LMFDB - Embedded newform 4004.2.a.d.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:	4.4.3981.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 2x + 1 $ x^4 - x^3 - 4x^2 + 2x + 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			$2.28400$ 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)

$f(q)$	$=$	$q-1.84617 q^{3} +1.77882 q^{5} +1.00000 q^{7} +0.408340 q^{9} +O(q^{10})$	$q-1.84617 q^{3} +1.77882 q^{5} +1.00000 q^{7} +0.408340 q^{9} +1.00000 q^{11} +1.00000 q^{13} -3.28400 q^{15} -3.77882 q^{17} -5.21665 q^{19} -1.84617 q^{21} +5.91935 q^{23} -1.83581 q^{25} +4.78464 q^{27} +3.15966 q^{29} +1.87566 q^{31} -1.84617 q^{33} +1.77882 q^{35} -7.31349 q^{37} -1.84617 q^{39} -10.3659 q^{41} -6.02495 q^{43} +0.726363 q^{45} -9.64994 q^{47} +1.00000 q^{49} +6.97634 q^{51} +0.678139 q^{53} +1.77882 q^{55} +9.63081 q^{57} +0.273637 q^{59} +7.42876 q^{61} +0.408340 q^{63} +1.77882 q^{65} -0.0103608 q^{67} -10.9281 q^{69} -3.75132 q^{71} +11.3148 q^{73} +3.38921 q^{75} +1.00000 q^{77} +6.29820 q^{79} -10.0583 q^{81} -11.1418 q^{83} -6.72183 q^{85} -5.83326 q^{87} -2.92971 q^{89} +1.00000 q^{91} -3.46278 q^{93} -9.27946 q^{95} -11.1930 q^{97} +0.408340 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} - q^{5} + 4 q^{7} + q^{9}+O(q^{10}) $ 4 * q - 3 * q^3 - q^5 + 4 * q^7 + q^9	$ 4 q - 3 q^{3} - q^{5} + 4 q^{7} + q^{9} + 4 q^{11} + 4 q^{13} - 5 q^{15} - 7 q^{17} - 9 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{25} - 9 q^{27} - 3 q^{29} - 3 q^{33} - q^{35} - 18 q^{37} - 3 q^{39} - 14 q^{41} - q^{43} + 11 q^{45} - 3 q^{47} + 4 q^{49} + 11 q^{51} + 4 q^{53} - q^{55} + 6 q^{57} - 7 q^{59} - 14 q^{61} + q^{63} - q^{65} - 20 q^{69} - 6 q^{73} + 16 q^{75} + 4 q^{77} + 7 q^{79} - 4 q^{81} + 8 q^{83} - 15 q^{85} + 11 q^{87} + 9 q^{89} + 4 q^{91} + 13 q^{93} - 9 q^{95} - 16 q^{97} + q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 - q^5 + 4 * q^7 + q^9 + 4 * q^11 + 4 * q^13 - 5 * q^15 - 7 * q^17 - 9 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^25 - 9 * q^27 - 3 * q^29 - 3 * q^33 - q^35 - 18 * q^37 - 3 * q^39 - 14 * q^41 - q^43 + 11 * q^45 - 3 * q^47 + 4 * q^49 + 11 * q^51 + 4 * q^53 - q^55 + 6 * q^57 - 7 * q^59 - 14 * q^61 + q^63 - q^65 - 20 * q^69 - 6 * q^73 + 16 * q^75 + 4 * q^77 + 7 * q^79 - 4 * q^81 + 8 * q^83 - 15 * q^85 + 11 * q^87 + 9 * q^89 + 4 * q^91 + 13 * q^93 - 9 * q^95 - 16 * q^97 + 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
$3$	−1.84617	−1.06589	−0.532943	−	0.846151i	$-0.678914\pi$
$3$	−1.84617	−1.06589	−0.532943	+	0.846151i	$0.678914\pi$
$4$	0	0
$5$	1.77882	0.795511	0.397756	−	0.917491i	$-0.369789\pi$
$5$	1.77882	0.795511	0.397756	+	0.917491i	$0.369789\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0.408340	0.136113
$10$	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
$15$	−3.28400	−0.847925
$16$	0	0
$17$	−3.77882	−0.916498	−0.458249	−	0.888824i	$-0.651523\pi$
$17$	−3.77882	−0.916498	−0.458249	+	0.888824i	$0.651523\pi$
$18$	0	0
$19$	−5.21665	−1.19678	−0.598390	−	0.801205i	$-0.704193\pi$
$19$	−5.21665	−1.19678	−0.598390	+	0.801205i	$0.704193\pi$
$20$	0	0
$21$	−1.84617	−0.402867
$22$	0	0
$23$	5.91935	1.23427	0.617134	−	0.786858i	$-0.288293\pi$
$23$	5.91935	1.23427	0.617134	+	0.786858i	$0.288293\pi$
$24$	0	0
$25$	−1.83581	−0.367162
$26$	0	0
$27$	4.78464	0.920805
$28$	0	0
$29$	3.15966	0.586733	0.293367	−	0.956000i	$-0.405224\pi$
$29$	3.15966	0.586733	0.293367	+	0.956000i	$0.405224\pi$
$30$	0	0
$31$	1.87566	0.336878	0.168439	−	0.985712i	$-0.446127\pi$
$31$	1.87566	0.336878	0.168439	+	0.985712i	$0.446127\pi$
$32$	0	0
$33$	−1.84617	−0.321377
$34$	0	0
$35$	1.77882	0.300675
$36$	0	0
$37$	−7.31349	−1.20233	−0.601165	−	0.799125i	$-0.705297\pi$
$37$	−7.31349	−1.20233	−0.601165	+	0.799125i	$0.705297\pi$
$38$	0	0
$39$	−1.84617	−0.295624
$40$	0	0
$41$	−10.3659	−1.61889	−0.809444	−	0.587197i	$-0.800231\pi$
$41$	−10.3659	−1.61889	−0.809444	+	0.587197i	$0.800231\pi$
$42$	0	0
$43$	−6.02495	−0.918797	−0.459398	−	0.888230i	$-0.651935\pi$
$43$	−6.02495	−0.918797	−0.459398	+	0.888230i	$0.651935\pi$
$44$	0	0
$45$	0.726363	0.108280
$46$	0	0
$47$	−9.64994	−1.40759	−0.703794	−	0.710404i	$-0.748512\pi$
$47$	−9.64994	−1.40759	−0.703794	+	0.710404i	$0.748512\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.97634	0.976882
$52$	0	0
$53$	0.678139	0.0931496	0.0465748	−	0.998915i	$-0.485169\pi$
$53$	0.678139	0.0931496	0.0465748	+	0.998915i	$0.485169\pi$
$54$	0	0
$55$	1.77882	0.239856
$56$	0	0
$57$	9.63081	1.27563
$58$	0	0
$59$	0.273637	0.0356245	0.0178123	−	0.999841i	$-0.494330\pi$
$59$	0.273637	0.0356245	0.0178123	+	0.999841i	$0.494330\pi$
$60$	0	0
$61$	7.42876	0.951155	0.475577	−	0.879674i	$-0.342239\pi$
$61$	7.42876	0.951155	0.475577	+	0.879674i	$0.342239\pi$
