LMFDB - Embedded newform 8008.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 12x^{7} + 20x^{6} + 47x^{5} - 55x^{4} - 68x^{3} + 37x^{2} + 21x - 9 $ x^9 - 2x^8 - 12x^7 + 20x^6 + 47x^5 - 55x^4 - 68x^3 + 37x^2 + 21x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.93909$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93909 q^{3} +2.25153 q^{5} +1.00000 q^{7} +0.760073 q^{9} +O(q^{10})$	$q-1.93909 q^{3} +2.25153 q^{5} +1.00000 q^{7} +0.760073 q^{9} -1.00000 q^{11} +1.00000 q^{13} -4.36592 q^{15} +1.65442 q^{17} -1.38606 q^{19} -1.93909 q^{21} +5.68682 q^{23} +0.0693775 q^{25} +4.34342 q^{27} -9.54649 q^{29} -0.746290 q^{31} +1.93909 q^{33} +2.25153 q^{35} -9.05788 q^{37} -1.93909 q^{39} -6.52541 q^{41} +9.58315 q^{43} +1.71133 q^{45} -4.54832 q^{47} +1.00000 q^{49} -3.20807 q^{51} -3.60830 q^{53} -2.25153 q^{55} +2.68769 q^{57} +0.294575 q^{59} +7.21914 q^{61} +0.760073 q^{63} +2.25153 q^{65} +8.06636 q^{67} -11.0273 q^{69} -11.0316 q^{71} -3.69721 q^{73} -0.134529 q^{75} -1.00000 q^{77} -10.6652 q^{79} -10.7025 q^{81} +6.88489 q^{83} +3.72497 q^{85} +18.5115 q^{87} +5.81991 q^{89} +1.00000 q^{91} +1.44712 q^{93} -3.12075 q^{95} -1.51209 q^{97} -0.760073 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) $ 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9	$ 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9} - 9 q^{11} + 9 q^{13} - 4 q^{15} - 9 q^{17} + 9 q^{19} - 2 q^{21} - 10 q^{23} - q^{25} - 8 q^{27} - 13 q^{29} + 7 q^{31} + 2 q^{33} - 4 q^{35} - 15 q^{37} - 2 q^{39} - 18 q^{41} + 7 q^{43} + 9 q^{45} - 11 q^{47} + 9 q^{49} + q^{51} - 16 q^{53} + 4 q^{55} - 26 q^{57} - 2 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + 13 q^{67} - 3 q^{69} - q^{71} - 6 q^{73} - 4 q^{75} - 9 q^{77} - 21 q^{79} - 23 q^{81} - 30 q^{83} + q^{85} + q^{87} - 36 q^{89} + 9 q^{91} - 22 q^{93} - 23 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) $ 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9 - 9 * q^11 + 9 * q^13 - 4 * q^15 - 9 * q^17 + 9 * q^19 - 2 * q^21 - 10 * q^23 - q^25 - 8 * q^27 - 13 * q^29 + 7 * q^31 + 2 * q^33 - 4 * q^35 - 15 * q^37 - 2 * q^39 - 18 * q^41 + 7 * q^43 + 9 * q^45 - 11 * q^47 + 9 * q^49 + q^51 - 16 * q^53 + 4 * q^55 - 26 * q^57 - 2 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + 13 * q^67 - 3 * q^69 - q^71 - 6 * q^73 - 4 * q^75 - 9 * q^77 - 21 * q^79 - 23 * q^81 - 30 * q^83 + q^85 + q^87 - 36 * q^89 + 9 * q^91 - 22 * q^93 - 23 * q^95 - 5 * 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
$3$	−1.93909	−1.11953	−0.559767	−	0.828650i	$-0.689109\pi$
$3$	−1.93909	−1.11953	−0.559767	+	0.828650i	$0.689109\pi$
$4$	0	0
$5$	2.25153	1.00691	0.503457	−	0.864020i	$-0.332061\pi$
$5$	2.25153	1.00691	0.503457	+	0.864020i	$0.332061\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0.760073	0.253358
$10$	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
$15$	−4.36592	−1.12727
$16$	0	0
$17$	1.65442	0.401255	0.200628	−	0.979668i	$-0.435702\pi$
$17$	1.65442	0.401255	0.200628	+	0.979668i	$0.435702\pi$
$18$	0	0
$19$	−1.38606	−0.317983	−0.158992	−	0.987280i	$-0.550824\pi$
$19$	−1.38606	−0.317983	−0.158992	+	0.987280i	$0.550824\pi$
$20$	0	0
$21$	−1.93909	−0.423144
$22$	0	0
$23$	5.68682	1.18578	0.592892	−	0.805282i	$-0.297986\pi$
