LMFDB - Embedded newform 4004.2.a.g.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:	$6$
Coefficient field:	6.6.246302029.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 9x^{4} + 14x^{3} + 15x^{2} - 13x - 10 $ x^6 - 2x^5 - 9x^4 + 14x^3 + 15x^2 - 13*x - 10
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			$3.12467$ 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)

$f(q)$	$=$	$q-3.12467 q^{3} -1.82266 q^{5} +1.00000 q^{7} +6.76356 q^{9} +O(q^{10})$	$q-3.12467 q^{3} -1.82266 q^{5} +1.00000 q^{7} +6.76356 q^{9} -1.00000 q^{11} -1.00000 q^{13} +5.69521 q^{15} -4.38678 q^{17} -0.0136540 q^{19} -3.12467 q^{21} +4.39320 q^{23} -1.67791 q^{25} -11.7599 q^{27} +2.69521 q^{29} -2.49576 q^{31} +3.12467 q^{33} -1.82266 q^{35} -2.23204 q^{37} +3.12467 q^{39} +6.14589 q^{41} +1.11824 q^{43} -12.3277 q^{45} +6.86701 q^{47} +1.00000 q^{49} +13.7072 q^{51} +5.30958 q^{53} +1.82266 q^{55} +0.0426643 q^{57} -7.05103 q^{59} -3.70611 q^{61} +6.76356 q^{63} +1.82266 q^{65} +4.94252 q^{67} -13.7273 q^{69} +7.14758 q^{71} +11.6509 q^{73} +5.24291 q^{75} -1.00000 q^{77} +5.87344 q^{79} +16.4551 q^{81} +3.12492 q^{83} +7.99560 q^{85} -8.42165 q^{87} -15.6308 q^{89} -1.00000 q^{91} +7.79844 q^{93} +0.0248866 q^{95} +4.38124 q^{97} -6.76356 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9	$ 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9} - 6 q^{11} - 6 q^{13} + 4 q^{15} - q^{17} - 12 q^{19} - 2 q^{21} + 5 q^{23} - 3 q^{25} - 8 q^{27} - 14 q^{29} - 4 q^{31} + 2 q^{33} - 3 q^{35} - 3 q^{37} + 2 q^{39} + 6 q^{41} - 14 q^{43} - 20 q^{45} + 2 q^{47} + 6 q^{49} - 5 q^{51} - 3 q^{53} + 3 q^{55} - 22 q^{57} + 2 q^{59} - 26 q^{61} + 4 q^{63} + 3 q^{65} + 9 q^{67} - 11 q^{69} + 3 q^{71} - 7 q^{73} - 6 q^{75} - 6 q^{77} + 6 q^{81} - 15 q^{83} + q^{85} - 23 q^{87} - q^{89} - 6 q^{91} + 8 q^{93} + 12 q^{95} - 16 q^{97} - 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9 - 6 * q^11 - 6 * q^13 + 4 * q^15 - q^17 - 12 * q^19 - 2 * q^21 + 5 * q^23 - 3 * q^25 - 8 * q^27 - 14 * q^29 - 4 * q^31 + 2 * q^33 - 3 * q^35 - 3 * q^37 + 2 * q^39 + 6 * q^41 - 14 * q^43 - 20 * q^45 + 2 * q^47 + 6 * q^49 - 5 * q^51 - 3 * q^53 + 3 * q^55 - 22 * q^57 + 2 * q^59 - 26 * q^61 + 4 * q^63 + 3 * q^65 + 9 * q^67 - 11 * q^69 + 3 * q^71 - 7 * q^73 - 6 * q^75 - 6 * q^77 + 6 * q^81 - 15 * q^83 + q^85 - 23 * q^87 - q^89 - 6 * q^91 + 8 * q^93 + 12 * q^95 - 16 * q^97 - 4 * 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
$3$	−3.12467	−1.80403	−0.902015	−	0.431705i	$-0.857912\pi$
$3$	−3.12467	−1.80403	−0.902015	+	0.431705i	$0.857912\pi$
$4$	0	0
$5$	−1.82266	−0.815118	−0.407559	−	0.913179i	$-0.633620\pi$
$5$	−1.82266	−0.815118	−0.407559	+	0.913179i	$0.633620\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	6.76356	2.25452
$10$	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
$15$	5.69521	1.47050
$16$	0	0
$17$	−4.38678	−1.06395	−0.531975	−	0.846760i	$-0.678550\pi$
$17$	−4.38678	−1.06395	−0.531975	+	0.846760i	$0.678550\pi$
$18$	0	0
$19$	−0.0136540	−0.00313244	−0.00156622	−	0.999999i	$-0.500499\pi$
$19$	−0.0136540	−0.00313244	−0.00156622	+	0.999999i	$0.500499\pi$
$20$	0	0
$21$	−3.12467	−0.681859
$22$	0	0
$23$	4.39320	0.916046	0.458023	−	0.888940i	$-0.348558\pi$
$23$	4.39320	0.916046	0.458023	+	0.888940i	$0.348558\pi$
$24$	0	0
$25$	−1.67791	−0.335582
$26$	0	0
$27$	−11.7599	−2.26319
$28$	0	0
$29$	2.69521	0.500488	0.250244	−	0.968183i	$-0.419489\pi$
$29$	2.69521	0.500488	0.250244	+	0.968183i	$0.419489\pi$
$30$	0	0
$31$	−2.49576	−0.448252	−0.224126	−	0.974560i	$-0.571953\pi$
$31$	−2.49576	−0.448252	−0.224126	+	0.974560i	$0.571953\pi$
$32$	0	0
$33$	3.12467	0.543935
$34$	0	0
$35$	−1.82266	−0.308086
$36$	0	0
$37$	−2.23204	−0.366945	−0.183472	−	0.983025i	$-0.558734\pi$
$37$	−2.23204	−0.366945	−0.183472	+	0.983025i	$0.558734\pi$
$38$	0	0
$39$	3.12467	0.500348
$40$	0	0
$41$	6.14589	0.959827	0.479913	−	0.877316i	$-0.340668\pi$
$41$	6.14589	0.959827	0.479913	+	0.877316i	$0.340668\pi$
$42$	0	0
$43$	1.11824	0.170531	0.0852653	−	0.996358i	$-0.472826\pi$
$43$	1.11824	0.170531	0.0852653	+	0.996358i	$0.472826\pi$
$44$	0	0
$45$	−12.3277	−1.83770
$46$	0	0
$47$	6.86701	1.00166	0.500829	−	0.865547i	$-0.333029\pi$
$47$	6.86701	1.00166	0.500829	+	0.865547i	$0.333029\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	13.7072	1.91940
$52$	0	0
$53$	5.30958	0.729326	0.364663	−	0.931139i	$-0.381184\pi$
$53$	5.30958	0.729326	0.364663	+	0.931139i	$0.381184\pi$
$54$	0	0
$55$	1.82266	0.245767
$56$	0	0
$57$	0.0426643	0.00565102
$58$	0	0
$59$	−7.05103	−0.917966	−0.458983	−	0.888445i	$-0.651786\pi$
$59$	−7.05103	−0.917966	−0.458983	+	0.888445i	$0.651786\pi$
$60$	0	0
$61$	−3.70611	−0.474518	−0.237259	−	0.971446i	$-0.576249\pi$