$62$	0	0
$63$	0.408340	0.0514460
$64$	0	0
$65$	1.77882	0.220635
$66$	0	0
$67$	−0.0103608	−0.00126577	−0.000632884	−	1.00000i	$-0.500201\pi$
$67$	−0.0103608	−0.00126577	−0.000632884		1.00000i	$0.500201\pi$
$68$	0	0
$69$	−10.9281	−1.31559
$70$	0	0
$71$	−3.75132	−0.445199	−0.222600	−	0.974910i	$-0.571454\pi$
$71$	−3.75132	−0.445199	−0.222600	+	0.974910i	$0.571454\pi$
$72$	0	0
$73$	11.3148	1.32429	0.662147	−	0.749374i	$-0.269645\pi$
$73$	11.3148	1.32429	0.662147	+	0.749374i	$0.269645\pi$
$74$	0	0
$75$	3.38921	0.391353
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	6.29820	0.708603	0.354301	−	0.935131i	$-0.384719\pi$
$79$	6.29820	0.708603	0.354301	+	0.935131i	$0.384719\pi$
$80$	0	0
$81$	−10.0583	−1.11759
$82$	0	0
$83$	−11.1418	−1.22297	−0.611487	−	0.791255i	$-0.709428\pi$
$83$	−11.1418	−1.22297	−0.611487	+	0.791255i	$0.709428\pi$
$84$	0	0
$85$	−6.72183	−0.729084
$86$	0	0
$87$	−5.83326	−0.625391
$88$	0	0
$89$	−2.92971	−0.310548	−0.155274	−	0.987871i	$-0.549626\pi$
$89$	−2.92971	−0.310548	−0.155274	+	0.987871i	$0.549626\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−3.46278	−0.359074
$94$	0	0
$95$	−9.27946	−0.952053
$96$	0	0
$97$	−11.1930	−1.13648	−0.568238	−	0.822865i	$-0.692375\pi$
$97$	−11.1930	−1.13648	−0.568238	+	0.822865i	$0.692375\pi$
$98$	0	0
$99$	0.408340	0.0410397
$100$	0	0
$101$	−10.4391	−1.03873	−0.519365	−	0.854552i	$-0.673832\pi$
$101$	−10.4391	−1.03873	−0.519365	+	0.854552i	$0.673832\pi$
$102$	0	0
$103$	7.64540	0.753324	0.376662	−	0.926351i	$-0.377072\pi$
$103$	7.64540	0.753324	0.376662	+	0.926351i	$0.377072\pi$
$104$	0	0
$105$	−3.28400	−0.320485
$106$	0	0
$107$	11.1069	1.07374	0.536872	−	0.843664i	$-0.319606\pi$
$107$	11.1069	1.07374	0.536872	+	0.843664i	$0.319606\pi$
$108$	0	0
$109$	−3.74095	−0.358318	−0.179159	−	0.983820i	$-0.557338\pi$
$109$	−3.74095	−0.358318	−0.179159	+	0.983820i	$0.557338\pi$
$110$	0	0
$111$	13.5019	1.28155
$112$	0	0
$113$	−2.51977	−0.237040	−0.118520	−	0.992952i	$-0.537815\pi$
$113$	−2.51977	−0.237040	−0.118520	+	0.992952i	$0.537815\pi$
$114$	0	0
$115$	10.5294	0.981875
$116$	0	0
$117$	0.408340	0.0377511
$118$	0	0
$119$	−3.77882	−0.346404
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	19.1373	1.72555
$124$	0	0
$125$	−12.1597	−1.08759
$126$	0	0
$127$	16.8001	1.49077	0.745385	−	0.666634i	$-0.232266\pi$
$127$	16.8001	1.49077	0.745385	+	0.666634i	$0.232266\pi$
$128$	0	0
$129$	11.1231	0.979333
$130$	0	0
$131$	13.5019	1.17967	0.589835	−	0.807524i	$-0.299193\pi$
$131$	13.5019	1.17967	0.589835	+	0.807524i	$0.299193\pi$
$132$	0	0
$133$	−5.21665	−0.452341
$134$	0	0
$135$	8.51100	0.732511
$136$	0	0
$137$	−3.34099	−0.285440	−0.142720	−	0.989763i	$-0.545585\pi$
$137$	−3.34099	−0.285440	−0.142720	+	0.989763i	$0.545585\pi$
$138$	0	0
$139$	17.4300	1.47840	0.739199	−	0.673488i	$-0.235205\pi$
$139$	17.4300	1.47840	0.739199	+	0.673488i	$0.235205\pi$
$140$	0	0
$141$	17.8154	1.50033
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	5.62045	0.466753
$146$	0	0
$147$	−1.84617	−0.152269
$148$	0	0
$149$	−8.60132	−0.704648	−0.352324	−	0.935878i	$-0.614608\pi$
$149$	−8.60132	−0.704648	−0.352324	+	0.935878i	$0.614608\pi$
$150$	0	0
$151$	−15.2425	−1.24042	−0.620208	−	0.784437i	$-0.712952\pi$
$151$	−15.2425	−1.24042	−0.620208	+	0.784437i	$0.712952\pi$
$152$	0	0
$153$	−1.54304	−0.124748
$154$	0	0
$155$	3.33645	0.267990
$156$	0	0
$157$	−22.2425	−1.77514	−0.887572	−	0.460669i	$-0.847610\pi$
$157$	−22.2425	−1.77514	−0.887572	+	0.460669i	$0.847610\pi$
$158$	0	0
$159$	−1.25196	−0.0992868
$160$	0	0
$161$	5.91935	0.466510
$162$	0	0
$163$	−2.84617	−0.222929	−0.111465	−	0.993768i	$-0.535554\pi$
$163$	−2.84617	−0.222929	−0.111465	+	0.993768i	$0.535554\pi$
$164$	0	0
$165$	−3.28400	−0.255659
$166$	0	0
$167$	−0.644812	−0.0498971	−0.0249485	−	0.999689i	$-0.507942\pi$
$167$	−0.644812	−0.0498971	−0.0249485	+	0.999689i	$0.507942\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.13017	−0.162898
$172$	0	0
$173$	6.55508	0.498374	0.249187	−	0.968455i	$-0.419837\pi$
$173$	6.55508	0.498374	0.249187	+	0.968455i	$0.419837\pi$
$174$	0	0
$175$	−1.83581	−0.138774
$176$	0	0
$177$	−0.505180	−0.0379717
$178$	0	0
$179$	3.70015	0.276562	0.138281	−	0.990393i	$-0.455842\pi$
$179$	3.70015	0.276562	0.138281	+	0.990393i	$0.455842\pi$
$180$	0	0
$181$	−19.0243	−1.41406	−0.707031	−	0.707183i	$-0.749966\pi$
$181$	−19.0243	−1.41406	−0.707031	+	0.707183i	$0.749966\pi$
$182$	0	0
$183$	−13.7147	−1.01382
$184$	0	0
$185$	−13.0094	−0.956467
$186$	0	0
$187$	−3.77882	−0.276335
$188$	0	0
$189$	4.78464	0.348031
$190$	0	0
$191$	5.09191	0.368438	0.184219	−	0.982885i	$-0.441024\pi$
$191$	5.09191	0.368438	0.184219	+	0.982885i	$0.441024\pi$
$192$	0	0
$193$	−25.4699	−1.83336	−0.916682	−	0.399619i	$-0.869143\pi$