$23$	5.68682	1.18578	0.592892	+	0.805282i	$0.297986\pi$
$24$	0	0
$25$	0.0693775	0.0138755
$26$	0	0
$27$	4.34342	0.835892
$28$	0	0
$29$	−9.54649	−1.77274	−0.886369	−	0.462979i	$-0.846780\pi$
$29$	−9.54649	−1.77274	−0.886369	+	0.462979i	$0.846780\pi$
$30$	0	0
$31$	−0.746290	−0.134038	−0.0670188	−	0.997752i	$-0.521349\pi$
$31$	−0.746290	−0.134038	−0.0670188	+	0.997752i	$0.521349\pi$
$32$	0	0
$33$	1.93909	0.337552
$34$	0	0
$35$	2.25153	0.380578
$36$	0	0
$37$	−9.05788	−1.48911	−0.744553	−	0.667563i	$-0.767337\pi$
$37$	−9.05788	−1.48911	−0.744553	+	0.667563i	$0.767337\pi$
$38$	0	0
$39$	−1.93909	−0.310503
$40$	0	0
$41$	−6.52541	−1.01910	−0.509549	−	0.860441i	$-0.670188\pi$
$41$	−6.52541	−1.01910	−0.509549	+	0.860441i	$0.670188\pi$
$42$	0	0
$43$	9.58315	1.46142	0.730709	−	0.682690i	$-0.239190\pi$
$43$	9.58315	1.46142	0.730709	+	0.682690i	$0.239190\pi$
$44$	0	0
$45$	1.71133	0.255109
$46$	0	0
$47$	−4.54832	−0.663440	−0.331720	−	0.943378i	$-0.607629\pi$
$47$	−4.54832	−0.663440	−0.331720	+	0.943378i	$0.607629\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.20807	−0.449219
$52$	0	0
$53$	−3.60830	−0.495638	−0.247819	−	0.968806i	$-0.579714\pi$
$53$	−3.60830	−0.495638	−0.247819	+	0.968806i	$0.579714\pi$
$54$	0	0
$55$	−2.25153	−0.303596
$56$	0	0
$57$	2.68769	0.355993
$58$	0	0
$59$	0.294575	0.0383504	0.0191752	−	0.999816i	$-0.493896\pi$
$59$	0.294575	0.0383504	0.0191752	+	0.999816i	$0.493896\pi$
$60$	0	0
$61$	7.21914	0.924317	0.462158	−	0.886797i	$-0.347075\pi$
$61$	7.21914	0.924317	0.462158	+	0.886797i	$0.347075\pi$
$62$	0	0
$63$	0.760073	0.0957602
$64$	0	0
$65$	2.25153	0.279268
$66$	0	0
$67$	8.06636	0.985463	0.492732	−	0.870181i	$-0.335998\pi$
$67$	8.06636	0.985463	0.492732	+	0.870181i	$0.335998\pi$
$68$	0	0
$69$	−11.0273	−1.32753
$70$	0	0
$71$	−11.0316	−1.30921	−0.654605	−	0.755971i	$-0.727165\pi$
$71$	−11.0316	−1.30921	−0.654605	+	0.755971i	$0.727165\pi$
$72$	0	0
$73$	−3.69721	−0.432726	−0.216363	−	0.976313i	$-0.569419\pi$
$73$	−3.69721	−0.432726	−0.216363	+	0.976313i	$0.569419\pi$
$74$	0	0
$75$	−0.134529	−0.0155341
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−10.6652	−1.19993	−0.599965	−	0.800027i	$-0.704819\pi$
$79$	−10.6652	−1.19993	−0.599965	+	0.800027i	$0.704819\pi$
$80$	0	0
$81$	−10.7025	−1.18917
$82$	0	0
$83$	6.88489	0.755715	0.377858	−	0.925864i	$-0.376661\pi$
$83$	6.88489	0.755715	0.377858	+	0.925864i	$0.376661\pi$
$84$	0	0
$85$	3.72497	0.404029
$86$	0	0
$87$	18.5115	1.98464
$88$	0	0
$89$	5.81991	0.616909	0.308455	−	0.951239i	$-0.400188\pi$
$89$	5.81991	0.616909	0.308455	+	0.951239i	$0.400188\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	1.44712	0.150060
$94$	0	0
$95$	−3.12075	−0.320182
$96$	0	0
$97$	−1.51209	−0.153529	−0.0767647	−	0.997049i	$-0.524459\pi$
$97$	−1.51209	−0.153529	−0.0767647	+	0.997049i	$0.524459\pi$
$98$	0	0
$99$	−0.760073	−0.0763902
$100$	0	0
$101$	−8.66409	−0.862109	−0.431055	−	0.902326i	$-0.641858\pi$
$101$	−8.66409	−0.862109	−0.431055	+	0.902326i	$0.641858\pi$
$102$	0	0
$103$	7.57112	0.746005	0.373002	−	0.927830i	$-0.378328\pi$
$103$	7.57112	0.746005	0.373002	+	0.927830i	$0.378328\pi$
$104$	0	0
$105$	−4.36592	−0.426070
$106$	0	0
$107$	7.76240	0.750419	0.375210	−	0.926940i	$-0.377571\pi$