$61$	−3.70611	−0.474518	−0.237259	+	0.971446i	$0.576249\pi$
$62$	0	0
$63$	6.76356	0.852129
$64$	0	0
$65$	1.82266	0.226073
$66$	0	0
$67$	4.94252	0.603825	0.301913	−	0.953336i	$-0.402375\pi$
$67$	4.94252	0.603825	0.301913	+	0.953336i	$0.402375\pi$
$68$	0	0
$69$	−13.7273	−1.65257
$70$	0	0
$71$	7.14758	0.848262	0.424131	−	0.905601i	$-0.360580\pi$
$71$	7.14758	0.848262	0.424131	+	0.905601i	$0.360580\pi$
$72$	0	0
$73$	11.6509	1.36364	0.681818	−	0.731522i	$-0.261190\pi$
$73$	11.6509	1.36364	0.681818	+	0.731522i	$0.261190\pi$
$74$	0	0
$75$	5.24291	0.605400
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	5.87344	0.660814	0.330407	−	0.943839i	$-0.392814\pi$
$79$	5.87344	0.660814	0.330407	+	0.943839i	$0.392814\pi$
$80$	0	0
$81$	16.4551	1.82835
$82$	0	0
$83$	3.12492	0.343005	0.171502	−	0.985184i	$-0.445138\pi$
$83$	3.12492	0.343005	0.171502	+	0.985184i	$0.445138\pi$
$84$	0	0
$85$	7.99560	0.867245
$86$	0	0
$87$	−8.42165	−0.902896
$88$	0	0
$89$	−15.6308	−1.65686	−0.828432	−	0.560089i	$-0.810767\pi$
$89$	−15.6308	−1.65686	−0.828432	+	0.560089i	$0.810767\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	7.79844	0.808660
$94$	0	0
$95$	0.0248866	0.00255331
$96$	0	0
$97$	4.38124	0.444847	0.222424	−	0.974950i	$-0.428603\pi$
$97$	4.38124	0.444847	0.222424	+	0.974950i	$0.428603\pi$
$98$	0	0
$99$	−6.76356	−0.679764
$100$	0	0
$101$	−9.04747	−0.900257	−0.450128	−	0.892964i	$-0.648622\pi$
$101$	−9.04747	−0.900257	−0.450128	+	0.892964i	$0.648622\pi$
$102$	0	0
$103$	19.2014	1.89197	0.945984	−	0.324214i	$-0.105100\pi$
$103$	19.2014	1.89197	0.945984	+	0.324214i	$0.105100\pi$
$104$	0	0
$105$	5.69521	0.555796
$106$	0	0
$107$	16.3842	1.58392	0.791958	−	0.610575i	$-0.209062\pi$
$107$	16.3842	1.58392	0.791958	+	0.610575i	$0.209062\pi$
$108$	0	0
$109$	−18.4599	−1.76814	−0.884070	−	0.467355i	$-0.845207\pi$
$109$	−18.4599	−1.76814	−0.884070	+	0.467355i	$0.845207\pi$
$110$	0	0
$111$	6.97438	0.661979
$112$	0	0
$113$	−4.15003	−0.390402	−0.195201	−	0.980763i	$-0.562536\pi$
$113$	−4.15003	−0.390402	−0.195201	+	0.980763i	$0.562536\pi$
$114$	0	0
$115$	−8.00732	−0.746686
$116$	0	0
$117$	−6.76356	−0.625292
$118$	0	0
$119$	−4.38678	−0.402135
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−19.2039	−1.73156
$124$	0	0
$125$	12.1716	1.08866
$126$	0	0
$127$	−11.0000	−0.976091	−0.488045	−	0.872818i	$-0.662290\pi$
$127$	−11.0000	−0.976091	−0.488045	+	0.872818i	$0.662290\pi$
$128$	0	0
$129$	−3.49414	−0.307642
$130$	0	0
$131$	−4.92678	−0.430454	−0.215227	−	0.976564i	$-0.569049\pi$
$131$	−4.92678	−0.430454	−0.215227	+	0.976564i	$0.569049\pi$
$132$	0	0
$133$	−0.0136540	−0.00118395
$134$	0	0
$135$	21.4343	1.84477
$136$	0	0
$137$	9.80006	0.837276	0.418638	−	0.908153i	$-0.362508\pi$
$137$	9.80006	0.837276	0.418638	+	0.908153i	$0.362508\pi$
$138$	0	0
$139$	−6.51180	−0.552324	−0.276162	−	0.961111i	$-0.589063\pi$
$139$	−6.51180	−0.552324	−0.276162	+	0.961111i	$0.589063\pi$
$140$	0	0
$141$	−21.4572	−1.80702
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−4.91246	−0.407957
$146$	0	0
$147$	−3.12467	−0.257718
$148$	0	0
$149$	3.55582	0.291304	0.145652	−	0.989336i	$-0.453472\pi$
$149$	3.55582	0.291304	0.145652	+	0.989336i	$0.453472\pi$
$150$	0	0
$151$	−3.21472	−0.261610	−0.130805	−	0.991408i	$-0.541756\pi$
$151$	−3.21472	−0.261610	−0.130805	+	0.991408i	$0.541756\pi$
$152$	0	0
$153$	−29.6702	−2.39870
$154$	0	0
$155$	4.54893	0.365379
$156$	0	0
$157$	−16.9654	−1.35399	−0.676995	−	0.735988i	$-0.736718\pi$
$157$	−16.9654	−1.35399	−0.676995	+	0.735988i	$0.736718\pi$
$158$	0	0
$159$	−16.5907	−1.31573
$160$	0	0
$161$	4.39320	0.346233
$162$	0	0
$163$	−3.66506	−0.287069	−0.143535	−	0.989645i	$-0.545847\pi$
$163$	−3.66506	−0.287069	−0.143535	+	0.989645i	$0.545847\pi$
$164$	0	0
$165$	−5.69521	−0.443372
$166$	0	0
$167$	−17.8860	−1.38406	−0.692030	−	0.721868i	$-0.743284\pi$
$167$	−17.8860	−1.38406	−0.692030	+	0.721868i	$0.743284\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.0923497	−0.00706216
$172$	0	0
$173$	−4.01675	−0.305388	−0.152694	−	0.988274i	$-0.548795\pi$
$173$	−4.01675	−0.305388	−0.152694	+	0.988274i	$0.548795\pi$
$174$	0	0
$175$	−1.67791	−0.126838
$176$	0	0
$177$	22.0321	1.65604
$178$	0	0
$179$	11.8144	0.883053	0.441526	−	0.897248i	$-0.354437\pi$
$179$	11.8144	0.883053	0.441526	+	0.897248i	$0.354437\pi$
$180$	0	0
$181$	−19.4850	−1.44831	−0.724155	−	0.689637i	$-0.757770\pi$
$181$	−19.4850	−1.44831	−0.724155	+	0.689637i	$0.757770\pi$
$182$	0	0
$183$	11.5804	0.856045
$184$	0	0
$185$	4.06825	0.299103
$186$	0	0
$187$	4.38678	0.320793
$188$	0	0
$189$	−11.7599	−0.855407
$190$	0	0
$191$	−7.49053	−0.541996	−0.270998	−	0.962580i	$-0.587354\pi$
$191$	−7.49053	−0.541996	−0.270998	+	0.962580i	$0.587354\pi$
$192$	0	0