$193$	−25.4699	−1.83336	−0.916682	+	0.399619i	$0.869143\pi$
$194$	0	0
$195$	−3.28400	−0.235172
$196$	0	0
$197$	−2.51848	−0.179435	−0.0897173	−	0.995967i	$-0.528596\pi$
$197$	−2.51848	−0.179435	−0.0897173	+	0.995967i	$0.528596\pi$
$198$	0	0
$199$	−24.1402	−1.71125	−0.855627	−	0.517592i	$-0.826828\pi$
$199$	−24.1402	−1.71125	−0.855627	+	0.517592i	$0.826828\pi$
$200$	0	0
$201$	0.0191277	0.00134916
$202$	0	0
$203$	3.15966	0.221764
$204$	0	0
$205$	−18.4391	−1.28784
$206$	0	0
$207$	2.41711	0.168001
$208$	0	0
$209$	−5.21665	−0.360843
$210$	0	0
$211$	−3.67973	−0.253323	−0.126662	−	0.991946i	$-0.540426\pi$
$211$	−3.67973	−0.253323	−0.126662	+	0.991946i	$0.540426\pi$
$212$	0	0
$213$	6.92556	0.474532
$214$	0	0
$215$	−10.7173	−0.730913
$216$	0	0
$217$	1.87566	0.127328
$218$	0	0
$219$	−20.8890	−1.41155
$220$	0	0
$221$	−3.77882	−0.254191
$222$	0	0
$223$	−4.01420	−0.268811	−0.134405	−	0.990926i	$-0.542912\pi$
$223$	−4.01420	−0.268811	−0.134405	+	0.990926i	$0.542912\pi$
$224$	0	0
$225$	−0.749635	−0.0499756
$226$	0	0
$227$	−15.5168	−1.02989	−0.514944	−	0.857224i	$-0.672187\pi$
$227$	−15.5168	−1.02989	−0.514944	+	0.857224i	$0.672187\pi$
$228$	0	0
$229$	−3.80209	−0.251249	−0.125625	−	0.992078i	$-0.540093\pi$
$229$	−3.80209	−0.251249	−0.125625	+	0.992078i	$0.540093\pi$
$230$	0	0
$231$	−1.84617	−0.121469
$232$	0	0
$233$	−6.51683	−0.426932	−0.213466	−	0.976951i	$-0.568475\pi$
$233$	−6.51683	−0.426932	−0.213466	+	0.976951i	$0.568475\pi$
$234$	0	0
$235$	−17.1655	−1.11975
$236$	0	0
$237$	−11.6275	−0.755290
$238$	0	0
$239$	0.244541	0.0158180	0.00790902	−	0.999969i	$-0.497482\pi$
$239$	0.244541	0.0158180	0.00790902	+	0.999969i	$0.497482\pi$
$240$	0	0
$241$	−24.7040	−1.59132	−0.795662	−	0.605741i	$-0.792877\pi$
$241$	−24.7040	−1.59132	−0.795662	+	0.605741i	$0.792877\pi$
$242$	0	0
$243$	4.21536	0.270415
$244$	0	0
$245$	1.77882	0.113644
$246$	0	0
$247$	−5.21665	−0.331927
$248$	0	0
$249$	20.5697	1.30355
$250$	0	0
$251$	−16.6346	−1.04997	−0.524985	−	0.851112i	$-0.675929\pi$
$251$	−16.6346	−1.04997	−0.524985	+	0.851112i	$0.675929\pi$
$252$	0	0
$253$	5.91935	0.372146
$254$	0	0
$255$	12.4096	0.777121
$256$	0	0
$257$	−1.05808	−0.0660013	−0.0330006	−	0.999455i	$-0.510506\pi$
$257$	−1.05808	−0.0660013	−0.0330006	+	0.999455i	$0.510506\pi$
$258$	0	0
$259$	−7.31349	−0.454438
$260$	0	0
$261$	1.29021	0.0798623
$262$	0	0
$263$	1.68466	0.103881	0.0519403	−	0.998650i	$-0.483459\pi$
$263$	1.68466	0.103881	0.0519403	+	0.998650i	$0.483459\pi$
$264$	0	0
$265$	1.20629	0.0741015
$266$	0	0
$267$	5.40873	0.331009
$268$	0	0
$269$	1.92881	0.117602	0.0588008	−	0.998270i	$-0.481272\pi$
$269$	1.92881	0.117602	0.0588008	+	0.998270i	$0.481272\pi$
$270$	0	0
$271$	15.8750	0.964335	0.482168	−	0.876079i	$-0.339850\pi$
$271$	15.8750	0.964335	0.482168	+	0.876079i	$0.339850\pi$
$272$	0	0
$273$	−1.84617	−0.111735
$274$	0	0
$275$	−1.83581	−0.110703
$276$	0	0
$277$	−6.60715	−0.396985	−0.198492	−	0.980102i	$-0.563605\pi$
$277$	−6.60715	−0.396985	−0.198492	+	0.980102i	$0.563605\pi$
$278$	0	0
$279$	0.765907	0.0458536
$280$	0	0
$281$	−1.98381	−0.118344	−0.0591722	−	0.998248i	$-0.518846\pi$
$281$	−1.98381	−0.118344	−0.0591722	+	0.998248i	$0.518846\pi$
$282$	0	0
$283$	−10.0894	−0.599750	−0.299875	−	0.953978i	$-0.596945\pi$
$283$	−10.0894	−0.599750	−0.299875	+	0.953978i	$0.596945\pi$
$284$	0	0
$285$	17.1315	1.01478
$286$	0	0
$287$	−10.3659	−0.611882
$288$	0	0
$289$	−2.72054	−0.160032
$290$	0	0
$291$	20.6641	1.21135
$292$	0	0
$293$	−27.0608	−1.58091	−0.790454	−	0.612522i	$-0.790155\pi$
$293$	−27.0608	−1.58091	−0.790454	+	0.612522i	$0.790155\pi$
$294$	0	0
$295$	0.486750	0.0283397
$296$	0	0
$297$	4.78464	0.277633
$298$	0	0
$299$	5.91935	0.342325
$300$	0	0
$301$	−6.02495	−0.347272
$302$	0	0
$303$	19.2724	1.10717
$304$	0	0
$305$	13.2144	0.756654
$306$	0	0
$307$	30.2384	1.72579	0.862897	−	0.505380i	$-0.168648\pi$
$307$	30.2384	1.72579	0.862897	+	0.505380i	$0.168648\pi$
$308$	0	0
$309$	−14.1147	−0.802958
$310$	0	0
$311$	−20.8180	−1.18048	−0.590239	−	0.807228i	$-0.700967\pi$
$311$	−20.8180	−1.18048	−0.590239	+	0.807228i	$0.700967\pi$
$312$	0	0
$313$	17.8869	1.01103	0.505514	−	0.862818i	$-0.331303\pi$
$313$	17.8869	1.01103	0.505514	+	0.862818i	$0.331303\pi$
$314$	0	0
$315$	0.726363	0.0409259
$316$	0	0
$317$	8.73927	0.490847	0.245423	−	0.969416i	$-0.421073\pi$
$317$	8.73927	0.490847	0.245423	+	0.969416i	$0.421073\pi$
$318$	0	0
$319$	3.15966	0.176907
$320$	0	0
$321$	−20.5052	−1.14449
$322$	0	0
$323$	19.7128	1.09685
$324$	0	0
$325$	−1.83581	−0.101832
$326$	0	0
$327$	6.90643	0.381927
$328$	0	0
$329$	−9.64994	−0.532018
$330$	0	0
$331$	−12.5690	−0.690852	−0.345426	−	0.938446i	$-0.612266\pi$
$331$	−12.5690	−0.690852	−0.345426	+	0.938446i	$0.612266\pi$