$107$	7.76240	0.750419	0.375210	+	0.926940i	$0.377571\pi$
$108$	0	0
$109$	−19.0796	−1.82749	−0.913746	−	0.406286i	$-0.866824\pi$
$109$	−19.0796	−1.82749	−0.913746	+	0.406286i	$0.866824\pi$
$110$	0	0
$111$	17.5641	1.66711
$112$	0	0
$113$	−14.7967	−1.39196	−0.695980	−	0.718061i	$-0.745030\pi$
$113$	−14.7967	−1.39196	−0.695980	+	0.718061i	$0.745030\pi$
$114$	0	0
$115$	12.8040	1.19398
$116$	0	0
$117$	0.760073	0.0702688
$118$	0	0
$119$	1.65442	0.151660
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	12.6534	1.14092
$124$	0	0
$125$	−11.1014	−0.992942
$126$	0	0
$127$	−15.2957	−1.35727	−0.678635	−	0.734475i	$-0.737428\pi$
$127$	−15.2957	−1.35727	−0.678635	+	0.734475i	$0.737428\pi$
$128$	0	0
$129$	−18.5826	−1.63611
$130$	0	0
$131$	−11.5725	−1.01109	−0.505546	−	0.862800i	$-0.668709\pi$
$131$	−11.5725	−1.01109	−0.505546	+	0.862800i	$0.668709\pi$
$132$	0	0
$133$	−1.38606	−0.120186
$134$	0	0
$135$	9.77933	0.841671
$136$	0	0
$137$	7.21055	0.616038	0.308019	−	0.951380i	$-0.400334\pi$
$137$	7.21055	0.616038	0.308019	+	0.951380i	$0.400334\pi$
$138$	0	0
$139$	2.79111	0.236739	0.118369	−	0.992970i	$-0.462233\pi$
$139$	2.79111	0.236739	0.118369	+	0.992970i	$0.462233\pi$
$140$	0	0
$141$	8.81960	0.742744
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−21.4942	−1.78499
$146$	0	0
$147$	−1.93909	−0.159934
$148$	0	0
$149$	9.33465	0.764724	0.382362	−	0.924013i	$-0.375111\pi$
$149$	9.33465	0.764724	0.382362	+	0.924013i	$0.375111\pi$
$150$	0	0
$151$	8.78750	0.715117	0.357558	−	0.933891i	$-0.383609\pi$
$151$	8.78750	0.715117	0.357558	+	0.933891i	$0.383609\pi$
$152$	0	0
$153$	1.25748	0.101661
$154$	0	0
$155$	−1.68029	−0.134964
$156$	0	0
$157$	−7.15463	−0.571002	−0.285501	−	0.958378i	$-0.592160\pi$
$157$	−7.15463	−0.571002	−0.285501	+	0.958378i	$0.592160\pi$
$158$	0	0
$159$	6.99681	0.554883
$160$	0	0
$161$	5.68682	0.448184
$162$	0	0
$163$	3.38749	0.265329	0.132664	−	0.991161i	$-0.457647\pi$
$163$	3.38749	0.265329	0.132664	+	0.991161i	$0.457647\pi$
$164$	0	0
$165$	4.36592	0.339886
$166$	0	0
$167$	6.39335	0.494732	0.247366	−	0.968922i	$-0.420435\pi$
$167$	6.39335	0.494732	0.247366	+	0.968922i	$0.420435\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.05350	−0.0805635
$172$	0	0
$173$	−13.1764	−1.00178	−0.500891	−	0.865510i	$-0.666994\pi$
$173$	−13.1764	−1.00178	−0.500891	+	0.865510i	$0.666994\pi$
$174$	0	0
$175$	0.0693775	0.00524445
$176$	0	0
$177$	−0.571207	−0.0429346
$178$	0	0
$179$	15.8942	1.18799	0.593993	−	0.804470i	$-0.297551\pi$
$179$	15.8942	1.18799	0.593993	+	0.804470i	$0.297551\pi$
$180$	0	0
$181$	−5.95298	−0.442482	−0.221241	−	0.975219i	$-0.571011\pi$
$181$	−5.95298	−0.442482	−0.221241	+	0.975219i	$0.571011\pi$
$182$	0	0
$183$	−13.9986	−1.03480
$184$	0	0
$185$	−20.3941	−1.49940
$186$	0	0
$187$	−1.65442	−0.120983
$188$	0	0
$189$	4.34342	0.315937
$190$	0	0
$191$	−1.10393	−0.0798773	−0.0399387	−	0.999202i	$-0.512716\pi$
$191$	−1.10393	−0.0798773	−0.0399387	+	0.999202i	$0.512716\pi$
$192$	0	0
$193$	23.8845	1.71925	0.859623	−	0.510929i	$-0.170698\pi$
$193$	23.8845	1.71925	0.859623	+	0.510929i	$0.170698\pi$
$194$	0	0
$195$	−4.36592	−0.312650
$196$	0	0
$197$	16.2939	1.16089	0.580446	−	0.814299i	$-0.302878\pi$
$197$	16.2939	1.16089	0.580446	+	0.814299i	$0.302878\pi$
