$	2.31592	0.561693	0.280847	+	0.959753i	$0.409385\pi$
$18$	0	0
$19$	3.06316	0.702737	0.351368	−	0.936237i	$-0.385716\pi$
$19$	3.06316	0.702737	0.351368	+	0.936237i	$0.385716\pi$
$20$	0	0
$21$	0.209057	0.0456200
$22$	0	0
$23$	−1.82709	−0.380975	−0.190487	−	0.981690i	$-0.561007\pi$
$23$	−1.82709	−0.380975	−0.190487	+	0.981690i	$0.561007\pi$
$24$	0	0
$25$	−3.20906	−0.641811
$26$	0	0
$27$	1.24520	0.239640
$28$	0	0
$29$	−5.86889	−1.08982	−0.544912	−	0.838493i	$-0.683437\pi$
$29$	−5.86889	−1.08982	−0.544912	+	0.838493i	$0.683437\pi$
$30$	0	0
$31$	−2.91259	−0.523117	−0.261558	−	0.965188i	$-0.584236\pi$
$31$	−2.91259	−0.523117	−0.261558	+	0.965188i	$0.584236\pi$
$32$	0	0
$33$	−0.209057	−0.0363921
$34$	0	0
$35$	−1.33826	−0.226207
$36$	0	0
$37$	9.05560	1.48873	0.744366	−	0.667772i	$-0.232752\pi$
$37$	9.05560	1.48873	0.744366	+	0.667772i	$0.232752\pi$
$38$	0	0
$39$	0.209057	0.0334759
$40$	0	0
$41$	−1.84846	−0.288680	−0.144340	−	0.989528i	$-0.546106\pi$
$41$	−1.84846	−0.288680	−0.144340	+	0.989528i	$0.546106\pi$
$42$	0	0
$43$	0.138341	0.0210968	0.0105484	−	0.999944i	$-0.496642\pi$
$43$	0.138341	0.0210968	0.0105484	+	0.999944i	$0.496642\pi$
$44$	0	0
$45$	−3.95630	−0.589770
$46$	0	0
$47$	−6.76202	−0.986342	−0.493171	−	0.869932i	$-0.664162\pi$
$47$	−6.76202	−0.986342	−0.493171	+	0.869932i	$0.664162\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.484159	−0.0677959
$52$	0	0
$53$	8.09740	1.11226	0.556131	−	0.831094i	$-0.312285\pi$
$53$	8.09740	1.11226	0.556131	+	0.831094i	$0.312285\pi$
$54$	0	0
$55$	1.33826	0.180451
$56$	0	0
$57$	−0.640375	−0.0848197
$58$	0	0
$59$	2.21064	0.287800	0.143900	−	0.989592i	$-0.454036\pi$
$59$	2.21064	0.287800	0.143900	+	0.989592i	$0.454036\pi$
$60$	0	0
$61$	−11.6375	−1.49003	−0.745014	−	0.667049i	$-0.767557\pi$
$61$	−11.6375	−1.49003	−0.745014	+	0.667049i	$0.767557\pi$
$62$	0	0
$63$	2.95630	0.372458
$64$	0	0
$65$	−1.33826	−0.165991
$66$	0	0
$67$	−12.8252	−1.56685	−0.783423	−	0.621489i	$-0.786528\pi$
$67$	−12.8252	−1.56685	−0.783423	+	0.621489i	$0.786528\pi$
$68$	0	0
$69$	0.381966	0.0459833
$70$	0	0
$71$	0.472136	0.0560322	0.0280161	−	0.999607i	$-0.491081\pi$
$71$	0.472136	0.0560322	0.0280161	+	0.999607i	$0.491081\pi$
$72$	0	0
$73$	−5.79719	−0.678510	−0.339255	−	0.940694i	$-0.610175\pi$
$73$	−5.79719	−0.678510	−0.339255	+	0.940694i	$0.610175\pi$
$74$	0	0
$75$	0.670876	0.0774660
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	4.19683	0.472180	0.236090	−	0.971731i	$-0.424134\pi$
$79$	4.19683	0.472180	0.236090	+	0.971731i	$0.424134\pi$
$80$	0	0
$81$	8.60857	0.956507
$82$	0	0
$83$	−5.30678	−0.582495	−0.291248	−	0.956648i	$-0.594070\pi$
$83$	−5.30678	−0.582495	−0.291248	+	0.956648i	$0.594070\pi$
$84$	0	0
$85$	3.09931	0.336167
$86$	0	0
$87$	1.22693	0.131541
$88$	0	0
$89$	−10.3622	−1.09839	−0.549195	−	0.835695i	$-0.685065\pi$
$89$	−10.3622	−1.09839	−0.549195	+	0.835695i	$0.685065\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0.608897	0.0631397
$94$	0	0
$95$	4.09931	0.420580
$96$	0	0
$97$	−8.30954	−0.843706	−0.421853	−	0.906664i	$-0.638620\pi$
$97$	−8.30954	−0.843706	−0.421853	+	0.906664i	$0.638620\pi$
$98$	0	0
$99$	−2.95630	−0.297119
$100$	0	0
$101$	0.810193	0.0806172	0.0403086	−	0.999187i	$-0.487166\pi$
$101$	0.810193	0.0806172	0.0403086	+	0.999187i	$0.487166\pi$
$102$	0	0
$103$	−10.1634	−1.00143	−0.500717	−	0.865611i	$-0.666930\pi$
$103$	−10.1634	−1.00143	−0.500717	+	0.865611i	$0.666930\pi$
$104$	0	0
$105$	0.279773	0.0273030
$106$	0	0
$107$	2.01011	0.194325	0.0971625	−	0.995269i	$-0.469023\pi$
$107$	2.01011	0.194325	0.0971625	+	0.995269i	$0.469023\pi$
$108$	0	0
$109$	−5.88091	−0.563289	−0.281644	−	0.959519i	$-0.590880\pi$
$109$	−5.88091	−0.563289	−0.281644	+	0.959519i	$0.590880\pi$
$110$	0	0
$111$	−1.89314	−0.179689
$112$	0	0
$113$	−5.49797	−0.517205	−0.258603	−	0.965984i	$-0.583262\pi$
$113$	−5.49797	−0.517205	−0.258603	+	0.965984i	$0.583262\pi$
$114$	0	0
$115$	−2.44512	−0.228009
$116$	0	0
$117$	2.95630	0.273310
$118$	0	0
$119$	−2.31592	−0.212300
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.386432	0.0348434
$124$	0	0
$125$	−10.9859	−0.982605
$126$	0	0
$127$	−7.09931	−0.629962	−0.314981	−	0.949098i	$-0.601998\pi$
$127$	−7.09931	−0.629962	−0.314981	+	0.949098i	$0.601998\pi$