$332$	0	0
$333$	−2.98639	−0.163653
$334$	0	0
$335$	−0.0184299	−0.00100693
$336$	0	0
$337$	−14.8753	−0.810307	−0.405154	−	0.914249i	$-0.632782\pi$
$337$	−14.8753	−0.810307	−0.405154	+	0.914249i	$0.632782\pi$
$338$	0	0
$339$	4.65192	0.252658
$340$	0	0
$341$	1.87566	0.101573
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−19.4391	−1.04657
$346$	0	0
$347$	−6.01006	−0.322637	−0.161318	−	0.986902i	$-0.551575\pi$
$347$	−6.01006	−0.322637	−0.161318	+	0.986902i	$0.551575\pi$
$348$	0	0
$349$	−22.7011	−1.21516	−0.607581	−	0.794258i	$-0.707860\pi$
$349$	−22.7011	−1.21516	−0.607581	+	0.794258i	$0.707860\pi$
$350$	0	0
$351$	4.78464	0.255385
$352$	0	0
$353$	24.6579	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$353$	24.6579	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$354$	0	0
$355$	−6.67290	−0.354161
$356$	0	0
$357$	6.97634	0.369227
$358$	0	0
$359$	−5.80212	−0.306224	−0.153112	−	0.988209i	$-0.548930\pi$
$359$	−5.80212	−0.306224	−0.153112	+	0.988209i	$0.548930\pi$
$360$	0	0
$361$	8.21340	0.432284
$362$	0	0
$363$	−1.84617	−0.0968987
$364$	0	0
$365$	20.1269	1.05349
$366$	0	0
$367$	−5.40420	−0.282097	−0.141048	−	0.990003i	$-0.545047\pi$
$367$	−5.40420	−0.282097	−0.141048	+	0.990003i	$0.545047\pi$
$368$	0	0
$369$	−4.23283	−0.220352
$370$	0	0
$371$	0.678139	0.0352072
$372$	0	0
$373$	−22.5440	−1.16729	−0.583643	−	0.812011i	$-0.698373\pi$
$373$	−22.5440	−1.16729	−0.583643	+	0.812011i	$0.698373\pi$
$374$	0	0
$375$	22.4488	1.15925
$376$	0	0
$377$	3.15966	0.162731
$378$	0	0
$379$	−11.1069	−0.570523	−0.285261	−	0.958450i	$-0.592080\pi$
$379$	−11.1069	−0.570523	−0.285261	+	0.958450i	$0.592080\pi$
$380$	0	0
$381$	−31.0159	−1.58899
$382$	0	0
$383$	−9.86818	−0.504240	−0.252120	−	0.967696i	$-0.581128\pi$
$383$	−9.86818	−0.504240	−0.252120	+	0.967696i	$0.581128\pi$
$384$	0	0
$385$	1.77882	0.0906569
$386$	0	0
$387$	−2.46023	−0.125061
$388$	0	0
$389$	23.6749	1.20036	0.600182	−	0.799863i	$-0.295095\pi$
$389$	23.6749	1.20036	0.600182	+	0.799863i	$0.295095\pi$
$390$	0	0
$391$	−22.3681	−1.13120
$392$	0	0
$393$	−24.9269	−1.25739
$394$	0	0
$395$	11.2033	0.563701
$396$	0	0
$397$	−22.8624	−1.14743	−0.573714	−	0.819055i	$-0.694498\pi$
$397$	−22.8624	−1.14743	−0.573714	+	0.819055i	$0.694498\pi$
$398$	0	0
$399$	9.63081	0.482144
$400$	0	0
$401$	32.4199	1.61897	0.809486	−	0.587139i	$-0.199746\pi$
$401$	32.4199	1.61897	0.809486	+	0.587139i	$0.199746\pi$
$402$	0	0
$403$	1.87566	0.0934332
$404$	0	0
$405$	−17.8918	−0.889053
$406$	0	0
$407$	−7.31349	−0.362516
$408$	0	0
$409$	7.15128	0.353608	0.176804	−	0.984246i	$-0.443424\pi$
$409$	7.15128	0.353608	0.176804	+	0.984246i	$0.443424\pi$
$410$	0	0
$411$	6.16803	0.304246
$412$	0	0
$413$	0.273637	0.0134648
$414$	0	0
$415$	−19.8193	−0.972889
$416$	0	0
$417$	−32.1788	−1.57580
$418$	0	0
$419$	−20.1848	−0.986092	−0.493046	−	0.870003i	$-0.664117\pi$
$419$	−20.1848	−0.986092	−0.493046	+	0.870003i	$0.664117\pi$
$420$	0	0
$421$	−11.9637	−0.583077	−0.291538	−	0.956559i	$-0.594167\pi$
$421$	−11.9637	−0.583077	−0.291538	+	0.956559i	$0.594167\pi$
$422$	0	0
$423$	−3.94046	−0.191592
$424$	0	0
$425$	6.93718	0.336503
$426$	0	0
$427$	7.42876	0.359503
$428$	0	0
$429$	−1.84617	−0.0891339
$430$	0	0
$431$	5.43813	0.261946	0.130973	−	0.991386i	$-0.458190\pi$
$431$	5.43813	0.261946	0.130973	+	0.991386i	$0.458190\pi$
$432$	0	0
$433$	22.8455	1.09788	0.548942	−	0.835860i	$-0.315031\pi$
$433$	22.8455	1.09788	0.548942	+	0.835860i	$0.315031\pi$
$434$	0	0
$435$	−10.3763	−0.497506
$436$	0	0
$437$	−30.8791	−1.47715
$438$	0	0
$439$	28.8100	1.37503	0.687513	−	0.726172i	$-0.258703\pi$
$439$	28.8100	1.37503	0.687513	+	0.726172i	$0.258703\pi$
$440$	0	0
$441$	0.408340	0.0194448
$442$	0	0
$443$	1.15128	0.0546990	0.0273495	−	0.999626i	$-0.491293\pi$
$443$	1.15128	0.0546990	0.0273495	+	0.999626i	$0.491293\pi$
$444$	0	0
$445$	−5.21141	−0.247045
$446$	0	0
$447$	15.8795	0.751074
$448$	0	0
$449$	7.75420	0.365943	0.182972	−	0.983118i	$-0.441428\pi$
$449$	7.75420	0.365943	0.182972	+	0.983118i	$0.441428\pi$
$450$	0	0
$451$	−10.3659	−0.488113
$452$	0	0
$453$	28.1402	1.32214
$454$	0	0
$455$	1.77882	0.0833923
$456$	0	0
$457$	26.7295	1.25035	0.625177	−	0.780483i	$-0.285027\pi$
$457$	26.7295	1.25035	0.625177	+	0.780483i	$0.285027\pi$
$458$	0	0
$459$	−18.0803	−0.843916
$460$	0	0
$461$	−10.7914	−0.502607	−0.251303	−	0.967908i	$-0.580859\pi$
$461$	−10.7914	−0.502607	−0.251303	+	0.967908i	$0.580859\pi$
$462$	0	0
$463$	32.7247	1.52085	0.760423	−	0.649428i	$-0.224992\pi$
$463$	32.7247	1.52085	0.760423	+	0.649428i	$0.224992\pi$
$464$	0	0
$465$	−6.15966	−0.285647
$466$	0	0
$467$	34.8775	1.61394	0.806970	−	0.590592i	$-0.201106\pi$
$467$	34.8775	1.61394	0.806970	+	0.590592i	$0.201106\pi$
$468$	0	0
$469$	−0.0103608	−0.000478415 0