$198$	0	0
$199$	−5.95812	−0.422360	−0.211180	−	0.977447i	$-0.567731\pi$
$199$	−5.95812	−0.422360	−0.211180	+	0.977447i	$0.567731\pi$
$200$	0	0
$201$	−15.6414	−1.10326
$202$	0	0
$203$	−9.54649	−0.670032
$204$	0	0
$205$	−14.6922	−1.02614
$206$	0	0
$207$	4.32240	0.300428
$208$	0	0
$209$	1.38606	0.0958756
$210$	0	0
$211$	−10.8006	−0.743541	−0.371771	−	0.928325i	$-0.621249\pi$
$211$	−10.8006	−0.743541	−0.371771	+	0.928325i	$0.621249\pi$
$212$	0	0
$213$	21.3913	1.46571
$214$	0	0
$215$	21.5767	1.47152
$216$	0	0
$217$	−0.746290	−0.0506615
$218$	0	0
$219$	7.16923	0.484451
$220$	0	0
$221$	1.65442	0.111288
$222$	0	0
$223$	−25.8209	−1.72910	−0.864549	−	0.502548i	$-0.832396\pi$
$223$	−25.8209	−1.72910	−0.864549	+	0.502548i	$0.832396\pi$
$224$	0	0
$225$	0.0527320	0.00351547
$226$	0	0
$227$	−3.62809	−0.240805	−0.120402	−	0.992725i	$-0.538418\pi$
$227$	−3.62809	−0.240805	−0.120402	+	0.992725i	$0.538418\pi$
$228$	0	0
$229$	−5.06618	−0.334782	−0.167391	−	0.985891i	$-0.553534\pi$
$229$	−5.06618	−0.334782	−0.167391	+	0.985891i	$0.553534\pi$
$230$	0	0
$231$	1.93909	0.127583
$232$	0	0
$233$	2.98190	0.195351	0.0976755	−	0.995218i	$-0.468859\pi$
$233$	2.98190	0.195351	0.0976755	+	0.995218i	$0.468859\pi$
$234$	0	0
$235$	−10.2407	−0.668027
$236$	0	0
$237$	20.6808	1.34336
$238$	0	0
$239$	−20.8150	−1.34641	−0.673206	−	0.739455i	$-0.735083\pi$
$239$	−20.8150	−1.34641	−0.673206	+	0.739455i	$0.735083\pi$
$240$	0	0
$241$	7.74720	0.499041	0.249521	−	0.968369i	$-0.419727\pi$
$241$	7.74720	0.499041	0.249521	+	0.968369i	$0.419727\pi$
$242$	0	0
$243$	7.72287	0.495422
$244$	0	0
$245$	2.25153	0.143845
$246$	0	0
$247$	−1.38606	−0.0881927
$248$	0	0
$249$	−13.3504	−0.846049
$250$	0	0
$251$	8.72583	0.550770	0.275385	−	0.961334i	$-0.411195\pi$
$251$	8.72583	0.550770	0.275385	+	0.961334i	$0.411195\pi$
$252$	0	0
$253$	−5.68682	−0.357527
$254$	0	0
$255$	−7.22305	−0.452325
$256$	0	0
$257$	−5.26515	−0.328431	−0.164215	−	0.986424i	$-0.552509\pi$
$257$	−5.26515	−0.328431	−0.164215	+	0.986424i	$0.552509\pi$
$258$	0	0
$259$	−9.05788	−0.562829
$260$	0	0
$261$	−7.25603	−0.449137
$262$	0	0
$263$	−11.4500	−0.706036	−0.353018	−	0.935617i	$-0.614845\pi$
$263$	−11.4500	−0.706036	−0.353018	+	0.935617i	$0.614845\pi$
$264$	0	0
$265$	−8.12418	−0.499064
$266$	0	0
$267$	−11.2853	−0.690651
$268$	0	0
$269$	6.46183	0.393985	0.196992	−	0.980405i	$-0.436883\pi$
$269$	6.46183	0.393985	0.196992	+	0.980405i	$0.436883\pi$
$270$	0	0
$271$	15.6061	0.948002	0.474001	−	0.880524i	$-0.342809\pi$
$271$	15.6061	0.948002	0.474001	+	0.880524i	$0.342809\pi$
$272$	0	0
$273$	−1.93909	−0.117359
$274$	0	0
$275$	−0.0693775	−0.00418362
$276$	0	0
$277$	24.4814	1.47095	0.735473	−	0.677554i	$-0.236960\pi$
$277$	24.4814	1.47095	0.735473	+	0.677554i	$0.236960\pi$
$278$	0	0
$279$	−0.567235	−0.0339595
$280$	0	0
$281$	−27.0970	−1.61647	−0.808236	−	0.588858i	$-0.799578\pi$
$281$	−27.0970	−1.61647	−0.808236	+	0.588858i	$0.799578\pi$
$282$	0	0
$283$	29.5224	1.75492	0.877462	−	0.479645i	$-0.159235\pi$
$283$	29.5224	1.75492	0.877462	+	0.479645i	$0.159235\pi$
$284$	0	0
$285$	6.05141	0.358455
$286$	0	0
$287$	−6.52541	−0.385183
$288$	0	0
$289$	−14.2629	−0.838994
$290$	0	0
$291$	2.93208	0.171882
$292$	0	0
$293$	32.9545	1.92522	0.962612	−	0.270885i	$-0.0873162\pi$