$128$	0	0
$129$	−0.0289212	−0.00254637
$130$	0	0
$131$	−6.59727	−0.576406	−0.288203	−	0.957569i	$-0.593058\pi$
$131$	−6.59727	−0.576406	−0.288203	+	0.957569i	$0.593058\pi$
$132$	0	0
$133$	−3.06316	−0.265610
$134$	0	0
$135$	1.66641	0.143422
$136$	0	0
$137$	−21.1061	−1.80322	−0.901609	−	0.432551i	$-0.857613\pi$
$137$	−21.1061	−1.80322	−0.901609	+	0.432551i	$0.857613\pi$
$138$	0	0
$139$	15.7170	1.33310	0.666550	−	0.745461i	$-0.267770\pi$
$139$	15.7170	1.33310	0.666550	+	0.745461i	$0.267770\pi$
$140$	0	0
$141$	1.41365	0.119051
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−7.85410	−0.652248
$146$	0	0
$147$	−0.209057	−0.0172427
$148$	0	0
$149$	5.33091	0.436725	0.218363	−	0.975868i	$-0.429928\pi$
$149$	5.33091	0.436725	0.218363	+	0.975868i	$0.429928\pi$
$150$	0	0
$151$	−13.9896	−1.13845	−0.569227	−	0.822180i	$-0.692757\pi$
$151$	−13.9896	−1.13845	−0.569227	+	0.822180i	$0.692757\pi$
$152$	0	0
$153$	−6.84655	−0.553510
$154$	0	0
$155$	−3.89781	−0.313079
$156$	0	0
$157$	−1.41344	−0.112805	−0.0564025	−	0.998408i	$-0.517963\pi$
$157$	−1.41344	−0.112805	−0.0564025	+	0.998408i	$0.517963\pi$
$158$	0	0
$159$	−1.69282	−0.134249
$160$	0	0
$161$	1.82709	0.143995
$162$	0	0
$163$	18.2816	1.43193	0.715963	−	0.698139i	$-0.245988\pi$
$163$	18.2816	1.43193	0.715963	+	0.698139i	$0.245988\pi$
$164$	0	0
$165$	−0.279773	−0.0217803
$166$	0	0
$167$	5.98233	0.462927	0.231463	−	0.972844i	$-0.425649\pi$
$167$	5.98233	0.462927	0.231463	+	0.972844i	$0.425649\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−9.05560	−0.692499
$172$	0	0
$173$	−10.9670	−0.833806	−0.416903	−	0.908951i	$-0.636885\pi$
$173$	−10.9670	−0.833806	−0.416903	+	0.908951i	$0.636885\pi$
$174$	0	0
$175$	3.20906	0.242582
$176$	0	0
$177$	−0.462149	−0.0347372
$178$	0	0
$179$	6.95808	0.520071	0.260036	−	0.965599i	$-0.416266\pi$
$179$	6.95808	0.520071	0.260036	+	0.965599i	$0.416266\pi$
$180$	0	0
$181$	14.2190	1.05689	0.528447	−	0.848967i	$-0.322775\pi$
$181$	14.2190	1.05689	0.528447	+	0.848967i	$0.322775\pi$
$182$	0	0
$183$	2.43290	0.179845
$184$	0	0
$185$	12.1188	0.890989
$186$	0	0
$187$	2.31592	0.169357
$188$	0	0
$189$	−1.24520	−0.0905753
$190$	0	0
$191$	5.10751	0.369566	0.184783	−	0.982779i	$-0.440842\pi$
$191$	5.10751	0.369566	0.184783	+	0.982779i	$0.440842\pi$
$192$	0	0
$193$	−23.6047	−1.69910	−0.849552	−	0.527505i	$-0.823128\pi$
$193$	−23.6047	−1.69910	−0.849552	+	0.527505i	$0.823128\pi$
$194$	0	0
$195$	0.279773	0.0200349
$196$	0	0
$197$	−10.9550	−0.780511	−0.390255	−	0.920707i	$-0.627613\pi$
$197$	−10.9550	−0.780511	−0.390255	+	0.920707i	$0.627613\pi$
$198$	0	0
$199$	1.25532	0.0889871	0.0444936	−	0.999010i	$-0.485833\pi$
$199$	1.25532	0.0889871	0.0444936	+	0.999010i	$0.485833\pi$
$200$	0	0
$201$	2.68119	0.189117
$202$	0	0
$203$	5.86889	0.411915
$204$	0	0
$205$	−2.47372	−0.172772
$206$	0	0
$207$	5.40142	0.375425
$208$	0	0
$209$	3.06316	0.211883
$210$	0	0
$211$	6.61166	0.455165	0.227583	−	0.973759i	$-0.426918\pi$
$211$	6.61166	0.455165	0.227583	+	0.973759i	$0.426918\pi$
$212$	0	0
$213$	−0.0987033	−0.00676304
$214$	0	0
$215$	0.185136	0.0126262
$216$	0	0
$217$	2.91259	0.197720
$218$	0	0
$219$	1.21194	0.0818955
$220$	0	0
$221$	−2.31592	−0.155786
$222$	0	0
$223$	24.1920	1.62002	0.810009	−	0.586417i	$-0.199462\pi$
$223$	24.1920	1.62002	0.810009	+	0.586417i	$0.199462\pi$
$224$	0	0
$225$	9.48692	0.632461
$226$	0	0
$227$	5.50301	0.365248	0.182624	−	0.983183i	$-0.441541\pi$
$227$	5.50301	0.365248	0.182624	+	0.983183i	$0.441541\pi$
$228$	0	0
$229$	8.63111	0.570360	0.285180	−	0.958474i	$-0.407947\pi$
$229$	8.63111	0.570360	0.285180	+	0.958474i	$0.407947\pi$
$230$	0	0
$231$	0.209057	0.0137549
$232$	0	0
$233$	−26.2106	−1.71711	−0.858555	−	0.512721i	$-0.828637\pi$
$233$	−26.2106	−1.71711	−0.858555	+	0.512721i	$0.828637\pi$
$234$	0	0
$235$	−9.04935	−0.590315
$236$	0	0
$237$	−0.877376	−0.0569917
$238$	0	0
$239$	15.4203	0.997454	0.498727	−	0.866759i	$-0.333801\pi$
$239$	15.4203	0.997454	0.498727	+	0.866759i	$0.333801\pi$
$240$	0	0
$241$	−13.5281	−0.871422	−0.435711	−	0.900087i	$-0.643503\pi$
$241$	−13.5281	−0.871422	−0.435711	+	0.900087i	$0.643503\pi$
$242$	0	0
$243$	−5.53529	−0.355089
$244$	0	0
$245$	1.33826	0.0854984
$246$	0	0
$247$	−3.06316	−0.194904
$248$	0	0