$470$	0	0
$471$	41.0634	1.89210
$472$	0	0
$473$	−6.02495	−0.277028
$474$	0	0
$475$	9.57676	0.439412
$476$	0	0
$477$	0.276911	0.0126789
$478$	0	0
$479$	3.71819	0.169888	0.0849441	−	0.996386i	$-0.472929\pi$
$479$	3.71819	0.169888	0.0849441	+	0.996386i	$0.472929\pi$
$480$	0	0
$481$	−7.31349	−0.333466
$482$	0	0
$483$	−10.9281	−0.497246
$484$	0	0
$485$	−19.9103	−0.904079
$486$	0	0
$487$	−21.2289	−0.961973	−0.480986	−	0.876728i	$-0.659721\pi$
$487$	−21.2289	−0.961973	−0.480986	+	0.876728i	$0.659721\pi$
$488$	0	0
$489$	5.25451	0.237617
$490$	0	0
$491$	4.98709	0.225064	0.112532	−	0.993648i	$-0.464104\pi$
$491$	4.98709	0.225064	0.112532	+	0.993648i	$0.464104\pi$
$492$	0	0
$493$	−11.9398	−0.537740
$494$	0	0
$495$	0.726363	0.0326476
$496$	0	0
$497$	−3.75132	−0.168269
$498$	0	0
$499$	9.66318	0.432583	0.216292	−	0.976329i	$-0.430604\pi$
$499$	9.66318	0.432583	0.216292	+	0.976329i	$0.430604\pi$
$500$	0	0
$501$	1.19043	0.0531846
$502$	0	0
$503$	−16.4254	−0.732373	−0.366187	−	0.930541i	$-0.619337\pi$
$503$	−16.4254	−0.732373	−0.366187	+	0.930541i	$0.619337\pi$
$504$	0	0
$505$	−18.5693	−0.826322
$506$	0	0
$507$	−1.84617	−0.0819912
$508$	0	0
$509$	14.0100	0.620981	0.310490	−	0.950576i	$-0.399507\pi$
$509$	14.0100	0.620981	0.310490	+	0.950576i	$0.399507\pi$
$510$	0	0
$511$	11.3148	0.500536
$512$	0	0
$513$	−24.9598	−1.10200
$514$	0	0
$515$	13.5998	0.599278
$516$	0	0
$517$	−9.64994	−0.424404
$518$	0	0
$519$	−12.1018	−0.531210
$520$	0	0
$521$	−40.2749	−1.76447	−0.882237	−	0.470806i	$-0.843963\pi$
$521$	−40.2749	−1.76447	−0.882237	+	0.470806i	$0.843963\pi$
$522$	0	0
$523$	−27.6852	−1.21059	−0.605294	−	0.796002i	$-0.706945\pi$
$523$	−27.6852	−1.21059	−0.605294	+	0.796002i	$0.706945\pi$
$524$	0	0
$525$	3.38921	0.147917
$526$	0	0
$527$	−7.08777	−0.308748
$528$	0	0
$529$	12.0386	0.523419
$530$	0	0
$531$	0.111737	0.00484898
$532$	0	0
$533$	−10.3659	−0.448999
$534$	0	0
$535$	19.7571	0.854176
$536$	0	0
$537$	−6.83110	−0.294784
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−1.01714	−0.0437303	−0.0218652	−	0.999761i	$-0.506960\pi$
$541$	−1.01714	−0.0437303	−0.0218652	+	0.999761i	$0.506960\pi$
$542$	0	0
$543$	35.1220	1.50723
$544$	0	0
$545$	−6.65448	−0.285046
$546$	0	0
$547$	10.4047	0.444873	0.222436	−	0.974947i	$-0.428599\pi$
$547$	10.4047	0.444873	0.222436	+	0.974947i	$0.428599\pi$
$548$	0	0
$549$	3.03346	0.129465
$550$	0	0
$551$	−16.4828	−0.702191
$552$	0	0
$553$	6.29820	0.267827
$554$	0	0
$555$	24.0175	1.01948
$556$	0	0
$557$	27.7461	1.17564	0.587820	−	0.808992i	$-0.299986\pi$
$557$	27.7461	1.17564	0.587820	+	0.808992i	$0.299986\pi$
$558$	0	0
$559$	−6.02495	−0.254828
$560$	0	0
$561$	6.97634	0.294541
$562$	0	0
$563$	8.11854	0.342156	0.171078	−	0.985258i	$-0.445275\pi$
$563$	8.11854	0.342156	0.171078	+	0.985258i	$0.445275\pi$
$564$	0	0
$565$	−4.48221	−0.188568
$566$	0	0
$567$	−10.0583	−0.422408
$568$	0	0
$569$	25.6943	1.07716	0.538581	−	0.842574i	$-0.318961\pi$
$569$	25.6943	1.07716	0.538581	+	0.842574i	$0.318961\pi$
$570$	0	0
$571$	10.4217	0.436136	0.218068	−	0.975934i	$-0.430025\pi$
$571$	10.4217	0.436136	0.218068	+	0.975934i	$0.430025\pi$
$572$	0	0
$573$	−9.40053	−0.392713
$574$	0	0
$575$	−10.8668	−0.453176
$576$	0	0
$577$	4.58534	0.190890	0.0954450	−	0.995435i	$-0.469573\pi$
$577$	4.58534	0.190890	0.0954450	+	0.995435i	$0.469573\pi$
$578$	0	0
$579$	47.0217	1.95416
$580$	0	0
$581$	−11.1418	−0.462240
$582$	0	0
$583$	0.678139	0.0280856
$584$	0	0
$585$	0.726363	0.0300314
$586$	0	0
$587$	−17.6481	−0.728414	−0.364207	−	0.931318i	$-0.618660\pi$
$587$	−17.6481	−0.728414	−0.364207	+	0.931318i	$0.618660\pi$
$588$	0	0
$589$	−9.78464	−0.403169
$590$	0	0
$591$	4.64955	0.191257
$592$	0	0
$593$	14.2163	0.583792	0.291896	−	0.956450i	$-0.405714\pi$
$593$	14.2163	0.583792	0.291896	+	0.956450i	$0.405714\pi$
$594$	0	0
$595$	−6.72183	−0.275568
$596$	0	0
$597$	44.5669	1.82400
$598$	0	0
$599$	8.45371	0.345409	0.172705	−	0.984974i	$-0.444749\pi$
$599$	8.45371	0.345409	0.172705	+	0.984974i	$0.444749\pi$
$600$	0	0
$601$	−2.25381	−0.0919349	−0.0459674	−	0.998943i	$-0.514637\pi$
$601$	−2.25381	−0.0919349	−0.0459674	+	0.998943i	$0.514637\pi$
$602$	0	0
$603$	−0.00423072	−0.000172288 0
$604$	0	0
$605$	1.77882	0.0723192
$606$	0	0
$607$	32.6055	1.32342	0.661708	−	0.749762i	$-0.269832\pi$
$607$	32.6055	1.32342	0.661708	+	0.749762i	$0.269832\pi$
$608$	0	0
$609$	−5.83326	−0.236376
$610$	0	0
$611$	−9.64994	−0.390395
$612$	0	0
$613$	37.8344	1.52812	0.764059	−	0.645146i	$-0.223203\pi$
$613$	37.8344	1.52812	0.764059	+	0.645146i	$0.223203\pi$
$614$	0	0
$615$	34.0417	1.37269
$616$	0	0
$617$	−30.7834	−1.23929	−0.619646	−	0.784881i	$-0.712724\pi$
$617$	−30.7834	−1.23929	−0.619646	+	0.784881i	$0.712724\pi$
$618$	0	0