$293$	32.9545	1.92522	0.962612	+	0.270885i	$0.0873162\pi$
$294$	0	0
$295$	0.663243	0.0386155
$296$	0	0
$297$	−4.34342	−0.252031
$298$	0	0
$299$	5.68682	0.328877
$300$	0	0
$301$	9.58315	0.552364
$302$	0	0
$303$	16.8005	0.965161
$304$	0	0
$305$	16.2541	0.930707
$306$	0	0
$307$	−25.4892	−1.45475	−0.727373	−	0.686242i	$-0.759259\pi$
$307$	−25.4892	−1.45475	−0.727373	+	0.686242i	$0.759259\pi$
$308$	0	0
$309$	−14.6811	−0.835178
$310$	0	0
$311$	−3.66426	−0.207781	−0.103891	−	0.994589i	$-0.533129\pi$
$311$	−3.66426	−0.207781	−0.103891	+	0.994589i	$0.533129\pi$
$312$	0	0
$313$	−0.689999	−0.0390010	−0.0195005	−	0.999810i	$-0.506208\pi$
$313$	−0.689999	−0.0390010	−0.0195005	+	0.999810i	$0.506208\pi$
$314$	0	0
$315$	1.71133	0.0964223
$316$	0	0
$317$	25.9797	1.45917	0.729583	−	0.683892i	$-0.239714\pi$
$317$	25.9797	1.45917	0.729583	+	0.683892i	$0.239714\pi$
$318$	0	0
$319$	9.54649	0.534501
$320$	0	0
$321$	−15.0520	−0.840120
$322$	0	0
$323$	−2.29312	−0.127592
$324$	0	0
$325$	0.0693775	0.00384837
$326$	0	0
$327$	36.9970	2.04594
$328$	0	0
$329$	−4.54832	−0.250757
$330$	0	0
$331$	3.09011	0.169848	0.0849239	−	0.996387i	$-0.472935\pi$
$331$	3.09011	0.169848	0.0849239	+	0.996387i	$0.472935\pi$
$332$	0	0
$333$	−6.88465	−0.377276
$334$	0	0
$335$	18.1616	0.992277
$336$	0	0
$337$	23.8647	1.29999	0.649997	−	0.759937i	$-0.274770\pi$
$337$	23.8647	1.29999	0.649997	+	0.759937i	$0.274770\pi$
$338$	0	0
$339$	28.6922	1.55835
$340$	0	0
$341$	0.746290	0.0404139
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−24.8282	−1.33670
$346$	0	0
$347$	−19.9197	−1.06934	−0.534672	−	0.845060i	$-0.679565\pi$
$347$	−19.9197	−1.06934	−0.534672	+	0.845060i	$0.679565\pi$
$348$	0	0
$349$	−31.4246	−1.68212	−0.841059	−	0.540943i	$-0.818067\pi$
$349$	−31.4246	−1.68212	−0.841059	+	0.540943i	$0.818067\pi$
$350$	0	0
$351$	4.34342	0.231835
$352$	0	0
$353$	16.9577	0.902566	0.451283	−	0.892381i	$-0.350967\pi$
$353$	16.9577	0.902566	0.451283	+	0.892381i	$0.350967\pi$
$354$	0	0
$355$	−24.8380	−1.31826
$356$	0	0
$357$	−3.20807	−0.169789
$358$	0	0
$359$	−1.08558	−0.0572946	−0.0286473	−	0.999590i	$-0.509120\pi$
$359$	−1.08558	−0.0572946	−0.0286473	+	0.999590i	$0.509120\pi$
$360$	0	0
$361$	−17.0788	−0.898887
$362$	0	0
$363$	−1.93909	−0.101776
$364$	0	0
$365$	−8.32437	−0.435718
$366$	0	0
$367$	19.0381	0.993783	0.496892	−	0.867813i	$-0.334475\pi$
$367$	19.0381	0.993783	0.496892	+	0.867813i	$0.334475\pi$
$368$	0	0
$369$	−4.95979	−0.258196
$370$	0	0
$371$	−3.60830	−0.187333
$372$	0	0
$373$	−19.0942	−0.988662	−0.494331	−	0.869274i	$-0.664587\pi$
$373$	−19.0942	−0.988662	−0.494331	+	0.869274i	$0.664587\pi$
$374$	0	0
$375$	21.5267	1.11163
$376$	0	0
$377$	−9.54649	−0.491669
$378$	0	0
$379$	−22.6887	−1.16544	−0.582721	−	0.812672i	$-0.698012\pi$
$379$	−22.6887	−1.16544	−0.582721	+	0.812672i	$0.698012\pi$
$380$	0	0
$381$	29.6597	1.51951
$382$	0	0
$383$	−20.1242	−1.02830	−0.514148	−	0.857701i	$-0.671892\pi$
$383$	−20.1242	−1.02830	−0.514148	+	0.857701i	$0.671892\pi$
$384$	0	0
$385$	−2.25153	−0.114748
$386$	0	0
$387$	7.28390	0.370261
$388$	0	0
$389$	−32.6478	−1.65531	−0.827653	−	0.561240i	$-0.810325\pi$
$389$	−32.6478	−1.65531	−0.827653	+	0.561240i	$0.810325\pi$
$390$	0	0
$391$	9.40838	0.475802
$392$	0	0