$249$	1.10942	0.0703066
$250$	0	0
$251$	−20.3362	−1.28361	−0.641806	−	0.766867i	$-0.721814\pi$
$251$	−20.3362	−1.28361	−0.641806	+	0.766867i	$0.721814\pi$
$252$	0	0
$253$	−1.82709	−0.114868
$254$	0	0
$255$	−0.647932	−0.0405750
$256$	0	0
$257$	−9.68001	−0.603823	−0.301911	−	0.953336i	$-0.597625\pi$
$257$	−9.68001	−0.603823	−0.301911	+	0.953336i	$0.597625\pi$
$258$	0	0
$259$	−9.05560	−0.562688
$260$	0	0
$261$	17.3502	1.07395
$262$	0	0
$263$	−9.80691	−0.604720	−0.302360	−	0.953194i	$-0.597774\pi$
$263$	−9.80691	−0.604720	−0.302360	+	0.953194i	$0.597774\pi$
$264$	0	0
$265$	10.8364	0.665677
$266$	0	0
$267$	2.16629	0.132575
$268$	0	0
$269$	−4.65260	−0.283674	−0.141837	−	0.989890i	$-0.545301\pi$
$269$	−4.65260	−0.283674	−0.141837	+	0.989890i	$0.545301\pi$
$270$	0	0
$271$	5.60541	0.340504	0.170252	−	0.985401i	$-0.445542\pi$
$271$	5.60541	0.340504	0.170252	+	0.985401i	$0.445542\pi$
$272$	0	0
$273$	−0.209057	−0.0126527
$274$	0	0
$275$	−3.20906	−0.193513
$276$	0	0
$277$	15.7770	0.947949	0.473974	−	0.880539i	$-0.342819\pi$
$277$	15.7770	0.947949	0.473974	+	0.880539i	$0.342819\pi$
$278$	0	0
$279$	8.61048	0.515496
$280$	0	0
$281$	−22.5684	−1.34632	−0.673158	−	0.739499i	$-0.735062\pi$
$281$	−22.5684	−1.34632	−0.673158	+	0.739499i	$0.735062\pi$
$282$	0	0
$283$	32.5641	1.93574	0.967868	−	0.251457i	$-0.0809098\pi$
$283$	32.5641	1.93574	0.967868	+	0.251457i	$0.0809098\pi$
$284$	0	0
$285$	−0.856988	−0.0507636
$286$	0	0
$287$	1.84846	0.109111
$288$	0	0
$289$	−11.6365	−0.684501
$290$	0	0
$291$	1.73717	0.101835
$292$	0	0
$293$	10.4983	0.613317	0.306659	−	0.951820i	$-0.400789\pi$
$293$	10.4983	0.613317	0.306659	+	0.951820i	$0.400789\pi$
$294$	0	0
$295$	2.95841	0.172245
$296$	0	0
$297$	1.24520	0.0722541
$298$	0	0
$299$	1.82709	0.105663
$300$	0	0
$301$	−0.138341	−0.00797385
$302$	0	0
$303$	−0.169376	−0.00973042
$304$	0	0
$305$	−15.5740	−0.891765
$306$	0	0
$307$	4.24581	0.242321	0.121161	−	0.992633i	$-0.461338\pi$
$307$	4.24581	0.242321	0.121161	+	0.992633i	$0.461338\pi$
$308$	0	0
$309$	2.12474	0.120872
$310$	0	0
$311$	−30.6201	−1.73631	−0.868154	−	0.496294i	$-0.834694\pi$
$311$	−30.6201	−1.73631	−0.868154	+	0.496294i	$0.834694\pi$
$312$	0	0
$313$	−16.3719	−0.925394	−0.462697	−	0.886516i	$-0.653118\pi$
$313$	−16.3719	−0.925394	−0.462697	+	0.886516i	$0.653118\pi$
$314$	0	0
$315$	3.95630	0.222912
$316$	0	0
$317$	−25.9643	−1.45830	−0.729151	−	0.684353i	$-0.760085\pi$
$317$	−25.9643	−1.45830	−0.729151	+	0.684353i	$0.760085\pi$
$318$	0	0
$319$	−5.86889	−0.328595
$320$	0	0
$321$	−0.420228	−0.0234548
$322$	0	0
$323$	7.09403	0.394723
$324$	0	0
$325$	3.20906	0.178006
$326$	0	0
$327$	1.22944	0.0679884
$328$	0	0
$329$	6.76202	0.372802
$330$	0	0
$331$	−23.3341	−1.28256	−0.641278	−	0.767308i	$-0.721596\pi$
$331$	−23.3341	−1.28256	−0.641278	+	0.767308i	$0.721596\pi$
$332$	0	0
$333$	−26.7710	−1.46704
$334$	0	0
$335$	−17.1634	−0.937739
$336$	0	0
$337$	−8.78976	−0.478809	−0.239404	−	0.970920i	$-0.576952\pi$
$337$	−8.78976	−0.478809	−0.239404	+	0.970920i	$0.576952\pi$
$338$	0	0
$339$	1.14939	0.0624262
$340$	0	0
$341$	−2.91259	−0.157726
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0.511170	0.0275205
$346$	0	0
$347$	30.5302	1.63895	0.819473	−	0.573118i	$-0.194266\pi$
$347$	30.5302	1.63895	0.819473	+	0.573118i	$0.194266\pi$
$348$	0	0
$349$	−8.75349	−0.468564	−0.234282	−	0.972169i	$-0.575274\pi$
$349$	−8.75349	−0.468564	−0.234282	+	0.972169i	$0.575274\pi$
$350$	0	0
$351$	−1.24520	−0.0664641
$352$	0	0
$353$	28.6191	1.52324	0.761620	−	0.648024i	$-0.224404\pi$
$353$	28.6191	1.52324	0.761620	+	0.648024i	$0.224404\pi$
$354$	0	0
$355$	0.631841	0.0335347
$356$	0	0
$357$	0.484159	0.0256244
$358$	0	0
$359$	−8.90503	−0.469990	−0.234995	−	0.971997i	$-0.575507\pi$
$359$	−8.90503	−0.469990	−0.234995	+	0.971997i	$0.575507\pi$
$360$	0	0
$361$	−9.61706	−0.506161
$362$	0	0
$363$	−0.209057	−0.0109726
$364$	0	0
$365$	−7.75816	−0.406081
$366$	0	0
$367$	−34.8257	−1.81789	−0.908943	−	0.416920i	$-0.863110\pi$
$367$	−34.8257	−1.81789	−0.908943	+	0.416920i	$0.863110\pi$
$368$	0	0
$369$	5.46458	0.284475
$370$	0	0
$371$	−8.09740	−0.420396
$372$	0	0
$373$	4.49817	0.232906	0.116453	−	0.993196i	$-0.462847\pi$
$373$	4.49817	0.232906	0.116453	+	0.993196i	$0.462847\pi$
$374$	0	0