$193$	−18.6964	−1.34580	−0.672899	−	0.739735i	$-0.734951\pi$
$193$	−18.6964	−1.34580	−0.672899	+	0.739735i	$0.734951\pi$
$194$	0	0
$195$	−5.69521	−0.407843
$196$	0	0
$197$	−7.36150	−0.524485	−0.262243	−	0.965002i	$-0.584462\pi$
$197$	−7.36150	−0.524485	−0.262243	+	0.965002i	$0.584462\pi$
$198$	0	0
$199$	8.59580	0.609340	0.304670	−	0.952458i	$-0.401454\pi$
$199$	8.59580	0.609340	0.304670	+	0.952458i	$0.401454\pi$
$200$	0	0
$201$	−15.4438	−1.08932
$202$	0	0
$203$	2.69521	0.189167
$204$	0	0
$205$	−11.2019	−0.782373
$206$	0	0
$207$	29.7137	2.06525
$208$	0	0
$209$	0.0136540	0.000944467 0
$210$	0	0
$211$	−11.3550	−0.781712	−0.390856	−	0.920452i	$-0.627821\pi$
$211$	−11.3550	−0.781712	−0.390856	+	0.920452i	$0.627821\pi$
$212$	0	0
$213$	−22.3338	−1.53029
$214$	0	0
$215$	−2.03818	−0.139003
$216$	0	0
$217$	−2.49576	−0.169423
$218$	0	0
$219$	−36.4053	−2.46004
$220$	0	0
$221$	4.38678	0.295086
$222$	0	0
$223$	−6.70728	−0.449152	−0.224576	−	0.974457i	$-0.572100\pi$
$223$	−6.70728	−0.449152	−0.224576	+	0.974457i	$0.572100\pi$
$224$	0	0
$225$	−11.3486	−0.756577
$226$	0	0
$227$	26.9474	1.78856	0.894280	−	0.447508i	$-0.147689\pi$
$227$	26.9474	1.78856	0.894280	+	0.447508i	$0.147689\pi$
$228$	0	0
$229$	−19.3734	−1.28023	−0.640114	−	0.768280i	$-0.721113\pi$
$229$	−19.3734	−1.28023	−0.640114	+	0.768280i	$0.721113\pi$
$230$	0	0
$231$	3.12467	0.205588
$232$	0	0
$233$	10.0831	0.660568	0.330284	−	0.943882i	$-0.392856\pi$
$233$	10.0831	0.660568	0.330284	+	0.943882i	$0.392856\pi$
$234$	0	0
$235$	−12.5162	−0.816469
$236$	0	0
$237$	−18.3526	−1.19213
$238$	0	0
$239$	2.97017	0.192124	0.0960622	−	0.995375i	$-0.469375\pi$
$239$	2.97017	0.192124	0.0960622	+	0.995375i	$0.469375\pi$
$240$	0	0
$241$	−13.9998	−0.901806	−0.450903	−	0.892573i	$-0.648898\pi$
$241$	−13.9998	−0.901806	−0.450903	+	0.892573i	$0.648898\pi$
$242$	0	0
$243$	−16.1371	−1.03520
$244$	0	0
$245$	−1.82266	−0.116445
$246$	0	0
$247$	0.0136540	0.000868783 0
$248$	0	0
$249$	−9.76435	−0.618790
$250$	0	0
$251$	0.244447	0.0154293	0.00771467	−	0.999970i	$-0.497544\pi$
$251$	0.244447	0.0154293	0.00771467	+	0.999970i	$0.497544\pi$
$252$	0	0
$253$	−4.39320	−0.276198
$254$	0	0
$255$	−24.9836	−1.56454
$256$	0	0
$257$	4.31842	0.269376	0.134688	−	0.990888i	$-0.456997\pi$
$257$	4.31842	0.269376	0.134688	+	0.990888i	$0.456997\pi$
$258$	0	0
$259$	−2.23204	−0.138692
$260$	0	0
$261$	18.2292	1.12836
$262$	0	0
$263$	−10.3268	−0.636777	−0.318389	−	0.947960i	$-0.603142\pi$
$263$	−10.3268	−0.636777	−0.318389	+	0.947960i	$0.603142\pi$
$264$	0	0
$265$	−9.67755	−0.594487
$266$	0	0
$267$	48.8412	2.98903
$268$	0	0
$269$	−24.9769	−1.52287	−0.761436	−	0.648241i	$-0.775505\pi$
$269$	−24.9769	−1.52287	−0.761436	+	0.648241i	$0.775505\pi$
$270$	0	0
$271$	−10.0374	−0.609727	−0.304864	−	0.952396i	$-0.598611\pi$
$271$	−10.0374	−0.609727	−0.304864	+	0.952396i	$0.598611\pi$
$272$	0	0
$273$	3.12467	0.189114
$274$	0	0
$275$	1.67791	0.101182
$276$	0	0
$277$	−24.4678	−1.47013	−0.735063	−	0.677999i	$-0.762848\pi$
$277$	−24.4678	−1.47013	−0.735063	+	0.677999i	$0.762848\pi$
$278$	0	0
$279$	−16.8803	−1.01059
$280$	0	0
$281$	−10.8865	−0.649432	−0.324716	−	0.945812i	$-0.605269\pi$
$281$	−10.8865	−0.649432	−0.324716	+	0.945812i	$0.605269\pi$
$282$	0	0
$283$	−14.3285	−0.851740	−0.425870	−	0.904784i	$-0.640032\pi$
$283$	−14.3285	−0.851740	−0.425870	+	0.904784i	$0.640032\pi$
$284$	0	0
$285$	−0.0777624	−0.00460625
$286$	0	0
$287$	6.14589	0.362780
$288$	0	0
$289$	2.24380	0.131988
$290$	0	0
$291$	−13.6899	−0.802518
$292$	0	0
$293$	−9.70852	−0.567178	−0.283589	−	0.958946i	$-0.591525\pi$
$293$	−9.70852	−0.567178	−0.283589	+	0.958946i	$0.591525\pi$
$294$	0	0
$295$	12.8516	0.748251
$296$	0	0
$297$	11.7599	0.682378
$298$	0	0
$299$	−4.39320	−0.254065
$300$	0	0
$301$	1.11824	0.0644545
$302$	0	0
$303$	28.2704	1.62409
$304$	0	0
$305$	6.75497	0.386789
$306$	0	0
$307$	4.55828	0.260155	0.130077	−	0.991504i	$-0.458477\pi$
$307$	4.55828	0.260155	0.130077	+	0.991504i	$0.458477\pi$
$308$	0	0
$309$	−59.9980	−3.41316
$310$	0	0
$311$	25.3552	1.43776	0.718881	−	0.695133i	$-0.244655\pi$
$311$	25.3552	1.43776	0.718881	+	0.695133i	$0.244655\pi$
$312$	0	0
$313$	−22.5339	−1.27369	−0.636845	−	0.770991i	$-0.719761\pi$
$313$	−22.5339	−1.27369	−0.636845	+	0.770991i	$0.719761\pi$
$314$	0	0
$315$	−12.3277	−0.694586
$316$	0	0
$317$	5.44009	0.305546	0.152773	−	0.988261i	$-0.451180\pi$
$317$	5.44009	0.305546	0.152773	+	0.988261i	$0.451180\pi$
$318$	0	0
$319$	−2.69521	−0.150903
$320$	0	0
$321$	−51.1951	−2.85743
$322$	0	0
$323$	0.0598970	0.00333276
$324$	0	0
$325$	1.67791	0.0930737
$326$	0	0
$327$	57.6812	3.18978
$328$	0	0
$329$	6.86701	0.378591
$330$	0	0
$331$	−0.707352	−0.0388796	−0.0194398	−	0.999811i	$-0.506188\pi$
$331$	−0.707352	−0.0388796	−0.0194398	+	0.999811i	$0.506188\pi$