$619$	−17.0758	−0.686332	−0.343166	−	0.939275i	$-0.611499\pi$
$619$	−17.0758	−0.686332	−0.343166	+	0.939275i	$0.611499\pi$
$620$	0	0
$621$	28.3219	1.13652
$622$	0	0
$623$	−2.92971	−0.117376
$624$	0	0
$625$	−12.4508	−0.498031
$626$	0	0
$627$	9.63081	0.384618
$628$	0	0
$629$	27.6363	1.10193
$630$	0	0
$631$	4.68231	0.186400	0.0931999	−	0.995647i	$-0.470290\pi$
$631$	4.68231	0.186400	0.0931999	+	0.995647i	$0.470290\pi$
$632$	0	0
$633$	6.79341	0.270014
$634$	0	0
$635$	29.8844	1.18592
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−1.53181	−0.0605976
$640$	0	0
$641$	−18.9287	−0.747639	−0.373819	−	0.927502i	$-0.621952\pi$
$641$	−18.9287	−0.747639	−0.373819	+	0.927502i	$0.621952\pi$
$642$	0	0
$643$	−47.2863	−1.86479	−0.932394	−	0.361443i	$-0.882284\pi$
$643$	−47.2863	−1.86479	−0.932394	+	0.361443i	$0.882284\pi$
$644$	0	0
$645$	19.7859	0.779070
$646$	0	0
$647$	27.2943	1.07305	0.536524	−	0.843885i	$-0.319737\pi$
$647$	27.2943	1.07305	0.536524	+	0.843885i	$0.319737\pi$
$648$	0	0
$649$	0.273637	0.0107412
$650$	0	0
$651$	−3.46278	−0.135717
$652$	0	0
$653$	41.0949	1.60817	0.804083	−	0.594516i	$-0.202656\pi$
$653$	41.0949	1.60817	0.804083	+	0.594516i	$0.202656\pi$
$654$	0	0
$655$	24.0175	0.938440
$656$	0	0
$657$	4.62028	0.180254
$658$	0	0
$659$	−1.64342	−0.0640184	−0.0320092	−	0.999488i	$-0.510191\pi$
$659$	−1.64342	−0.0640184	−0.0320092	+	0.999488i	$0.510191\pi$
$660$	0	0
$661$	−8.64283	−0.336167	−0.168083	−	0.985773i	$-0.553758\pi$
$661$	−8.64283	−0.336167	−0.168083	+	0.985773i	$0.553758\pi$
$662$	0	0
$663$	6.97634	0.270938
$664$	0	0
$665$	−9.27946	−0.359842
$666$	0	0
$667$	18.7031	0.724187
$668$	0	0
$669$	7.41089	0.286522
$670$	0	0
$671$	7.42876	0.286784
$672$	0	0
$673$	50.9778	1.96505	0.982525	−	0.186129i	$-0.0595942\pi$
$673$	50.9778	1.96505	0.982525	+	0.186129i	$0.0595942\pi$
$674$	0	0
$675$	−8.78369	−0.338084
$676$	0	0
$677$	−38.7192	−1.48810	−0.744049	−	0.668125i	$-0.767097\pi$
$677$	−38.7192	−1.48810	−0.744049	+	0.668125i	$0.767097\pi$
$678$	0	0
$679$	−11.1930	−0.429547
$680$	0	0
$681$	28.6467	1.09774
$682$	0	0
$683$	−35.0171	−1.33989	−0.669946	−	0.742410i	$-0.733683\pi$
$683$	−35.0171	−1.33989	−0.669946	+	0.742410i	$0.733683\pi$
$684$	0	0
$685$	−5.94301	−0.227071
$686$	0	0
$687$	7.01930	0.267803
$688$	0	0
$689$	0.678139	0.0258350
$690$	0	0
$691$	−9.36952	−0.356433	−0.178217	−	0.983991i	$-0.557033\pi$
$691$	−9.36952	−0.356433	−0.178217	+	0.983991i	$0.557033\pi$
$692$	0	0
$693$	0.408340	0.0155116
$694$	0	0
$695$	31.0049	1.17608
$696$	0	0
$697$	39.1710	1.48371
$698$	0	0
$699$	12.0312	0.455061
$700$	0	0
$701$	−12.5764	−0.475003	−0.237501	−	0.971387i	$-0.576328\pi$
$701$	−12.5764	−0.475003	−0.237501	+	0.971387i	$0.576328\pi$
$702$	0	0
$703$	38.1519	1.43893
$704$	0	0
$705$	31.6904	1.19353
$706$	0	0
$707$	−10.4391	−0.392603
$708$	0	0
$709$	15.3582	0.576788	0.288394	−	0.957512i	$-0.406879\pi$
$709$	15.3582	0.576788	0.288394	+	0.957512i	$0.406879\pi$
$710$	0	0
$711$	2.57181	0.0964503
$712$	0	0
$713$	11.1027	0.415798
$714$	0	0
$715$	1.77882	0.0665240
$716$	0	0
$717$	−0.451464	−0.0168602
$718$	0	0
$719$	−6.76207	−0.252183	−0.126091	−	0.992019i	$-0.540243\pi$
$719$	−6.76207	−0.252183	−0.126091	+	0.992019i	$0.540243\pi$
$720$	0	0
$721$	7.64540	0.284730
$722$	0	0
$723$	45.6077	1.69617
$724$	0	0
$725$	−5.80052	−0.215426
$726$	0	0
$727$	−6.66963	−0.247363	−0.123681	−	0.992322i	$-0.539470\pi$
$727$	−6.66963	−0.247363	−0.123681	+	0.992322i	$0.539470\pi$
$728$	0	0
$729$	22.3926	0.829355
$730$	0	0
$731$	22.7672	0.842075
$732$	0	0
$733$	−43.4673	−1.60550	−0.802751	−	0.596314i	$-0.796631\pi$
$733$	−43.4673	−1.60550	−0.802751	+	0.596314i	$0.796631\pi$
$734$	0	0
$735$	−3.28400	−0.121132
$736$	0	0
$737$	−0.0103608	−0.000381643 0
$738$	0	0
$739$	18.6395	0.685665	0.342832	−	0.939397i	$-0.388614\pi$
$739$	18.6395	0.685665	0.342832	+	0.939397i	$0.388614\pi$
$740$	0	0
$741$	9.63081	0.353797
$742$	0	0
$743$	9.87400	0.362242	0.181121	−	0.983461i	$-0.442027\pi$
$743$	9.87400	0.362242	0.181121	+	0.983461i	$0.442027\pi$
$744$	0	0
$745$	−15.3002	−0.560555
$746$	0	0
$747$	−4.54965	−0.166463
$748$	0	0
$749$	11.1069	0.405837
$750$	0	0
$751$	−30.0951	−1.09819	−0.549093	−	0.835762i	$-0.685027\pi$
$751$	−30.0951	−1.09819	−0.549093	+	0.835762i	$0.685027\pi$
$752$	0	0
$753$	30.7104	1.11915
$754$	0	0
$755$	−27.1136	−0.986766
$756$	0	0
$757$	28.0521	1.01957	0.509786	−	0.860301i	$-0.329724\pi$
$757$	28.0521	1.01957	0.509786	+	0.860301i	$0.329724\pi$
$758$	0	0
$759$	−10.9281	−0.396665
$760$	0	0
$761$	−13.1304	−0.475975	−0.237988	−	0.971268i	$-0.576488\pi$
$761$	−13.1304	−0.475975	−0.237988	+	0.971268i	$0.576488\pi$
$762$	0	0
$763$	−3.74095	−0.135432
$764$	0	0
$765$	−2.74479	−0.0992382
$766$	0	0
$767$	0.273637	0.00988046
$768$	0	0