$375$	2.29667	0.118600
$376$	0	0
$377$	5.86889	0.302263
$378$	0	0
$379$	−24.8857	−1.27829	−0.639145	−	0.769086i	$-0.720712\pi$
$379$	−24.8857	−1.27829	−0.639145	+	0.769086i	$0.720712\pi$
$380$	0	0
$381$	1.48416	0.0760358
$382$	0	0
$383$	2.49363	0.127418	0.0637092	−	0.997969i	$-0.479707\pi$
$383$	2.49363	0.127418	0.0637092	+	0.997969i	$0.479707\pi$
$384$	0	0
$385$	−1.33826	−0.0682041
$386$	0	0
$387$	−0.408977	−0.0207895
$388$	0	0
$389$	−21.5139	−1.09080	−0.545400	−	0.838176i	$-0.683622\pi$
$389$	−21.5139	−1.09080	−0.545400	+	0.838176i	$0.683622\pi$
$390$	0	0
$391$	−4.23140	−0.213991
$392$	0	0
$393$	1.37921	0.0695717
$394$	0	0
$395$	5.61645	0.282594
$396$	0	0
$397$	−29.7145	−1.49133	−0.745663	−	0.666323i	$-0.767867\pi$
$397$	−29.7145	−1.49133	−0.745663	+	0.666323i	$0.767867\pi$
$398$	0	0
$399$	0.640375	0.0320588
$400$	0	0
$401$	22.8765	1.14240	0.571199	−	0.820811i	$-0.306478\pi$
$401$	22.8765	1.14240	0.571199	+	0.820811i	$0.306478\pi$
$402$	0	0
$403$	2.91259	0.145086
$404$	0	0
$405$	11.5205	0.572459
$406$	0	0
$407$	9.05560	0.448870
$408$	0	0
$409$	33.8877	1.67564	0.837819	−	0.545948i	$-0.183830\pi$
$409$	33.8877	1.67564	0.837819	+	0.545948i	$0.183830\pi$
$410$	0	0
$411$	4.41238	0.217647
$412$	0	0
$413$	−2.21064	−0.108778
$414$	0	0
$415$	−7.10186	−0.348617
$416$	0	0
$417$	−3.28575	−0.160904
$418$	0	0
$419$	6.89808	0.336993	0.168497	−	0.985702i	$-0.446109\pi$
$419$	6.89808	0.336993	0.168497	+	0.985702i	$0.446109\pi$
$420$	0	0
$421$	−23.6747	−1.15383	−0.576917	−	0.816803i	$-0.695744\pi$
$421$	−23.6747	−1.15383	−0.576917	+	0.816803i	$0.695744\pi$
$422$	0	0
$423$	19.9905	0.971973
$424$	0	0
$425$	−7.43192	−0.360501
$426$	0	0
$427$	11.6375	0.563178
$428$	0	0
$429$	0.209057	0.0100934
$430$	0	0
$431$	31.1897	1.50236	0.751179	−	0.660099i	$-0.229486\pi$
$431$	31.1897	1.50236	0.751179	+	0.660099i	$0.229486\pi$
$432$	0	0
$433$	18.6473	0.896131	0.448065	−	0.894001i	$-0.352113\pi$
$433$	18.6473	0.896131	0.448065	+	0.894001i	$0.352113\pi$
$434$	0	0
$435$	1.64195	0.0787257
$436$	0	0
$437$	−5.59667	−0.267725
$438$	0	0
$439$	10.6353	0.507595	0.253798	−	0.967257i	$-0.418320\pi$
$439$	10.6353	0.507595	0.253798	+	0.967257i	$0.418320\pi$
$440$	0	0
$441$	−2.95630	−0.140776
$442$	0	0
$443$	−6.11100	−0.290342	−0.145171	−	0.989407i	$-0.546373\pi$
$443$	−6.11100	−0.290342	−0.145171	+	0.989407i	$0.546373\pi$
$444$	0	0
$445$	−13.8673	−0.657373
$446$	0	0
$447$	−1.11446	−0.0527123
$448$	0	0
$449$	19.9855	0.943174	0.471587	−	0.881820i	$-0.343681\pi$
$449$	19.9855	0.943174	0.471587	+	0.881820i	$0.343681\pi$
$450$	0	0
$451$	−1.84846	−0.0870404
$452$	0	0
$453$	2.92461	0.137410
$454$	0	0
$455$	1.33826	0.0627387
$456$	0	0
$457$	−10.2654	−0.480197	−0.240098	−	0.970749i	$-0.577180\pi$
$457$	−10.2654	−0.480197	−0.240098	+	0.970749i	$0.577180\pi$
$458$	0	0
$459$	2.88380	0.134604
$460$	0	0
$461$	8.10044	0.377275	0.188638	−	0.982047i	$-0.439593\pi$
$461$	8.10044	0.377275	0.188638	+	0.982047i	$0.439593\pi$
$462$	0	0
$463$	30.0737	1.39765	0.698823	−	0.715295i	$-0.253708\pi$
$463$	30.0737	1.39765	0.698823	+	0.715295i	$0.253708\pi$
$464$	0	0
$465$	0.814864	0.0377884
$466$	0	0
$467$	20.8612	0.965341	0.482671	−	0.875802i	$-0.339667\pi$
$467$	20.8612	0.965341	0.482671	+	0.875802i	$0.339667\pi$
$468$	0	0
$469$	12.8252	0.592212
$470$	0	0
$471$	0.295490	0.0136155
$472$	0	0
$473$	0.138341	0.00636093
$474$	0	0
$475$	−9.82985	−0.451025
$476$	0	0
$477$	−23.9383	−1.09606
$478$	0	0
$479$	7.16458	0.327358	0.163679	−	0.986514i	$-0.447664\pi$
$479$	7.16458	0.327358	0.163679	+	0.986514i	$0.447664\pi$
$480$	0	0
$481$	−9.05560	−0.412900
$482$	0	0
$483$	−0.381966	−0.0173801
$484$	0	0
$485$	−11.1203	−0.504949
$486$	0	0
$487$	23.6809	1.07308	0.536542	−	0.843874i	$-0.319730\pi$
$487$	23.6809	1.07308	0.536542	+	0.843874i	$0.319730\pi$
$488$	0	0
$489$	−3.82189	−0.172832
$490$	0	0
$491$	−8.76287	−0.395463	−0.197731	−	0.980256i	$-0.563357\pi$
$491$	−8.76287	−0.395463	−0.197731	+	0.980256i	$0.563357\pi$
$492$	0	0
$493$	−13.5919	−0.612147
$494$	0	0
$495$	−3.95630	−0.177822
$496$	0	0
$497$	−0.472136	−0.0211782
$498$	0	0
$499$	−34.1322	−1.52797	−0.763984	−	0.645235i	$-0.776760\pi$
$499$	−34.1322	−1.52797	−0.763984	+	0.645235i	$0.776760\pi$
$500$	0	0
$501$	−1.25065	−0.0558748
$502$	0	0