$332$	0	0
$333$	−15.0965	−0.827285
$334$	0	0
$335$	−9.00854	−0.492189
$336$	0	0
$337$	17.7811	0.968598	0.484299	−	0.874903i	$-0.339075\pi$
$337$	17.7811	0.968598	0.484299	+	0.874903i	$0.339075\pi$
$338$	0	0
$339$	12.9675	0.704297
$340$	0	0
$341$	2.49576	0.135153
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	25.0202	1.34704
$346$	0	0
$347$	0.161077	0.00864706	0.00432353	−	0.999991i	$-0.498624\pi$
$347$	0.161077	0.00864706	0.00432353	+	0.999991i	$0.498624\pi$
$348$	0	0
$349$	24.6632	1.32019	0.660096	−	0.751181i	$-0.270515\pi$
$349$	24.6632	1.32019	0.660096	+	0.751181i	$0.270515\pi$
$350$	0	0
$351$	11.7599	0.627697
$352$	0	0
$353$	−2.54700	−0.135563	−0.0677816	−	0.997700i	$-0.521592\pi$
$353$	−2.54700	−0.135563	−0.0677816	+	0.997700i	$0.521592\pi$
$354$	0	0
$355$	−13.0276	−0.691434
$356$	0	0
$357$	13.7072	0.725463
$358$	0	0
$359$	−14.4737	−0.763893	−0.381947	−	0.924184i	$-0.624746\pi$
$359$	−14.4737	−0.763893	−0.381947	+	0.924184i	$0.624746\pi$
$360$	0	0
$361$	−18.9998	−0.999990
$362$	0	0
$363$	−3.12467	−0.164003
$364$	0	0
$365$	−21.2357	−1.11152
$366$	0	0
$367$	−6.42732	−0.335504	−0.167752	−	0.985829i	$-0.553651\pi$
$367$	−6.42732	−0.335504	−0.167752	+	0.985829i	$0.553651\pi$
$368$	0	0
$369$	41.5681	2.16395
$370$	0	0
$371$	5.30958	0.275659
$372$	0	0
$373$	−5.82498	−0.301606	−0.150803	−	0.988564i	$-0.548186\pi$
$373$	−5.82498	−0.301606	−0.150803	+	0.988564i	$0.548186\pi$
$374$	0	0
$375$	−38.0321	−1.96397
$376$	0	0
$377$	−2.69521	−0.138810
$378$	0	0
$379$	−25.7697	−1.32370	−0.661850	−	0.749636i	$-0.730228\pi$
$379$	−25.7697	−1.32370	−0.661850	+	0.749636i	$0.730228\pi$
$380$	0	0
$381$	34.3713	1.76090
$382$	0	0
$383$	−6.45974	−0.330077	−0.165039	−	0.986287i	$-0.552775\pi$
$383$	−6.45974	−0.330077	−0.165039	+	0.986287i	$0.552775\pi$
$384$	0	0
$385$	1.82266	0.0928914
$386$	0	0
$387$	7.56331	0.384465
$388$	0	0
$389$	−14.8355	−0.752190	−0.376095	−	0.926581i	$-0.622733\pi$
$389$	−14.8355	−0.752190	−0.376095	+	0.926581i	$0.622733\pi$
$390$	0	0
$391$	−19.2720	−0.974627
$392$	0	0
$393$	15.3945	0.776552
$394$	0	0
$395$	−10.7053	−0.538641
$396$	0	0
$397$	32.4019	1.62620	0.813102	−	0.582122i	$-0.197777\pi$
$397$	32.4019	1.62620	0.813102	+	0.582122i	$0.197777\pi$
$398$	0	0
$399$	0.0426643	0.00213588
$400$	0	0
$401$	−15.8620	−0.792111	−0.396055	−	0.918227i	$-0.629621\pi$
$401$	−15.8620	−0.792111	−0.396055	+	0.918227i	$0.629621\pi$
$402$	0	0
$403$	2.49576	0.124323
$404$	0	0
$405$	−29.9921	−1.49032
$406$	0	0
$407$	2.23204	0.110638
$408$	0	0
$409$	−2.97147	−0.146930	−0.0734649	−	0.997298i	$-0.523406\pi$
$409$	−2.97147	−0.146930	−0.0734649	+	0.997298i	$0.523406\pi$
$410$	0	0
$411$	−30.6220	−1.51047
$412$	0	0
$413$	−7.05103	−0.346959
$414$	0	0
$415$	−5.69567	−0.279589
$416$	0	0
$417$	20.3472	0.996408
$418$	0	0
$419$	2.44455	0.119424	0.0597119	−	0.998216i	$-0.480982\pi$
$419$	2.44455	0.119424	0.0597119	+	0.998216i	$0.480982\pi$
$420$	0	0
$421$	4.17732	0.203590	0.101795	−	0.994805i	$-0.467541\pi$
$421$	4.17732	0.203590	0.101795	+	0.994805i	$0.467541\pi$
$422$	0	0
$423$	46.4455	2.25826
$424$	0	0
$425$	7.36061	0.357042
$426$	0	0
$427$	−3.70611	−0.179351
$428$	0	0
$429$	−3.12467	−0.150861
$430$	0	0
$431$	6.48872	0.312550	0.156275	−	0.987714i	$-0.450051\pi$
$431$	6.48872	0.312550	0.156275	+	0.987714i	$0.450051\pi$
$432$	0	0
$433$	11.8198	0.568024	0.284012	−	0.958821i	$-0.408334\pi$
$433$	11.8198	0.568024	0.284012	+	0.958821i	$0.408334\pi$
$434$	0	0
$435$	15.3498	0.735967
$436$	0	0
$437$	−0.0599848	−0.00286946
$438$	0	0
$439$	33.4060	1.59438	0.797191	−	0.603728i	$-0.206319\pi$
$439$	33.4060	1.59438	0.797191	+	0.603728i	$0.206319\pi$
$440$	0	0
$441$	6.76356	0.322074
$442$	0	0
$443$	−4.94400	−0.234897	−0.117448	−	0.993079i	$-0.537471\pi$
$443$	−4.94400	−0.234897	−0.117448	+	0.993079i	$0.537471\pi$
$444$	0	0
$445$	28.4897	1.35054
$446$	0	0
$447$	−11.1108	−0.525521
$448$	0	0
$449$	27.3588	1.29114	0.645570	−	0.763701i	$-0.276620\pi$
$449$	27.3588	1.29114	0.645570	+	0.763701i	$0.276620\pi$
$450$	0	0
$451$	−6.14589	−0.289399
$452$	0	0
$453$	10.0449	0.471952
$454$	0	0
$455$	1.82266	0.0854476
$456$	0	0
$457$	3.53798	0.165500	0.0827499	−	0.996570i	$-0.473630\pi$
$457$	3.53798	0.165500	0.0827499	+	0.996570i	$0.473630\pi$
$458$	0	0
$459$	51.5880	2.40792
$460$	0	0
$461$	22.4209	1.04424	0.522122	−	0.852871i	$-0.325140\pi$
$461$	22.4209	1.04424	0.522122	+	0.852871i	$0.325140\pi$
$462$	0	0
$463$	−0.171252	−0.00795875	−0.00397937	−	0.999992i	$-0.501267\pi$
$463$	−0.171252	−0.00795875	−0.00397937	+	0.999992i	$0.501267\pi$
$464$	0	0
$465$	−14.2139	−0.659154
$466$	0	0
$467$	5.67011	0.262382	0.131191	−	0.991357i	$-0.458120\pi$
$467$	5.67011	0.262382	0.131191	+	0.991357i	$0.458120\pi$
$468$	0	0