$393$	22.4401	1.13195
$394$	0	0
$395$	−24.0130	−1.20823
$396$	0	0
$397$	−12.6906	−0.636924	−0.318462	−	0.947936i	$-0.603166\pi$
$397$	−12.6906	−0.636924	−0.318462	+	0.947936i	$0.603166\pi$
$398$	0	0
$399$	2.68769	0.134553
$400$	0	0
$401$	−10.1384	−0.506286	−0.253143	−	0.967429i	$-0.581464\pi$
$401$	−10.1384	−0.506286	−0.253143	+	0.967429i	$0.581464\pi$
$402$	0	0
$403$	−0.746290	−0.0371753
$404$	0	0
$405$	−24.0970	−1.19739
$406$	0	0
$407$	9.05788	0.448982
$408$	0	0
$409$	2.70452	0.133730	0.0668650	−	0.997762i	$-0.478700\pi$
$409$	2.70452	0.133730	0.0668650	+	0.997762i	$0.478700\pi$
$410$	0	0
$411$	−13.9819	−0.689676
$412$	0	0
$413$	0.294575	0.0144951
$414$	0	0
$415$	15.5015	0.760940
$416$	0	0
$417$	−5.41222	−0.265037
$418$	0	0
$419$	−0.706172	−0.0344987	−0.0172494	−	0.999851i	$-0.505491\pi$
$419$	−0.706172	−0.0344987	−0.0172494	+	0.999851i	$0.505491\pi$
$420$	0	0
$421$	−2.69933	−0.131557	−0.0657786	−	0.997834i	$-0.520953\pi$
$421$	−2.69933	−0.131557	−0.0657786	+	0.997834i	$0.520953\pi$
$422$	0	0
$423$	−3.45705	−0.168088
$424$	0	0
$425$	0.114779	0.00556762
$426$	0	0
$427$	7.21914	0.349359
$428$	0	0
$429$	1.93909	0.0936202
$430$	0	0
$431$	−38.9193	−1.87468	−0.937339	−	0.348420i	$-0.886718\pi$
$431$	−38.9193	−1.87468	−0.937339	+	0.348420i	$0.886718\pi$
$432$	0	0
$433$	−33.4861	−1.60924	−0.804619	−	0.593792i	$-0.797630\pi$
$433$	−33.4861	−1.60924	−0.804619	+	0.593792i	$0.797630\pi$
$434$	0	0
$435$	41.6792	1.99836
$436$	0	0
$437$	−7.88226	−0.377060
$438$	0	0
$439$	−32.6126	−1.55652	−0.778258	−	0.627944i	$-0.783897\pi$
$439$	−32.6126	−1.55652	−0.778258	+	0.627944i	$0.783897\pi$
$440$	0	0
$441$	0.760073	0.0361940
$442$	0	0
$443$	2.05393	0.0975851	0.0487925	−	0.998809i	$-0.484463\pi$
$443$	2.05393	0.0975851	0.0487925	+	0.998809i	$0.484463\pi$
$444$	0	0
$445$	13.1037	0.621174
$446$	0	0
$447$	−18.1007	−0.856135
$448$	0	0
$449$	12.3092	0.580906	0.290453	−	0.956889i	$-0.406194\pi$
$449$	12.3092	0.580906	0.290453	+	0.956889i	$0.406194\pi$
$450$	0	0
$451$	6.52541	0.307270
$452$	0	0
$453$	−17.0398	−0.800598
$454$	0	0
$455$	2.25153	0.105553
$456$	0	0
$457$	−6.78452	−0.317367	−0.158683	−	0.987330i	$-0.550725\pi$
$457$	−6.78452	−0.317367	−0.158683	+	0.987330i	$0.550725\pi$
$458$	0	0
$459$	7.18583	0.335406
$460$	0	0
$461$	−10.9392	−0.509490	−0.254745	−	0.967008i	$-0.581991\pi$
$461$	−10.9392	−0.509490	−0.254745	+	0.967008i	$0.581991\pi$
$462$	0	0
$463$	−13.9638	−0.648954	−0.324477	−	0.945894i	$-0.605188\pi$
$463$	−13.9638	−0.648954	−0.324477	+	0.945894i	$0.605188\pi$
$464$	0	0
$465$	3.25824	0.151097
$466$	0	0
$467$	−22.6210	−1.04678	−0.523388	−	0.852095i	$-0.675332\pi$
$467$	−22.6210	−1.04678	−0.523388	+	0.852095i	$0.675332\pi$
$468$	0	0
$469$	8.06636	0.372470
$470$	0	0
$471$	13.8735	0.639257
$472$	0	0
$473$	−9.58315	−0.440634
$474$	0	0
$475$	−0.0961612	−0.00441218
$476$	0	0
$477$	−2.74257	−0.125574
$478$	0	0
$479$	13.6238	0.622488	0.311244	−	0.950330i	$-0.399254\pi$
$479$	13.6238	0.622488	0.311244	+	0.950330i	$0.399254\pi$
$480$	0	0
$481$	−9.05788	−0.413004
$482$	0	0
$483$	−11.0273	−0.501758
$484$	0	0
$485$	−3.40451	−0.154591
$486$	0	0
$487$	−2.21132	−0.100204	−0.0501022	−	0.998744i	$-0.515955\pi$
$487$	−2.21132	−0.100204	−0.0501022	+	0.998744i	$0.515955\pi$
$488$	0	0