$503$	−4.87104	−0.217189	−0.108595	−	0.994086i	$-0.534635\pi$
$503$	−4.87104	−0.217189	−0.108595	+	0.994086i	$0.534635\pi$
$504$	0	0
$505$	1.08425	0.0482485
$506$	0	0
$507$	−0.209057	−0.00928454
$508$	0	0
$509$	3.56710	0.158109	0.0790545	−	0.996870i	$-0.474810\pi$
$509$	3.56710	0.158109	0.0790545	+	0.996870i	$0.474810\pi$
$510$	0	0
$511$	5.79719	0.256453
$512$	0	0
$513$	3.81426	0.168404
$514$	0	0
$515$	−13.6013	−0.599347
$516$	0	0
$517$	−6.76202	−0.297393
$518$	0	0
$519$	2.29273	0.100640
$520$	0	0
$521$	−1.98233	−0.0868474	−0.0434237	−	0.999057i	$-0.513827\pi$
$521$	−1.98233	−0.0868474	−0.0434237	+	0.999057i	$0.513827\pi$
$522$	0	0
$523$	−8.55467	−0.374070	−0.187035	−	0.982353i	$-0.559888\pi$
$523$	−8.55467	−0.374070	−0.187035	+	0.982353i	$0.559888\pi$
$524$	0	0
$525$	−0.670876	−0.0292794
$526$	0	0
$527$	−6.74533	−0.293831
$528$	0	0
$529$	−19.6617	−0.854858
$530$	0	0
$531$	−6.53529	−0.283608
$532$	0	0
$533$	1.84846	0.0800655
$534$	0	0
$535$	2.69006	0.116301
$536$	0	0
$537$	−1.45463	−0.0627721
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−8.08197	−0.347471	−0.173735	−	0.984792i	$-0.555584\pi$
$541$	−8.08197	−0.347471	−0.173735	+	0.984792i	$0.555584\pi$
$542$	0	0
$543$	−2.97259	−0.127566
$544$	0	0
$545$	−7.87019	−0.337122
$546$	0	0
$547$	16.8264	0.719443	0.359722	−	0.933060i	$-0.382872\pi$
$547$	16.8264	0.719443	0.359722	+	0.933060i	$0.382872\pi$
$548$	0	0
$549$	34.4039	1.46832
$550$	0	0
$551$	−17.9773	−0.765860
$552$	0	0
$553$	−4.19683	−0.178467
$554$	0	0
$555$	−2.53351	−0.107542
$556$	0	0
$557$	27.7629	1.17635	0.588175	−	0.808733i	$-0.299846\pi$
$557$	27.7629	1.17635	0.588175	+	0.808733i	$0.299846\pi$
$558$	0	0
$559$	−0.138341	−0.00585120
$560$	0	0
$561$	−0.484159	−0.0204412
$562$	0	0
$563$	45.9315	1.93578	0.967892	−	0.251367i	$-0.0808800\pi$
$563$	45.9315	1.93578	0.967892	+	0.251367i	$0.0808800\pi$
$564$	0	0
$565$	−7.35772	−0.309541
$566$	0	0
$567$	−8.60857	−0.361526
$568$	0	0
$569$	−12.2166	−0.512147	−0.256074	−	0.966657i	$-0.582429\pi$
$569$	−12.2166	−0.512147	−0.256074	+	0.966657i	$0.582429\pi$
$570$	0	0
$571$	−22.4570	−0.939796	−0.469898	−	0.882721i	$-0.655709\pi$
$571$	−22.4570	−0.939796	−0.469898	+	0.882721i	$0.655709\pi$
$572$	0	0
$573$	−1.06776	−0.0446063
$574$	0	0
$575$	5.86324	0.244514
$576$	0	0
$577$	−27.5398	−1.14650	−0.573249	−	0.819381i	$-0.694317\pi$
$577$	−27.5398	−1.14650	−0.573249	+	0.819381i	$0.694317\pi$
$578$	0	0
$579$	4.93473	0.205080
$580$	0	0
$581$	5.30678	0.220162
$582$	0	0
$583$	8.09740	0.335360
$584$	0	0
$585$	3.95630	0.163573
$586$	0	0
$587$	5.41251	0.223398	0.111699	−	0.993742i	$-0.464371\pi$
$587$	5.41251	0.223398	0.111699	+	0.993742i	$0.464371\pi$
$588$	0	0
$589$	−8.92173	−0.367613
$590$	0	0
$591$	2.29022	0.0942069
$592$	0	0
$593$	−22.8251	−0.937313	−0.468656	−	0.883381i	$-0.655262\pi$
$593$	−22.8251	−0.937313	−0.468656	+	0.883381i	$0.655262\pi$
$594$	0	0
$595$	−3.09931	−0.127059
$596$	0	0
$597$	−0.262433	−0.0107407
$598$	0	0
$599$	8.79606	0.359397	0.179699	−	0.983722i	$-0.442488\pi$
$599$	8.79606	0.359397	0.179699	+	0.983722i	$0.442488\pi$
$600$	0	0
$601$	24.3926	0.994995	0.497497	−	0.867465i	$-0.334252\pi$
$601$	24.3926	0.994995	0.497497	+	0.867465i	$0.334252\pi$
$602$	0	0
$603$	37.9150	1.54402
$604$	0	0
$605$	1.33826	0.0544081
$606$	0	0
$607$	−30.8149	−1.25074	−0.625370	−	0.780328i	$-0.715052\pi$
$607$	−30.8149	−1.25074	−0.625370	+	0.780328i	$0.715052\pi$
$608$	0	0
$609$	−1.22693	−0.0497178
$610$	0	0
$611$	6.76202	0.273562
$612$	0	0
$613$	6.51084	0.262970	0.131485	−	0.991318i	$-0.458025\pi$
$613$	6.51084	0.262970	0.131485	+	0.991318i	$0.458025\pi$
$614$	0	0
$615$	0.517147	0.0208534
$616$	0	0
$617$	20.1695	0.811993	0.405997	−	0.913875i	$-0.366925\pi$
$617$	20.1695	0.811993	0.405997	+	0.913875i	$0.366925\pi$
$618$	0	0
$619$	8.54508	0.343456	0.171728	−	0.985144i	$-0.445065\pi$
$619$	8.54508	0.343456	0.171728	+	0.985144i	$0.445065\pi$
$620$	0	0
$621$	−2.27510	−0.0912967
$622$	0	0
$623$	10.3622	0.415152
$624$	0	0
$625$	1.34333	0.0537332
$626$	0	0
$627$	−0.640375	−0.0255741
$628$	0	0
$629$	20.9721	0.836211
$630$	0	0
$631$	10.7969	0.429816	0.214908	−	0.976634i	$-0.431055\pi$
$631$	10.7969	0.429816	0.214908	+	0.976634i	$0.431055\pi$
$632$	0	0
$633$	−1.38221	−0.0549380