$469$	4.94252	0.228225
$470$	0	0
$471$	53.0114	2.44264
$472$	0	0
$473$	−1.11824	−0.0514169
$474$	0	0
$475$	0.0229102	0.00105119
$476$	0	0
$477$	35.9117	1.64428
$478$	0	0
$479$	−8.66786	−0.396044	−0.198022	−	0.980198i	$-0.563452\pi$
$479$	−8.66786	−0.396044	−0.198022	+	0.980198i	$0.563452\pi$
$480$	0	0
$481$	2.23204	0.101772
$482$	0	0
$483$	−13.7273	−0.624614
$484$	0	0
$485$	−7.98551	−0.362603
$486$	0	0
$487$	28.1801	1.27696	0.638482	−	0.769637i	$-0.279563\pi$
$487$	28.1801	1.27696	0.638482	+	0.769637i	$0.279563\pi$
$488$	0	0
$489$	11.4521	0.517882
$490$	0	0
$491$	−16.5831	−0.748385	−0.374193	−	0.927351i	$-0.622080\pi$
$491$	−16.5831	−0.748385	−0.374193	+	0.927351i	$0.622080\pi$
$492$	0	0
$493$	−11.8233	−0.532494
$494$	0	0
$495$	12.3277	0.554088
$496$	0	0
$497$	7.14758	0.320613
$498$	0	0
$499$	−41.9268	−1.87690	−0.938451	−	0.345411i	$-0.887739\pi$
$499$	−41.9268	−1.87690	−0.938451	+	0.345411i	$0.887739\pi$
$500$	0	0
$501$	55.8879	2.49689
$502$	0	0
$503$	−40.4707	−1.80450	−0.902249	−	0.431215i	$-0.858085\pi$
$503$	−40.4707	−1.80450	−0.902249	+	0.431215i	$0.858085\pi$
$504$	0	0
$505$	16.4905	0.733816
$506$	0	0
$507$	−3.12467	−0.138771
$508$	0	0
$509$	7.59349	0.336576	0.168288	−	0.985738i	$-0.446176\pi$
$509$	7.59349	0.336576	0.168288	+	0.985738i	$0.446176\pi$
$510$	0	0
$511$	11.6509	0.515406
$512$	0	0
$513$	0.160570	0.00708932
$514$	0	0
$515$	−34.9976	−1.54218
$516$	0	0
$517$	−6.86701	−0.302011
$518$	0	0
$519$	12.5510	0.550929
$520$	0	0
$521$	−12.2302	−0.535816	−0.267908	−	0.963444i	$-0.586332\pi$
$521$	−12.2302	−0.535816	−0.267908	+	0.963444i	$0.586332\pi$
$522$	0	0
$523$	−12.8192	−0.560545	−0.280273	−	0.959920i	$-0.590425\pi$
$523$	−12.8192	−0.560545	−0.280273	+	0.959920i	$0.590425\pi$
$524$	0	0
$525$	5.24291	0.228820
$526$	0	0
$527$	10.9484	0.476918
$528$	0	0
$529$	−3.69977	−0.160860
$530$	0	0
$531$	−47.6901	−2.06957
$532$	0	0
$533$	−6.14589	−0.266208
$534$	0	0
$535$	−29.8628	−1.29108
$536$	0	0
$537$	−36.9162	−1.59305
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−18.7889	−0.807800	−0.403900	−	0.914803i	$-0.632346\pi$
$541$	−18.7889	−0.807800	−0.403900	+	0.914803i	$0.632346\pi$
$542$	0	0
$543$	60.8843	2.61279
$544$	0	0
$545$	33.6462	1.44124
$546$	0	0
$547$	3.75253	0.160446	0.0802232	−	0.996777i	$-0.474437\pi$
$547$	3.75253	0.160446	0.0802232	+	0.996777i	$0.474437\pi$
$548$	0	0
$549$	−25.0665	−1.06981
$550$	0	0
$551$	−0.0368004	−0.00156775
$552$	0	0
$553$	5.87344	0.249764
$554$	0	0
$555$	−12.7119	−0.539591
$556$	0	0
$557$	26.2043	1.11031	0.555156	−	0.831746i	$-0.312659\pi$
$557$	26.2043	1.11031	0.555156	+	0.831746i	$0.312659\pi$
$558$	0	0
$559$	−1.11824	−0.0472967
$560$	0	0
$561$	−13.7072	−0.578720
$562$	0	0
$563$	32.4757	1.36869	0.684344	−	0.729159i	$-0.260089\pi$
$563$	32.4757	1.36869	0.684344	+	0.729159i	$0.260089\pi$
$564$	0	0
$565$	7.56410	0.318224
$566$	0	0
$567$	16.4551	0.691050
$568$	0	0
$569$	15.3802	0.644772	0.322386	−	0.946608i	$-0.395515\pi$
$569$	15.3802	0.644772	0.322386	+	0.946608i	$0.395515\pi$
$570$	0	0
$571$	−5.71809	−0.239295	−0.119647	−	0.992816i	$-0.538176\pi$
$571$	−5.71809	−0.239295	−0.119647	+	0.992816i	$0.538176\pi$
$572$	0	0
$573$	23.4055	0.977777
$574$	0	0
$575$	−7.37140	−0.307408
$576$	0	0
$577$	−39.7445	−1.65458	−0.827292	−	0.561773i	$-0.810120\pi$
$577$	−39.7445	−1.65458	−0.827292	+	0.561773i	$0.810120\pi$
$578$	0	0
$579$	58.4201	2.42786
$580$	0	0
$581$	3.12492	0.129644
$582$	0	0
$583$	−5.30958	−0.219900
$584$	0	0
$585$	12.3277	0.509687
$586$	0	0
$587$	−47.3947	−1.95619	−0.978095	−	0.208159i	$-0.933253\pi$
$587$	−47.3947	−1.95619	−0.978095	+	0.208159i	$0.933253\pi$
$588$	0	0
$589$	0.0340772	0.00140412
$590$	0	0
$591$	23.0023	0.946187
$592$	0	0
$593$	33.8272	1.38912	0.694559	−	0.719436i	$-0.255600\pi$
$593$	33.8272	1.38912	0.694559	+	0.719436i	$0.255600\pi$
$594$	0	0
$595$	7.99560	0.327788
$596$	0	0
$597$	−26.8590	−1.09927
$598$	0	0
$599$	28.1775	1.15130	0.575650	−	0.817696i	$-0.304749\pi$
$599$	28.1775	1.15130	0.575650	+	0.817696i	$0.304749\pi$
$600$	0	0
$601$	−26.3721	−1.07574	−0.537871	−	0.843027i	$-0.680771\pi$
$601$	−26.3721	−1.07574	−0.537871	+	0.843027i	$0.680771\pi$
$602$	0	0
$603$	33.4291	1.36134
$604$	0	0
$605$	−1.82266	−0.0741017
$606$	0	0
$607$	−33.6484	−1.36575	−0.682874	−	0.730536i	$-0.739270\pi$
$607$	−33.6484	−1.36575	−0.682874	+	0.730536i	$0.739270\pi$
$608$	0	0
$609$	−8.42165	−0.341262
$610$	0	0
$611$	−6.86701	−0.277810
$612$	0	0
$613$	−1.20526	−0.0486801	−0.0243400	−	0.999704i	$-0.507748\pi$
$613$	−1.20526	−0.0486801	−0.0243400	+	0.999704i	$0.507748\pi$
$614$	0	0
$615$	35.0022	1.41142
$616$	0	0
$617$	2.56665	0.103330	0.0516648	−	0.998664i	$-0.483547\pi$
$617$	2.56665	0.103330	0.0516648	+	0.998664i	$0.483547\pi$
$618$	0	0