$769$	−34.0369	−1.22740	−0.613702	−	0.789538i	$-0.710320\pi$
$769$	−34.0369	−1.22740	−0.613702	+	0.789538i	$0.710320\pi$
$770$	0	0
$771$	1.95340	0.0703499
$772$	0	0
$773$	27.6234	0.993545	0.496773	−	0.867881i	$-0.334518\pi$
$773$	27.6234	0.993545	0.496773	+	0.867881i	$0.334518\pi$
$774$	0	0
$775$	−3.44335	−0.123689
$776$	0	0
$777$	13.5019	0.484379
$778$	0	0
$779$	54.0754	1.93745
$780$	0	0
$781$	−3.75132	−0.134233
$782$	0	0
$783$	15.1178	0.540267
$784$	0	0
$785$	−39.5653	−1.41215
$786$	0	0
$787$	−16.0865	−0.573421	−0.286711	−	0.958017i	$-0.592562\pi$
$787$	−16.0865	−0.573421	−0.286711	+	0.958017i	$0.592562\pi$
$788$	0	0
$789$	−3.11017	−0.110725
$790$	0	0
$791$	−2.51977	−0.0895928
$792$	0	0
$793$	7.42876	0.263803
$794$	0	0
$795$	−2.22701	−0.0789838
$796$	0	0
$797$	−16.6841	−0.590979	−0.295490	−	0.955346i	$-0.595483\pi$
$797$	−16.6841	−0.590979	−0.295490	+	0.955346i	$0.595483\pi$
$798$	0	0
$799$	36.4654	1.29005
$800$	0	0
$801$	−1.19632	−0.0422698
$802$	0	0
$803$	11.3148	0.399290
$804$	0	0
$805$	10.5294	0.371114
$806$	0	0
$807$	−3.56091	−0.125350
$808$	0	0
$809$	32.8365	1.15447	0.577234	−	0.816578i	$-0.304132\pi$
$809$	32.8365	1.15447	0.577234	+	0.816578i	$0.304132\pi$
$810$	0	0
$811$	10.8151	0.379770	0.189885	−	0.981806i	$-0.439188\pi$
$811$	10.8151	0.379770	0.189885	+	0.981806i	$0.439188\pi$
$812$	0	0
$813$	−29.3079	−1.02787
$814$	0	0
$815$	−5.06282	−0.177343
$816$	0	0
$817$	31.4300	1.09960
$818$	0	0
$819$	0.408340	0.0142686
$820$	0	0
$821$	36.5091	1.27418	0.637089	−	0.770790i	$-0.280138\pi$
$821$	36.5091	1.27418	0.637089	+	0.770790i	$0.280138\pi$
$822$	0	0
$823$	−32.7537	−1.14172	−0.570861	−	0.821047i	$-0.693391\pi$
$823$	−32.7537	−1.14172	−0.570861	+	0.821047i	$0.693391\pi$
$824$	0	0
$825$	3.38921	0.117997
$826$	0	0
$827$	46.2224	1.60731	0.803655	−	0.595096i	$-0.202886\pi$
$827$	46.2224	1.60731	0.803655	+	0.595096i	$0.202886\pi$
$828$	0	0
$829$	−40.0279	−1.39023	−0.695114	−	0.718900i	$-0.744646\pi$
$829$	−40.0279	−1.39023	−0.695114	+	0.718900i	$0.744646\pi$
$830$	0	0
$831$	12.1979	0.423141
$832$	0	0
$833$	−3.77882	−0.130928
$834$	0	0
$835$	−1.14700	−0.0396937
$836$	0	0
$837$	8.97435	0.310199
$838$	0	0
$839$	−25.6713	−0.886270	−0.443135	−	0.896455i	$-0.646134\pi$
$839$	−25.6713	−0.886270	−0.443135	+	0.896455i	$0.646134\pi$
$840$	0	0
$841$	−19.0166	−0.655744
$842$	0	0
$843$	3.66246	0.126142
$844$	0	0
$845$	1.77882	0.0611932
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	18.6267	0.639266
$850$	0	0
$851$	−43.2911	−1.48400
$852$	0	0
$853$	−19.9553	−0.683257	−0.341628	−	0.939835i	$-0.610978\pi$
$853$	−19.9553	−0.683257	−0.341628	+	0.939835i	$0.610978\pi$
$854$	0	0
$855$	−3.78918	−0.129587
$856$	0	0
$857$	40.9724	1.39959	0.699795	−	0.714344i	$-0.253275\pi$
$857$	40.9724	1.39959	0.699795	+	0.714344i	$0.253275\pi$
$858$	0	0
$859$	15.4391	0.526774	0.263387	−	0.964690i	$-0.415160\pi$
$859$	15.4391	0.526774	0.263387	+	0.964690i	$0.415160\pi$
$860$	0	0
$861$	19.1373	0.652197
$862$	0	0
$863$	40.8519	1.39061	0.695307	−	0.718713i	$-0.255268\pi$
$863$	40.8519	1.39061	0.695307	+	0.718713i	$0.255268\pi$
$864$	0	0
$865$	11.6603	0.396462
$866$	0	0
$867$	5.02257	0.170576
$868$	0	0
$869$	6.29820	0.213652
$870$	0	0
$871$	−0.0103608	−0.000351061 0
$872$	0	0
$873$	−4.57055	−0.154690
$874$	0	0
$875$	−12.1597	−0.411071
$876$	0	0
$877$	2.90578	0.0981213	0.0490607	−	0.998796i	$-0.484377\pi$
$877$	2.90578	0.0981213	0.0490607	+	0.998796i	$0.484377\pi$
$878$	0	0
$879$	49.9588	1.68507
$880$	0	0
$881$	−19.3869	−0.653162	−0.326581	−	0.945169i	$-0.605897\pi$
$881$	−19.3869	−0.653162	−0.326581	+	0.945169i	$0.605897\pi$
$882$	0	0
$883$	−22.7749	−0.766435	−0.383218	−	0.923658i	$-0.625184\pi$
$883$	−22.7749	−0.766435	−0.383218	+	0.923658i	$0.625184\pi$
$884$	0	0
$885$	−0.898624	−0.0302069
$886$	0	0
$887$	−23.0712	−0.774656	−0.387328	−	0.921942i	$-0.626602\pi$
$887$	−23.0712	−0.774656	−0.387328	+	0.921942i	$0.626602\pi$
$888$	0	0
$889$	16.8001	0.563458
$890$	0	0
$891$	−10.0583	−0.336965
$892$	0	0
$893$	50.3403	1.68457
$894$	0	0
$895$	6.58189	0.220008
$896$	0	0
$897$	−10.9281	−0.364879
$898$	0	0
$899$	5.92643	0.197658
$900$	0	0
$901$	−2.56256	−0.0853714
$902$	0	0
$903$	11.1231	0.370153
$904$	0	0
$905$	−33.8407	−1.12490
$906$	0	0
$907$	7.46010	0.247708	0.123854	−	0.992300i	$-0.460474\pi$
$907$	7.46010	0.247708	0.123854	+	0.992300i	$0.460474\pi$
$908$	0	0
$909$	−4.26271	−0.141385
$910$	0	0
$911$	−7.52126	−0.249190	−0.124595	−	0.992208i	$-0.539763\pi$
$911$	−7.52126	−0.249190	−0.124595	+	0.992208i	$0.539763\pi$
$912$	0	0
$913$	−11.1418	−0.368740
$914$	0	0
$915$	−24.3960	−0.806508
$916$	0	0
$917$	13.5019	0.445873
$918$	0	0
$919$	−19.1735	−0.632475	−0.316238	−	0.948680i	$-0.602420\pi$
$919$	−19.1735	−0.632475	−0.316238	+	0.948680i	$0.602420\pi$