$634$	0	0
$635$	−9.50073	−0.377025
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−1.39577	−0.0552159
$640$	0	0
$641$	4.45715	0.176047	0.0880234	−	0.996118i	$-0.471945\pi$
$641$	4.45715	0.176047	0.0880234	+	0.996118i	$0.471945\pi$
$642$	0	0
$643$	24.5481	0.968084	0.484042	−	0.875045i	$-0.339168\pi$
$643$	24.5481	0.968084	0.484042	+	0.875045i	$0.339168\pi$
$644$	0	0
$645$	−0.0387041	−0.00152397
$646$	0	0
$647$	47.0718	1.85058	0.925292	−	0.379256i	$-0.123820\pi$
$647$	47.0718	1.85058	0.925292	+	0.379256i	$0.123820\pi$
$648$	0	0
$649$	2.21064	0.0867751
$650$	0	0
$651$	−0.608897	−0.0238646
$652$	0	0
$653$	39.8337	1.55881	0.779406	−	0.626520i	$-0.215521\pi$
$653$	39.8337	1.55881	0.779406	+	0.626520i	$0.215521\pi$
$654$	0	0
$655$	−8.82887	−0.344973
$656$	0	0
$657$	17.1382	0.668625
$658$	0	0
$659$	29.3112	1.14180	0.570900	−	0.821019i	$-0.306594\pi$
$659$	29.3112	1.14180	0.570900	+	0.821019i	$0.306594\pi$
$660$	0	0
$661$	−38.9992	−1.51689	−0.758446	−	0.651736i	$-0.774041\pi$
$661$	−38.9992	−1.51689	−0.758446	+	0.651736i	$0.774041\pi$
$662$	0	0
$663$	0.484159	0.0188032
$664$	0	0
$665$	−4.09931	−0.158964
$666$	0	0
$667$	10.7230	0.415196
$668$	0	0
$669$	−5.05751	−0.195535
$670$	0	0
$671$	−11.6375	−0.449260
$672$	0	0
$673$	37.5069	1.44578	0.722891	−	0.690962i	$-0.242813\pi$
$673$	37.5069	1.44578	0.722891	+	0.690962i	$0.242813\pi$
$674$	0	0
$675$	−3.99593	−0.153804
$676$	0	0
$677$	−0.433432	−0.0166581	−0.00832907	−	0.999965i	$-0.502651\pi$
$677$	−0.433432	−0.0166581	−0.00832907	+	0.999965i	$0.502651\pi$
$678$	0	0
$679$	8.30954	0.318891
$680$	0	0
$681$	−1.15044	−0.0440850
$682$	0	0
$683$	8.59183	0.328757	0.164379	−	0.986397i	$-0.447438\pi$
$683$	8.59183	0.328757	0.164379	+	0.986397i	$0.447438\pi$
$684$	0	0
$685$	−28.2455	−1.07921
$686$	0	0
$687$	−1.80439	−0.0688419
$688$	0	0
$689$	−8.09740	−0.308486
$690$	0	0
$691$	13.4199	0.510516	0.255258	−	0.966873i	$-0.417840\pi$
$691$	13.4199	0.510516	0.255258	+	0.966873i	$0.417840\pi$
$692$	0	0
$693$	2.95630	0.112300
$694$	0	0
$695$	21.0335	0.797845
$696$	0	0
$697$	−4.28088	−0.162150
$698$	0	0
$699$	5.47950	0.207254
$700$	0	0
$701$	29.8194	1.12626	0.563131	−	0.826367i	$-0.309597\pi$
$701$	29.8194	1.12626	0.563131	+	0.826367i	$0.309597\pi$
$702$	0	0
$703$	27.7387	1.04619
$704$	0	0
$705$	1.89183	0.0712504
$706$	0	0
$707$	−0.810193	−0.0304704
$708$	0	0
$709$	11.3165	0.425001	0.212500	−	0.977161i	$-0.431839\pi$
$709$	11.3165	0.425001	0.212500	+	0.977161i	$0.431839\pi$
$710$	0	0
$711$	−12.4071	−0.465301
$712$	0	0
$713$	5.32157	0.199294
$714$	0	0
$715$	−1.33826	−0.0500481
$716$	0	0
$717$	−3.22371	−0.120392
$718$	0	0
$719$	−27.3063	−1.01835	−0.509177	−	0.860662i	$-0.670050\pi$
$719$	−27.3063	−1.01835	−0.509177	+	0.860662i	$0.670050\pi$
$720$	0	0
$721$	10.1634	0.378506
$722$	0	0
$723$	2.82815	0.105180
$724$	0	0
$725$	18.8336	0.699462
$726$	0	0
$727$	−10.3852	−0.385167	−0.192584	−	0.981281i	$-0.561687\pi$
$727$	−10.3852	−0.385167	−0.192584	+	0.981281i	$0.561687\pi$
$728$	0	0
$729$	−24.6685	−0.913648
$730$	0	0
$731$	0.320387	0.0118499
$732$	0	0
$733$	28.9807	1.07043	0.535214	−	0.844717i	$-0.320231\pi$
$733$	28.9807	1.07043	0.535214	+	0.844717i	$0.320231\pi$
$734$	0	0
$735$	−0.279773	−0.0103196
$736$	0	0
$737$	−12.8252	−0.472422
$738$	0	0
$739$	−12.5344	−0.461084	−0.230542	−	0.973062i	$-0.574050\pi$
$739$	−12.5344	−0.461084	−0.230542	+	0.973062i	$0.574050\pi$
$740$	0	0
$741$	0.640375	0.0235247
$742$	0	0
$743$	26.1273	0.958517	0.479259	−	0.877674i	$-0.340906\pi$
$743$	26.1273	0.958517	0.479259	+	0.877674i	$0.340906\pi$
$744$	0	0
$745$	7.13415	0.261375
$746$	0	0
$747$	15.6884	0.574009
$748$	0	0
$749$	−2.01011	−0.0734479
$750$	0	0
$751$	−21.0785	−0.769164	−0.384582	−	0.923091i	$-0.625654\pi$
$751$	−21.0785	−0.769164	−0.384582	+	0.923091i	$0.625654\pi$
$752$	0	0
$753$	4.25143	0.154931
$754$	0	0
$755$	−18.7217	−0.681352
$756$	0	0
$757$	−8.90793	−0.323764	−0.161882	−	0.986810i	$-0.551756\pi$
$757$	−8.90793	−0.323764	−0.161882	+	0.986810i	$0.551756\pi$
$758$	0	0
$759$	0.381966	0.0138645
$760$	0	0
$761$	50.7789	1.84073	0.920366	−	0.391058i	$-0.127891\pi$
$761$	50.7789	1.84073	0.920366	+	0.391058i	$0.127891\pi$
$762$	0	0
$763$	5.88091	0.212903
$764$	0	0
$765$	−9.16247	−0.331270
$766$	0	0