$619$	44.0572	1.77081	0.885405	−	0.464820i	$-0.153881\pi$
$619$	44.0572	1.77081	0.885405	+	0.464820i	$0.153881\pi$
$620$	0	0
$621$	−51.6636	−2.07319
$622$	0	0
$623$	−15.6308	−0.626236
$624$	0	0
$625$	−13.7951	−0.551803
$626$	0	0
$627$	−0.0426643	−0.00170385
$628$	0	0
$629$	9.79145	0.390411
$630$	0	0
$631$	17.0871	0.680228	0.340114	−	0.940384i	$-0.389534\pi$
$631$	17.0871	0.680228	0.340114	+	0.940384i	$0.389534\pi$
$632$	0	0
$633$	35.4807	1.41023
$634$	0	0
$635$	20.0492	0.795629
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	48.3431	1.91242
$640$	0	0
$641$	−1.94587	−0.0768574	−0.0384287	−	0.999261i	$-0.512235\pi$
$641$	−1.94587	−0.0768574	−0.0384287	+	0.999261i	$0.512235\pi$
$642$	0	0
$643$	−34.9480	−1.37822	−0.689108	−	0.724659i	$-0.741997\pi$
$643$	−34.9480	−1.37822	−0.689108	+	0.724659i	$0.741997\pi$
$644$	0	0
$645$	6.36864	0.250765
$646$	0	0
$647$	−24.3754	−0.958298	−0.479149	−	0.877734i	$-0.659055\pi$
$647$	−24.3754	−0.958298	−0.479149	+	0.877734i	$0.659055\pi$
$648$	0	0
$649$	7.05103	0.276777
$650$	0	0
$651$	7.79844	0.305645
$652$	0	0
$653$	−6.71198	−0.262660	−0.131330	−	0.991339i	$-0.541925\pi$
$653$	−6.71198	−0.262660	−0.131330	+	0.991339i	$0.541925\pi$
$654$	0	0
$655$	8.97984	0.350871
$656$	0	0
$657$	78.8017	3.07435
$658$	0	0
$659$	−8.24764	−0.321282	−0.160641	−	0.987013i	$-0.551356\pi$
$659$	−8.24764	−0.321282	−0.160641	+	0.987013i	$0.551356\pi$
$660$	0	0
$661$	−10.0054	−0.389163	−0.194582	−	0.980886i	$-0.562335\pi$
$661$	−10.0054	−0.389163	−0.194582	+	0.980886i	$0.562335\pi$
$662$	0	0
$663$	−13.7072	−0.532345
$664$	0	0
$665$	0.0248866	0.000965061 0
$666$	0	0
$667$	11.8406	0.458470
$668$	0	0
$669$	20.9580	0.810284
$670$	0	0
$671$	3.70611	0.143073
$672$	0	0
$673$	−22.5687	−0.869961	−0.434981	−	0.900440i	$-0.643245\pi$
$673$	−22.5687	−0.869961	−0.434981	+	0.900440i	$0.643245\pi$
$674$	0	0
$675$	19.7320	0.759487
$676$	0	0
$677$	41.6884	1.60221	0.801107	−	0.598521i	$-0.204245\pi$
$677$	41.6884	1.60221	0.801107	+	0.598521i	$0.204245\pi$
$678$	0	0
$679$	4.38124	0.168137
$680$	0	0
$681$	−84.2016	−3.22661
$682$	0	0
$683$	0.549745	0.0210354	0.0105177	−	0.999945i	$-0.496652\pi$
$683$	0.549745	0.0210354	0.0105177	+	0.999945i	$0.496652\pi$
$684$	0	0
$685$	−17.8622	−0.682479
$686$	0	0
$687$	60.5354	2.30957
$688$	0	0
$689$	−5.30958	−0.202279
$690$	0	0
$691$	0.204361	0.00777427	0.00388714	−	0.999992i	$-0.498763\pi$
$691$	0.204361	0.00777427	0.00388714	+	0.999992i	$0.498763\pi$
$692$	0	0
$693$	−6.76356	−0.256927
$694$	0	0
$695$	11.8688	0.450209
$696$	0	0
$697$	−26.9606	−1.02121
$698$	0	0
$699$	−31.5065	−1.19168
$700$	0	0
$701$	−45.7851	−1.72928	−0.864639	−	0.502393i	$-0.832453\pi$
$701$	−45.7851	−1.72928	−0.864639	+	0.502393i	$0.832453\pi$
$702$	0	0
$703$	0.0304762	0.00114943
$704$	0	0
$705$	39.1091	1.47293
$706$	0	0
$707$	−9.04747	−0.340265
$708$	0	0
$709$	3.66929	0.137803	0.0689015	−	0.997623i	$-0.478051\pi$
$709$	3.66929	0.137803	0.0689015	+	0.997623i	$0.478051\pi$
$710$	0	0
$711$	39.7254	1.48982
$712$	0	0
$713$	−10.9644	−0.410620
$714$	0	0
$715$	−1.82266	−0.0681636
$716$	0	0
$717$	−9.28080	−0.346598
$718$	0	0
$719$	−1.17000	−0.0436336	−0.0218168	−	0.999762i	$-0.506945\pi$
$719$	−1.17000	−0.0436336	−0.0218168	+	0.999762i	$0.506945\pi$
$720$	0	0
$721$	19.2014	0.715096
$722$	0	0
$723$	43.7448	1.62689
$724$	0	0
$725$	−4.52232	−0.167955
$726$	0	0
$727$	−3.97603	−0.147463	−0.0737315	−	0.997278i	$-0.523491\pi$
$727$	−3.97603	−0.147463	−0.0737315	+	0.997278i	$0.523491\pi$
$728$	0	0
$729$	1.05779	0.0391773
$730$	0	0
$731$	−4.90549	−0.181436
$732$	0	0
$733$	34.6403	1.27947	0.639735	−	0.768596i	$-0.279044\pi$
$733$	34.6403	1.27947	0.639735	+	0.768596i	$0.279044\pi$
$734$	0	0
$735$	5.69521	0.210071
$736$	0	0
$737$	−4.94252	−0.182060
$738$	0	0
$739$	−32.7617	−1.20516	−0.602579	−	0.798059i	$-0.705860\pi$
$739$	−32.7617	−1.20516	−0.602579	+	0.798059i	$0.705860\pi$
$740$	0	0
$741$	−0.0426643	−0.00156731
$742$	0	0
$743$	1.38728	0.0508944	0.0254472	−	0.999676i	$-0.491899\pi$
$743$	1.38728	0.0508944	0.0254472	+	0.999676i	$0.491899\pi$
$744$	0	0
$745$	−6.48105	−0.237447
$746$	0	0
$747$	21.1356	0.773311
$748$	0	0
$749$	16.3842	0.598664
$750$	0	0
$751$	36.1089	1.31763	0.658816	−	0.752304i	$-0.271058\pi$
$751$	36.1089	1.31763	0.658816	+	0.752304i	$0.271058\pi$
$752$	0	0
$753$	−0.763815	−0.0278350
$754$	0	0
$755$	5.85934	0.213243
$756$	0	0
$757$	−44.9415	−1.63343	−0.816713	−	0.577044i	$-0.804206\pi$
$757$	−44.9415	−1.63343	−0.816713	+	0.577044i	$0.804206\pi$
$758$	0	0
$759$	13.7273	0.498270
$760$	0	0
$761$	33.3115	1.20754	0.603771	−	0.797158i	$-0.293664\pi$
$761$	33.3115	1.20754	0.603771	+	0.797158i	$0.293664\pi$
$762$	0	0
$763$	−18.4599	−0.668294
$764$	0	0
$765$	54.0788	1.95522
$766$	0	0