$489$	−6.56865	−0.297045
$490$	0	0
$491$	31.4608	1.41981	0.709904	−	0.704299i	$-0.248739\pi$
$491$	31.4608	1.41981	0.709904	+	0.704299i	$0.248739\pi$
$492$	0	0
$493$	−15.7939	−0.711320
$494$	0	0
$495$	−1.71133	−0.0769184
$496$	0	0
$497$	−11.0316	−0.494835
$498$	0	0
$499$	−5.93293	−0.265594	−0.132797	−	0.991143i	$-0.542396\pi$
$499$	−5.93293	−0.265594	−0.132797	+	0.991143i	$0.542396\pi$
$500$	0	0
$501$	−12.3973	−0.553870
$502$	0	0
$503$	−23.0603	−1.02821	−0.514103	−	0.857728i	$-0.671875\pi$
$503$	−23.0603	−1.02821	−0.514103	+	0.857728i	$0.671875\pi$
$504$	0	0
$505$	−19.5074	−0.868070
$506$	0	0
$507$	−1.93909	−0.0861180
$508$	0	0
$509$	5.99390	0.265675	0.132837	−	0.991138i	$-0.457591\pi$
$509$	5.99390	0.265675	0.132837	+	0.991138i	$0.457591\pi$
$510$	0	0
$511$	−3.69721	−0.163555
$512$	0	0
$513$	−6.02023	−0.265800
$514$	0	0
$515$	17.0466	0.751162
$516$	0	0
$517$	4.54832	0.200035
$518$	0	0
$519$	25.5502	1.12153
$520$	0	0
$521$	−39.5062	−1.73080	−0.865400	−	0.501082i	$-0.832936\pi$
$521$	−39.5062	−1.73080	−0.865400	+	0.501082i	$0.832936\pi$
$522$	0	0
$523$	−36.3244	−1.58836	−0.794178	−	0.607685i	$-0.792098\pi$
$523$	−36.3244	−1.58836	−0.794178	+	0.607685i	$0.792098\pi$
$524$	0	0
$525$	−0.134529	−0.00587134
$526$	0	0
$527$	−1.23467	−0.0537833
$528$	0	0
$529$	9.33995	0.406085
$530$	0	0
$531$	0.223898	0.00971636
$532$	0	0
$533$	−6.52541	−0.282647
$534$	0	0
$535$	17.4773	0.755607
$536$	0	0
$537$	−30.8202	−1.32999
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	22.4191	0.963872	0.481936	−	0.876206i	$-0.339934\pi$
$541$	22.4191	0.963872	0.481936	+	0.876206i	$0.339934\pi$
$542$	0	0
$543$	11.5434	0.495373
$544$	0	0
$545$	−42.9582	−1.84013
$546$	0	0
$547$	−0.00644170	−0.000275427 0	−0.000137714	−	1.00000i	$-0.500044\pi$
$547$	−0.00644170	−0.000275427 0	−0.000137714		1.00000i	$0.500044\pi$
$548$	0	0
$549$	5.48708	0.234183
$550$	0	0
$551$	13.2320	0.563701
$552$	0	0
$553$	−10.6652	−0.453531
$554$	0	0
$555$	39.5459	1.67863
$556$	0	0
$557$	−25.6413	−1.08646	−0.543228	−	0.839585i	$-0.682798\pi$
$557$	−25.6413	−1.08646	−0.543228	+	0.839585i	$0.682798\pi$
$558$	0	0
$559$	9.58315	0.405324
$560$	0	0
$561$	3.20807	0.135445
$562$	0	0
$563$	14.1896	0.598018	0.299009	−	0.954250i	$-0.403344\pi$
$563$	14.1896	0.598018	0.299009	+	0.954250i	$0.403344\pi$
$564$	0	0
$565$	−33.3153	−1.40158
$566$	0	0
$567$	−10.7025	−0.449463
$568$	0	0
$569$	6.84547	0.286977	0.143489	−	0.989652i	$-0.454168\pi$
$569$	6.84547	0.286977	0.143489	+	0.989652i	$0.454168\pi$
$570$	0	0
$571$	−23.7086	−0.992174	−0.496087	−	0.868273i	$-0.665230\pi$
$571$	−23.7086	−0.992174	−0.496087	+	0.868273i	$0.665230\pi$
$572$	0	0
$573$	2.14061	0.0894254
$574$	0	0
$575$	0.394538	0.0164534
$576$	0	0
$577$	34.7780	1.44783	0.723914	−	0.689890i	$-0.242341\pi$
$577$	34.7780	1.44783	0.723914	+	0.689890i	$0.242341\pi$
$578$	0	0
$579$	−46.3143	−1.92476
$580$	0	0
$581$	6.88489	0.285633
$582$	0	0
$583$	3.60830	0.149440
$584$	0	0
$585$	1.71133	0.0707546
$586$	0	0
$587$	−16.2459	−0.670541	−0.335271	−	0.942122i	$-0.608828\pi$
$587$	−16.2459	−0.670541	−0.335271	+	0.942122i	$0.608828\pi$
$588$	0	0
$589$	1.03440	0.0426217
$590$	0	0
$591$	−31.5953	−1.29966
$592$	0	0
$593$	−25.2496	−1.03688	−0.518438	−	0.855115i	$-0.673486\pi$