$767$	−2.21064	−0.0798215
$768$	0	0
$769$	14.9603	0.539483	0.269741	−	0.962933i	$-0.413062\pi$
$769$	14.9603	0.539483	0.269741	+	0.962933i	$0.413062\pi$
$770$	0	0
$771$	2.02367	0.0728808
$772$	0	0
$773$	−33.4973	−1.20481	−0.602407	−	0.798189i	$-0.705792\pi$
$773$	−33.4973	−1.20481	−0.602407	+	0.798189i	$0.705792\pi$
$774$	0	0
$775$	9.34667	0.335742
$776$	0	0
$777$	1.89314	0.0679159
$778$	0	0
$779$	−5.66211	−0.202866
$780$	0	0
$781$	0.472136	0.0168944
$782$	0	0
$783$	−7.30796	−0.261165
$784$	0	0
$785$	−1.89156	−0.0675125
$786$	0	0
$787$	9.00138	0.320864	0.160432	−	0.987047i	$-0.448711\pi$
$787$	9.00138	0.320864	0.160432	+	0.987047i	$0.448711\pi$
$788$	0	0
$789$	2.05020	0.0729891
$790$	0	0
$791$	5.49797	0.195485
$792$	0	0
$793$	11.6375	0.413259
$794$	0	0
$795$	−2.26543	−0.0803466
$796$	0	0
$797$	37.5585	1.33039	0.665195	−	0.746670i	$-0.268349\pi$
$797$	37.5585	1.33039	0.665195	+	0.746670i	$0.268349\pi$
$798$	0	0
$799$	−15.6603	−0.554022
$800$	0	0
$801$	30.6337	1.08239
$802$	0	0
$803$	−5.79719	−0.204579
$804$	0	0
$805$	2.44512	0.0861793
$806$	0	0
$807$	0.972659	0.0342392
$808$	0	0
$809$	−49.2941	−1.73309	−0.866544	−	0.499100i	$-0.833664\pi$
$809$	−49.2941	−1.73309	−0.866544	+	0.499100i	$0.833664\pi$
$810$	0	0
$811$	−0.0463390	−0.00162718	−0.000813591	−	1.00000i	$-0.500259\pi$
$811$	−0.0463390	−0.00162718	−0.000813591		1.00000i	$0.500259\pi$
$812$	0	0
$813$	−1.17185	−0.0410985
$814$	0	0
$815$	24.4656	0.856991
$816$	0	0
$817$	0.423761	0.0148255
$818$	0	0
$819$	−2.95630	−0.103301
$820$	0	0
$821$	38.7691	1.35305	0.676525	−	0.736420i	$-0.263485\pi$
$821$	38.7691	1.35305	0.676525	+	0.736420i	$0.263485\pi$
$822$	0	0
$823$	−9.10093	−0.317238	−0.158619	−	0.987340i	$-0.550704\pi$
$823$	−9.10093	−0.317238	−0.158619	+	0.987340i	$0.550704\pi$
$824$	0	0
$825$	0.670876	0.0233569
$826$	0	0
$827$	−20.4905	−0.712524	−0.356262	−	0.934386i	$-0.615949\pi$
$827$	−20.4905	−0.712524	−0.356262	+	0.934386i	$0.615949\pi$
$828$	0	0
$829$	9.24410	0.321061	0.160530	−	0.987031i	$-0.448680\pi$
$829$	9.24410	0.321061	0.160530	+	0.987031i	$0.448680\pi$
$830$	0	0
$831$	−3.29829	−0.114417
$832$	0	0
$833$	2.31592	0.0802419
$834$	0	0
$835$	8.00592	0.277056
$836$	0	0
$837$	−3.62677	−0.125360
$838$	0	0
$839$	51.7443	1.78641	0.893205	−	0.449649i	$-0.148451\pi$
$839$	51.7443	1.78641	0.893205	+	0.449649i	$0.148451\pi$
$840$	0	0
$841$	5.44382	0.187718
$842$	0	0
$843$	4.71807	0.162499
$844$	0	0
$845$	1.33826	0.0460376
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−6.80776	−0.233642
$850$	0	0
$851$	−16.5454	−0.567169
$852$	0	0
$853$	26.7416	0.915613	0.457807	−	0.889052i	$-0.348635\pi$
$853$	26.7416	0.915613	0.457807	+	0.889052i	$0.348635\pi$
$854$	0	0
$855$	−12.1188	−0.414453
$856$	0	0
$857$	11.6732	0.398749	0.199374	−	0.979923i	$-0.436109\pi$
$857$	11.6732	0.398749	0.199374	+	0.979923i	$0.436109\pi$
$858$	0	0
$859$	6.06507	0.206937	0.103469	−	0.994633i	$-0.467006\pi$
$859$	6.06507	0.206937	0.103469	+	0.994633i	$0.467006\pi$
$860$	0	0
$861$	−0.386432	−0.0131696
$862$	0	0
$863$	−21.6473	−0.736882	−0.368441	−	0.929651i	$-0.620108\pi$
$863$	−21.6473	−0.736882	−0.368441	+	0.929651i	$0.620108\pi$
$864$	0	0
$865$	−14.6767	−0.499024
$866$	0	0
$867$	2.43269	0.0826186
$868$	0	0
$869$	4.19683	0.142368
$870$	0	0
$871$	12.8252	0.434565
$872$	0	0
$873$	24.5655	0.831415
$874$	0	0
$875$	10.9859	0.371390
$876$	0	0
$877$	23.7632	0.802427	0.401214	−	0.915985i	$-0.368589\pi$
$877$	23.7632	0.802427	0.401214	+	0.915985i	$0.368589\pi$
$878$	0	0
$879$	−2.19474	−0.0740268
$880$	0	0
$881$	14.0947	0.474862	0.237431	−	0.971404i	$-0.423695\pi$
$881$	14.0947	0.474862	0.237431	+	0.971404i	$0.423695\pi$
$882$	0	0
$883$	51.8819	1.74596	0.872982	−	0.487752i	$-0.162183\pi$
$883$	51.8819	1.74596	0.872982	+	0.487752i	$0.162183\pi$
$884$	0	0
$885$	−0.618476	−0.0207898
$886$	0	0
$887$	10.1702	0.341483	0.170741	−	0.985316i	$-0.445384\pi$
$887$	10.1702	0.341483	0.170741	+	0.985316i	$0.445384\pi$
$888$	0	0
$889$	7.09931	0.238103
$890$	0	0
$891$	8.60857	0.288398
$892$	0	0
$893$	−20.7131	−0.693139
$894$	0	0
$895$	9.31173	0.311257
$896$	0	0
$897$	−0.381966	−0.0127535
$898$	0	0
$899$	17.0937	0.570105
$900$	0	0
$901$	18.7529	0.624751
$902$	0	0
$903$	0.0289212	0.000962436 0
$904$	0	0