$767$	7.05103	0.254598
$768$	0	0
$769$	−28.3272	−1.02151	−0.510753	−	0.859727i	$-0.670633\pi$
$769$	−28.3272	−1.02151	−0.510753	+	0.859727i	$0.670633\pi$
$770$	0	0
$771$	−13.4937	−0.485962
$772$	0	0
$773$	−44.1261	−1.58710	−0.793552	−	0.608502i	$-0.791771\pi$
$773$	−44.1261	−1.58710	−0.793552	+	0.608502i	$0.791771\pi$
$774$	0	0
$775$	4.18767	0.150425
$776$	0	0
$777$	6.97438	0.250205
$778$	0	0
$779$	−0.0839160	−0.00300660
$780$	0	0
$781$	−7.14758	−0.255761
$782$	0	0
$783$	−31.6954	−1.13270
$784$	0	0
$785$	30.9222	1.10366
$786$	0	0
$787$	13.7658	0.490696	0.245348	−	0.969435i	$-0.421098\pi$
$787$	13.7658	0.490696	0.245348	+	0.969435i	$0.421098\pi$
$788$	0	0
$789$	32.2678	1.14876
$790$	0	0
$791$	−4.15003	−0.147558
$792$	0	0
$793$	3.70611	0.131608
$794$	0	0
$795$	30.2392	1.07247
$796$	0	0
$797$	−49.7871	−1.76355	−0.881774	−	0.471672i	$-0.843651\pi$
$797$	−49.7871	−1.76355	−0.881774	+	0.471672i	$0.843651\pi$
$798$	0	0
$799$	−30.1241	−1.06571
$800$	0	0
$801$	−105.720	−3.73544
$802$	0	0
$803$	−11.6509	−0.411152
$804$	0	0
$805$	−8.00732	−0.282221
$806$	0	0
$807$	78.0447	2.74730
$808$	0	0
$809$	−8.79766	−0.309309	−0.154655	−	0.987969i	$-0.549426\pi$
$809$	−8.79766	−0.309309	−0.154655	+	0.987969i	$0.549426\pi$
$810$	0	0
$811$	7.19309	0.252584	0.126292	−	0.991993i	$-0.459692\pi$
$811$	7.19309	0.252584	0.126292	+	0.991993i	$0.459692\pi$
$812$	0	0
$813$	31.3635	1.09997
$814$	0	0
$815$	6.68015	0.233996
$816$	0	0
$817$	−0.0152685	−0.000534177 0
$818$	0	0
$819$	−6.76356	−0.236338
$820$	0	0
$821$	−52.9406	−1.84764	−0.923820	−	0.382826i	$-0.874951\pi$
$821$	−52.9406	−1.84764	−0.923820	+	0.382826i	$0.874951\pi$
$822$	0	0
$823$	−39.9359	−1.39208	−0.696040	−	0.718003i	$-0.745056\pi$
$823$	−39.9359	−1.39208	−0.696040	+	0.718003i	$0.745056\pi$
$824$	0	0
$825$	−5.24291	−0.182535
$826$	0	0
$827$	−3.25252	−0.113101	−0.0565507	−	0.998400i	$-0.518010\pi$
$827$	−3.25252	−0.113101	−0.0565507	+	0.998400i	$0.518010\pi$
$828$	0	0
$829$	37.6199	1.30659	0.653297	−	0.757102i	$-0.273385\pi$
$829$	37.6199	1.30659	0.653297	+	0.757102i	$0.273385\pi$
$830$	0	0
$831$	76.4538	2.65215
$832$	0	0
$833$	−4.38678	−0.151993
$834$	0	0
$835$	32.6001	1.12817
$836$	0	0
$837$	29.3499	1.01448
$838$	0	0
$839$	−32.3905	−1.11824	−0.559121	−	0.829086i	$-0.688861\pi$
$839$	−32.3905	−1.11824	−0.559121	+	0.829086i	$0.688861\pi$
$840$	0	0
$841$	−21.7358	−0.749511
$842$	0	0
$843$	34.0166	1.17159
$844$	0	0
$845$	−1.82266	−0.0627014
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	44.7718	1.53656
$850$	0	0
$851$	−9.80579	−0.336138
$852$	0	0
$853$	−23.7785	−0.814160	−0.407080	−	0.913392i	$-0.633453\pi$
$853$	−23.7785	−0.814160	−0.407080	+	0.913392i	$0.633453\pi$
$854$	0	0
$855$	0.168322	0.00575650
$856$	0	0
$857$	−24.0940	−0.823036	−0.411518	−	0.911402i	$-0.635001\pi$
$857$	−24.0940	−0.823036	−0.411518	+	0.911402i	$0.635001\pi$
$858$	0	0
$859$	14.3143	0.488398	0.244199	−	0.969725i	$-0.421475\pi$
$859$	14.3143	0.488398	0.244199	+	0.969725i	$0.421475\pi$
$860$	0	0
$861$	−19.2039	−0.654467
$862$	0	0
$863$	−35.7621	−1.21736	−0.608678	−	0.793417i	$-0.708300\pi$
$863$	−35.7621	−1.21736	−0.608678	+	0.793417i	$0.708300\pi$
$864$	0	0
$865$	7.32117	0.248927
$866$	0	0
$867$	−7.01114	−0.238111
$868$	0	0
$869$	−5.87344	−0.199243
$870$	0	0
$871$	−4.94252	−0.167471
$872$	0	0
$873$	29.6328	1.00292
$874$	0	0
$875$	12.1716	0.411474
$876$	0	0
$877$	−8.86718	−0.299423	−0.149712	−	0.988730i	$-0.547835\pi$
$877$	−8.86718	−0.299423	−0.149712	+	0.988730i	$0.547835\pi$
$878$	0	0
$879$	30.3359	1.02321
$880$	0	0
$881$	17.8036	0.599819	0.299909	−	0.953968i	$-0.403044\pi$
$881$	17.8036	0.599819	0.299909	+	0.953968i	$0.403044\pi$
$882$	0	0
$883$	19.9471	0.671273	0.335637	−	0.941992i	$-0.391049\pi$
$883$	19.9471	0.671273	0.335637	+	0.941992i	$0.391049\pi$
$884$	0	0
$885$	−40.1571	−1.34987
$886$	0	0
$887$	−15.9006	−0.533890	−0.266945	−	0.963712i	$-0.586014\pi$
$887$	−15.9006	−0.533890	−0.266945	+	0.963712i	$0.586014\pi$
$888$	0	0
$889$	−11.0000	−0.368928
$890$	0	0
$891$	−16.4551	−0.551267
$892$	0	0
$893$	−0.0937622	−0.00313763
$894$	0	0
$895$	−21.5337	−0.719792
$896$	0	0
$897$	13.7273	0.458342
$898$	0	0
$899$	−6.72661	−0.224345
$900$	0	0
$901$	−23.2919	−0.775966
$902$	0	0
$903$	−3.49414	−0.116278
$904$	0	0
$905$	35.5146	1.18054
$906$	0	0
$907$	−1.42371	−0.0472735	−0.0236368	−	0.999721i	$-0.507525\pi$
$907$	−1.42371	−0.0472735	−0.0236368	+	0.999721i	$0.507525\pi$
$908$	0	0
$909$	−61.1931	−2.02965
$910$	0	0
$911$	24.1305	0.799479	0.399739	−	0.916629i	$-0.369101\pi$
$911$	24.1305	0.799479	0.399739	+	0.916629i	$0.369101\pi$
$912$	0	0
$913$	−3.12492	−0.103420
$914$	0	0
$915$	−21.1071	−0.697778
$916$	0	0
$917$	−4.92678	−0.162696
$918$	0	0
$919$	18.9315	0.624492	0.312246	−	0.950001i	$-0.398919\pi$