$920$	0	0
$921$	−55.8251	−1.83950
$922$	0	0
$923$	−3.75132	−0.123476
$924$	0	0
$925$	13.4262	0.441449
$926$	0	0
$927$	3.12193	0.102538
$928$	0	0
$929$	34.2195	1.12270	0.561352	−	0.827577i	$-0.310281\pi$
$929$	34.2195	1.12270	0.561352	+	0.827577i	$0.310281\pi$
$930$	0	0
$931$	−5.21665	−0.170969
$932$	0	0
$933$	38.4335	1.25826
$934$	0	0
$935$	−6.72183	−0.219827
$936$	0	0
$937$	10.2157	0.333732	0.166866	−	0.985980i	$-0.446635\pi$
$937$	10.2157	0.333732	0.166866	+	0.985980i	$0.446635\pi$
$938$	0	0
$939$	−33.0223	−1.07764
$940$	0	0
$941$	−40.7923	−1.32979	−0.664895	−	0.746937i	$-0.731524\pi$
$941$	−40.7923	−1.32979	−0.664895	+	0.746937i	$0.731524\pi$
$942$	0	0
$943$	−61.3596	−1.99814
$944$	0	0
$945$	8.51100	0.276863
$946$	0	0
$947$	43.9122	1.42696	0.713478	−	0.700678i	$-0.247119\pi$
$947$	43.9122	1.42696	0.713478	+	0.700678i	$0.247119\pi$
$948$	0	0
$949$	11.3148	0.367293
$950$	0	0
$951$	−16.1342	−0.523187
$952$	0	0
$953$	36.3078	1.17612	0.588062	−	0.808815i	$-0.299891\pi$
$953$	36.3078	1.17612	0.588062	+	0.808815i	$0.299891\pi$
$954$	0	0
$955$	9.05758	0.293096
$956$	0	0
$957$	−5.83326	−0.188562
$958$	0	0
$959$	−3.34099	−0.107886
$960$	0	0
$961$	−27.4819	−0.886513
$962$	0	0
$963$	4.53539	0.146151
$964$	0	0
$965$	−45.3063	−1.45846
$966$	0	0
$967$	−14.4850	−0.465808	−0.232904	−	0.972500i	$-0.574823\pi$
$967$	−14.4850	−0.465808	−0.232904	+	0.972500i	$0.574823\pi$
$968$	0	0
$969$	−36.3931	−1.16911
$970$	0	0
$971$	44.6201	1.43193	0.715963	−	0.698138i	$-0.245988\pi$
$971$	44.6201	1.43193	0.715963	+	0.698138i	$0.245988\pi$
$972$	0	0
$973$	17.4300	0.558782
$974$	0	0
$975$	3.38921	0.108542
$976$	0	0
$977$	6.69509	0.214195	0.107097	−	0.994249i	$-0.465844\pi$
$977$	6.69509	0.214195	0.107097	+	0.994249i	$0.465844\pi$
$978$	0	0
$979$	−2.92971	−0.0936338
$980$	0	0
$981$	−1.52758	−0.0487720
$982$	0	0
$983$	27.3255	0.871547	0.435774	−	0.900056i	$-0.356475\pi$
$983$	27.3255	0.871547	0.435774	+	0.900056i	$0.356475\pi$
$984$	0	0
$985$	−4.47992	−0.142742
$986$	0	0
$987$	17.8154	0.567071
$988$	0	0
$989$	−35.6638	−1.13404
$990$	0	0
$991$	−1.16181	−0.0369062	−0.0184531	−	0.999830i	$-0.505874\pi$
$991$	−1.16181	−0.0369062	−0.0184531	+	0.999830i	$0.505874\pi$
$992$	0	0
$993$	23.2044	0.736370
$994$	0	0
$995$	−42.9410	−1.36132
$996$	0	0
$997$	11.7060	0.370732	0.185366	−	0.982670i	$-0.440653\pi$
$997$	11.7060	0.370732	0.185366	+	0.982670i	$0.440653\pi$
$998$	0	0
$999$	−34.9924	−1.10711

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.d.1.2	✓	4	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q-1.84617 q^{3} +1.77882 q^{5} +1.00000 q^{7} +0.408340 q^{9} +O(q^{10})\)	\(q-1.84617 q^{3} +1.77882 q^{5} +1.00000 q^{7} +0.408340 q^{9} +1.00000 q^{11} +1.00000 q^{13} -3.28400 q^{15} -3.77882 q^{17} -5.21665 q^{19} -1.84617 q^{21} +5.91935 q^{23} -1.83581 q^{25} +4.78464 q^{27} +3.15966 q^{29} +1.87566 q^{31} -1.84617 q^{33} +1.77882 q^{35} -7.31349 q^{37} -1.84617 q^{39} -10.3659 q^{41} -6.02495 q^{43} +0.726363 q^{45} -9.64994 q^{47} +1.00000 q^{49} +6.97634 q^{51} +0.678139 q^{53} +1.77882 q^{55} +9.63081 q^{57} +0.273637 q^{59} +7.42876 q^{61} +0.408340 q^{63} +1.77882 q^{65} -0.0103608 q^{67} -10.9281 q^{69} -3.75132 q^{71} +11.3148 q^{73} +3.38921 q^{75} +1.00000 q^{77} +6.29820 q^{79} -10.0583 q^{81} -11.1418 q^{83} -6.72183 q^{85} -5.83326 q^{87} -2.92971 q^{89} +1.00000 q^{91} -3.46278 q^{93} -9.27946 q^{95} -11.1930 q^{97} +0.408340 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} - q^{5} + 4 q^{7} + q^{9}+O(q^{10}) \) 4 * q - 3 * q^3 - q^5 + 4 * q^7 + q^9	\( 4 q - 3 q^{3} - q^{5} + 4 q^{7} + q^{9} + 4 q^{11} + 4 q^{13} - 5 q^{15} - 7 q^{17} - 9 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{25} - 9 q^{27} - 3 q^{29} - 3 q^{33} - q^{35} - 18 q^{37} - 3 q^{39} - 14 q^{41} - q^{43} + 11 q^{45} - 3 q^{47} + 4 q^{49} + 11 q^{51} + 4 q^{53} - q^{55} + 6 q^{57} - 7 q^{59} - 14 q^{61} + q^{63} - q^{65} - 20 q^{69} - 6 q^{73} + 16 q^{75} + 4 q^{77} + 7 q^{79} - 4 q^{81} + 8 q^{83} - 15 q^{85} + 11 q^{87} + 9 q^{89} + 4 q^{91} + 13 q^{93} - 9 q^{95} - 16 q^{97} + q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 - q^5 + 4 * q^7 + q^9 + 4 * q^11 + 4 * q^13 - 5 * q^15 - 7 * q^17 - 9 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^25 - 9 * q^27 - 3 * q^29 - 3 * q^33 - q^35 - 18 * q^37 - 3 * q^39 - 14 * q^41 - q^43 + 11 * q^45 - 3 * q^47 + 4 * q^49 + 11 * q^51 + 4 * q^53 - q^55 + 6 * q^57 - 7 * q^59 - 14 * q^61 + q^63 - q^65 - 20 * q^69 - 6 * q^73 + 16 * q^75 + 4 * q^77 + 7 * q^79 - 4 * q^81 + 8 * q^83 - 15 * q^85 + 11 * q^87 + 9 * q^89 + 4 * q^91 + 13 * q^93 - 9 * q^95 - 16 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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