$905$	19.0288	0.632539
$906$	0	0
$907$	−44.1332	−1.46542	−0.732709	−	0.680542i	$-0.761744\pi$
$907$	−44.1332	−1.46542	−0.732709	+	0.680542i	$0.761744\pi$
$908$	0	0
$909$	−2.39517	−0.0794428
$910$	0	0
$911$	−2.31246	−0.0766151	−0.0383076	−	0.999266i	$-0.512197\pi$
$911$	−2.31246	−0.0766151	−0.0383076	+	0.999266i	$0.512197\pi$
$912$	0	0
$913$	−5.30678	−0.175629
$914$	0	0
$915$	3.25585	0.107635
$916$	0	0
$917$	6.59727	0.217861
$918$	0	0
$919$	−34.2071	−1.12839	−0.564195	−	0.825642i	$-0.690813\pi$
$919$	−34.2071	−1.12839	−0.564195	+	0.825642i	$0.690813\pi$
$920$	0	0
$921$	−0.887616	−0.0292479
$922$	0	0
$923$	−0.472136	−0.0155405
$924$	0	0
$925$	−29.0599	−0.955485
$926$	0	0
$927$	30.0461	0.986845
$928$	0	0
$929$	−10.9829	−0.360337	−0.180169	−	0.983636i	$-0.557664\pi$
$929$	−10.9829	−0.360337	−0.180169	+	0.983636i	$0.557664\pi$
$930$	0	0
$931$	3.06316	0.100391
$932$	0	0
$933$	6.40135	0.209571
$934$	0	0
$935$	3.09931	0.101358
$936$	0	0
$937$	37.1909	1.21497	0.607487	−	0.794330i	$-0.292178\pi$
$937$	37.1909	1.21497	0.607487	+	0.794330i	$0.292178\pi$
$938$	0	0
$939$	3.42266	0.111694
$940$	0	0
$941$	17.9984	0.586730	0.293365	−	0.956000i	$-0.405225\pi$
$941$	17.9984	0.586730	0.293365	+	0.956000i	$0.405225\pi$
$942$	0	0
$943$	3.37730	0.109980
$944$	0	0
$945$	−1.66641	−0.0542083
$946$	0	0
$947$	−27.3181	−0.887720	−0.443860	−	0.896096i	$-0.646391\pi$
$947$	−27.3181	−0.887720	−0.443860	+	0.896096i	$0.646391\pi$
$948$	0	0
$949$	5.79719	0.188185
$950$	0	0
$951$	5.42802	0.176016
$952$	0	0
$953$	55.3555	1.79314	0.896570	−	0.442902i	$-0.146051\pi$
$953$	55.3555	1.79314	0.896570	+	0.442902i	$0.146051\pi$
$954$	0	0
$955$	6.83518	0.221181
$956$	0	0
$957$	1.22693	0.0396611
$958$	0	0
$959$	21.1061	0.681553
$960$	0	0
$961$	−22.5168	−0.726349
$962$	0	0
$963$	−5.94249	−0.191494
$964$	0	0
$965$	−31.5893	−1.01689
$966$	0	0
$967$	14.5794	0.468842	0.234421	−	0.972135i	$-0.424681\pi$
$967$	14.5794	0.468842	0.234421	+	0.972135i	$0.424681\pi$
$968$	0	0
$969$	−1.48306	−0.0476426
$970$	0	0
$971$	−8.78399	−0.281892	−0.140946	−	0.990017i	$-0.545014\pi$
$971$	−8.78399	−0.281892	−0.140946	+	0.990017i	$0.545014\pi$
$972$	0	0
$973$	−15.7170	−0.503864
$974$	0	0
$975$	−0.670876	−0.0214852
$976$	0	0
$977$	−47.7152	−1.52655	−0.763273	−	0.646077i	$-0.776409\pi$
$977$	−47.7152	−1.52655	−0.763273	+	0.646077i	$0.776409\pi$
$978$	0	0
$979$	−10.3622	−0.331177
$980$	0	0
$981$	17.3857	0.555083
$982$	0	0
$983$	−33.7415	−1.07619	−0.538093	−	0.842885i	$-0.680855\pi$
$983$	−33.7415	−1.07619	−0.538093	+	0.842885i	$0.680855\pi$
$984$	0	0
$985$	−14.6606	−0.467127
$986$	0	0
$987$	−1.41365	−0.0449969
$988$	0	0
$989$	−0.252762	−0.00803736
$990$	0	0
$991$	20.2726	0.643981	0.321990	−	0.946743i	$-0.395648\pi$
$991$	20.2726	0.643981	0.321990	+	0.946743i	$0.395648\pi$
$992$	0	0
$993$	4.87815	0.154803
$994$	0	0
$995$	1.67994	0.0532578
$996$	0	0
$997$	1.92059	0.0608257	0.0304128	−	0.999537i	$-0.490318\pi$
$997$	1.92059	0.0608257	0.0304128	+	0.999537i	$0.490318\pi$
$998$	0	0
$999$	11.2761	0.356759

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.e.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.e.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-0.209057 q^{3} +1.33826 q^{5} -1.00000 q^{7} -2.95630 q^{9} +O(q^{10})\)	\(q-0.209057 q^{3} +1.33826 q^{5} -1.00000 q^{7} -2.95630 q^{9} +1.00000 q^{11} -1.00000 q^{13} -0.279773 q^{15} +2.31592 q^{17} +3.06316 q^{19} +0.209057 q^{21} -1.82709 q^{23} -3.20906 q^{25} +1.24520 q^{27} -5.86889 q^{29} -2.91259 q^{31} -0.209057 q^{33} -1.33826 q^{35} +9.05560 q^{37} +0.209057 q^{39} -1.84846 q^{41} +0.138341 q^{43} -3.95630 q^{45} -6.76202 q^{47} +1.00000 q^{49} -0.484159 q^{51} +8.09740 q^{53} +1.33826 q^{55} -0.640375 q^{57} +2.21064 q^{59} -11.6375 q^{61} +2.95630 q^{63} -1.33826 q^{65} -12.8252 q^{67} +0.381966 q^{69} +0.472136 q^{71} -5.79719 q^{73} +0.670876 q^{75} -1.00000 q^{77} +4.19683 q^{79} +8.60857 q^{81} -5.30678 q^{83} +3.09931 q^{85} +1.22693 q^{87} -10.3622 q^{89} +1.00000 q^{91} +0.608897 q^{93} +4.09931 q^{95} -8.30954 q^{97} -2.95630 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + q^{5} - 4 q^{7} - 3 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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