$919$	18.9315	0.624492	0.312246	+	0.950001i	$0.398919\pi$
$920$	0	0
$921$	−14.2431	−0.469327
$922$	0	0
$923$	−7.14758	−0.235265
$924$	0	0
$925$	3.74516	0.123140
$926$	0	0
$927$	129.870	4.26548
$928$	0	0
$929$	16.2805	0.534145	0.267073	−	0.963676i	$-0.413944\pi$
$929$	16.2805	0.534145	0.267073	+	0.963676i	$0.413944\pi$
$930$	0	0
$931$	−0.0136540	−0.000447492 0
$932$	0	0
$933$	−79.2267	−2.59377
$934$	0	0
$935$	−7.99560	−0.261484
$936$	0	0
$937$	30.2142	0.987056	0.493528	−	0.869730i	$-0.335707\pi$
$937$	30.2142	0.987056	0.493528	+	0.869730i	$0.335707\pi$
$938$	0	0
$939$	70.4110	2.29778
$940$	0	0
$941$	46.9452	1.53037	0.765186	−	0.643810i	$-0.222647\pi$
$941$	46.9452	1.53037	0.765186	+	0.643810i	$0.222647\pi$
$942$	0	0
$943$	27.0001	0.879246
$944$	0	0
$945$	21.4343	0.697258
$946$	0	0
$947$	38.2350	1.24247	0.621235	−	0.783624i	$-0.286631\pi$
$947$	38.2350	1.24247	0.621235	+	0.783624i	$0.286631\pi$
$948$	0	0
$949$	−11.6509	−0.378205
$950$	0	0
$951$	−16.9985	−0.551214
$952$	0	0
$953$	59.0668	1.91336	0.956681	−	0.291139i	$-0.0940342\pi$
$953$	59.0668	1.91336	0.956681	+	0.291139i	$0.0940342\pi$
$954$	0	0
$955$	13.6527	0.441791
$956$	0	0
$957$	8.42165	0.272233
$958$	0	0
$959$	9.80006	0.316460
$960$	0	0
$961$	−24.7712	−0.799070
$962$	0	0
$963$	110.815	3.57097
$964$	0	0
$965$	34.0772	1.09698
$966$	0	0
$967$	10.1685	0.326998	0.163499	−	0.986544i	$-0.447722\pi$
$967$	10.1685	0.326998	0.163499	+	0.986544i	$0.447722\pi$
$968$	0	0
$969$	−0.187159	−0.00601240
$970$	0	0
$971$	−58.2420	−1.86907	−0.934537	−	0.355866i	$-0.884186\pi$
$971$	−58.2420	−1.86907	−0.934537	+	0.355866i	$0.884186\pi$
$972$	0	0
$973$	−6.51180	−0.208759
$974$	0	0
$975$	−5.24291	−0.167908
$976$	0	0
$977$	3.73286	0.119425	0.0597124	−	0.998216i	$-0.480982\pi$
$977$	3.73286	0.119425	0.0597124	+	0.998216i	$0.480982\pi$
$978$	0	0
$979$	15.6308	0.499563
$980$	0	0
$981$	−124.855	−3.98631
$982$	0	0
$983$	24.9773	0.796651	0.398326	−	0.917244i	$-0.369591\pi$
$983$	24.9773	0.796651	0.398326	+	0.917244i	$0.369591\pi$
$984$	0	0
$985$	13.4175	0.427518
$986$	0	0
$987$	−21.4572	−0.682989
$988$	0	0
$989$	4.91267	0.156214
$990$	0	0
$991$	8.70612	0.276559	0.138279	−	0.990393i	$-0.455843\pi$
$991$	8.70612	0.276559	0.138279	+	0.990393i	$0.455843\pi$
$992$	0	0
$993$	2.21024	0.0701399
$994$	0	0
$995$	−15.6672	−0.496684
$996$	0	0
$997$	45.2513	1.43312	0.716561	−	0.697524i	$-0.245715\pi$
$997$	45.2513	1.43312	0.716561	+	0.697524i	$0.245715\pi$
$998$	0	0
$999$	26.2485	0.830467

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.g.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.g.1.1	✓	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.12467 q^{3} -1.82266 q^{5} +1.00000 q^{7} +6.76356 q^{9} +O(q^{10})\)	\(q-3.12467 q^{3} -1.82266 q^{5} +1.00000 q^{7} +6.76356 q^{9} -1.00000 q^{11} -1.00000 q^{13} +5.69521 q^{15} -4.38678 q^{17} -0.0136540 q^{19} -3.12467 q^{21} +4.39320 q^{23} -1.67791 q^{25} -11.7599 q^{27} +2.69521 q^{29} -2.49576 q^{31} +3.12467 q^{33} -1.82266 q^{35} -2.23204 q^{37} +3.12467 q^{39} +6.14589 q^{41} +1.11824 q^{43} -12.3277 q^{45} +6.86701 q^{47} +1.00000 q^{49} +13.7072 q^{51} +5.30958 q^{53} +1.82266 q^{55} +0.0426643 q^{57} -7.05103 q^{59} -3.70611 q^{61} +6.76356 q^{63} +1.82266 q^{65} +4.94252 q^{67} -13.7273 q^{69} +7.14758 q^{71} +11.6509 q^{73} +5.24291 q^{75} -1.00000 q^{77} +5.87344 q^{79} +16.4551 q^{81} +3.12492 q^{83} +7.99560 q^{85} -8.42165 q^{87} -15.6308 q^{89} -1.00000 q^{91} +7.79844 q^{93} +0.0248866 q^{95} +4.38124 q^{97} -6.76356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9	\( 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9} - 6 q^{11} - 6 q^{13} + 4 q^{15} - q^{17} - 12 q^{19} - 2 q^{21} + 5 q^{23} - 3 q^{25} - 8 q^{27} - 14 q^{29} - 4 q^{31} + 2 q^{33} - 3 q^{35} - 3 q^{37} + 2 q^{39} + 6 q^{41} - 14 q^{43} - 20 q^{45} + 2 q^{47} + 6 q^{49} - 5 q^{51} - 3 q^{53} + 3 q^{55} - 22 q^{57} + 2 q^{59} - 26 q^{61} + 4 q^{63} + 3 q^{65} + 9 q^{67} - 11 q^{69} + 3 q^{71} - 7 q^{73} - 6 q^{75} - 6 q^{77} + 6 q^{81} - 15 q^{83} + q^{85} - 23 q^{87} - q^{89} - 6 q^{91} + 8 q^{93} + 12 q^{95} - 16 q^{97} - 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9 - 6 * q^11 - 6 * q^13 + 4 * q^15 - q^17 - 12 * q^19 - 2 * q^21 + 5 * q^23 - 3 * q^25 - 8 * q^27 - 14 * q^29 - 4 * q^31 + 2 * q^33 - 3 * q^35 - 3 * q^37 + 2 * q^39 + 6 * q^41 - 14 * q^43 - 20 * q^45 + 2 * q^47 + 6 * q^49 - 5 * q^51 - 3 * q^53 + 3 * q^55 - 22 * q^57 + 2 * q^59 - 26 * q^61 + 4 * q^63 + 3 * q^65 + 9 * q^67 - 11 * q^69 + 3 * q^71 - 7 * q^73 - 6 * q^75 - 6 * q^77 + 6 * q^81 - 15 * q^83 + q^85 - 23 * q^87 - q^89 - 6 * q^91 + 8 * q^93 + 12 * q^95 - 16 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more