$593$	−25.2496	−1.03688	−0.518438	+	0.855115i	$0.673486\pi$
$594$	0	0
$595$	3.72497	0.152709
$596$	0	0
$597$	11.5533	0.472847
$598$	0	0
$599$	29.0353	1.18635	0.593176	−	0.805073i	$-0.297874\pi$
$599$	29.0353	1.18635	0.593176	+	0.805073i	$0.297874\pi$
$600$	0	0
$601$	−41.4578	−1.69110	−0.845550	−	0.533896i	$-0.820727\pi$
$601$	−41.4578	−1.69110	−0.845550	+	0.533896i	$0.820727\pi$
$602$	0	0
$603$	6.13103	0.249675
$604$	0	0
$605$	2.25153	0.0915376
$606$	0	0
$607$	−1.26269	−0.0512510	−0.0256255	−	0.999672i	$-0.508158\pi$
$607$	−1.26269	−0.0512510	−0.0256255	+	0.999672i	$0.508158\pi$
$608$	0	0
$609$	18.5115	0.750124
$610$	0	0
$611$	−4.54832	−0.184005
$612$	0	0
$613$	−12.1703	−0.491553	−0.245776	−	0.969327i	$-0.579043\pi$
$613$	−12.1703	−0.491553	−0.245776	+	0.969327i	$0.579043\pi$
$614$	0	0
$615$	28.4894	1.14880
$616$	0	0
$617$	−17.0004	−0.684411	−0.342205	−	0.939625i	$-0.611174\pi$
$617$	−17.0004	−0.684411	−0.342205	+	0.939625i	$0.611174\pi$
$618$	0	0
$619$	35.9068	1.44321	0.721607	−	0.692303i	$-0.243404\pi$
$619$	35.9068	1.44321	0.721607	+	0.692303i	$0.243404\pi$
$620$	0	0
$621$	24.7003	0.991188
$622$	0	0
$623$	5.81991	0.233170
$624$	0	0
$625$	−25.3421	−1.01368
$626$	0	0
$627$	−2.68769	−0.107336
$628$	0	0
$629$	−14.9855	−0.597512
$630$	0	0
$631$	18.9718	0.755255	0.377627	−	0.925958i	$-0.376740\pi$
$631$	18.9718	0.755255	0.377627	+	0.925958i	$0.376740\pi$
$632$	0	0
$633$	20.9433	0.832420
$634$	0	0
$635$	−34.4386	−1.36665
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−8.38482	−0.331698
$640$	0	0
$641$	9.53981	0.376800	0.188400	−	0.982092i	$-0.439670\pi$
$641$	9.53981	0.376800	0.188400	+	0.982092i	$0.439670\pi$
$642$	0	0
$643$	−20.4275	−0.805581	−0.402790	−	0.915292i	$-0.631960\pi$
$643$	−20.4275	−0.805581	−0.402790	+	0.915292i	$0.631960\pi$
$644$	0	0
$645$	−41.8393	−1.64742
$646$	0	0
$647$	−41.5394	−1.63308	−0.816541	−	0.577287i	$-0.804111\pi$
$647$	−41.5394	−1.63308	−0.816541	+	0.577287i	$0.804111\pi$
$648$	0	0
$649$	−0.294575	−0.0115631
$650$	0	0
$651$	1.44712	0.0567173
$652$	0	0
$653$	−11.2776	−0.441326	−0.220663	−	0.975350i	$-0.570822\pi$
$653$	−11.2776	−0.441326	−0.220663	+	0.975350i	$0.570822\pi$
$654$	0	0
$655$	−26.0557	−1.01808
$656$	0	0
$657$	−2.81015	−0.109634
$658$	0	0
$659$	−9.66149	−0.376358	−0.188179	−	0.982135i	$-0.560259\pi$
$659$	−9.66149	−0.376358	−0.188179	+	0.982135i	$0.560259\pi$
$660$	0	0
$661$	−6.80277	−0.264597	−0.132299	−	0.991210i	$-0.542236\pi$
$661$	−6.80277	−0.264597	−0.132299	+	0.991210i	$0.542236\pi$
$662$	0	0
$663$	−3.20807	−0.124591
$664$	0	0
$665$	−3.12075	−0.121017
$666$	0	0
$667$	−54.2892	−2.10209
$668$	0	0
$669$	50.0692	1.93579
$670$	0	0
$671$	−7.21914	−0.278692
$672$	0	0
$673$	39.4575	1.52098	0.760488	−	0.649352i	$-0.224960\pi$
$673$	39.4575	1.52098	0.760488	+	0.649352i	$0.224960\pi$
$674$	0	0
$675$	0.301336	0.0115984
$676$	0	0
$677$	−40.9738	−1.57475	−0.787375	−	0.616475i	$-0.788560\pi$
$677$	−40.9738	−1.57475	−0.787375	+	0.616475i	$0.788560\pi$
$678$	0	0
$679$	−1.51209	−0.0580287
$680$	0	0
$681$	7.03520	0.269589
$682$	0	0
$683$	3.09514	0.118432	0.0592161	−	0.998245i	$-0.481140\pi$
$683$	3.09514	0.118432	0.0592161	+	0.998245i	$0.481140\pi$
$684$	0	0
$685$	16.2347	0.620298
$686$	0	0
$687$	9.82377	0.374800
$688$	0	0
$689$	−3.60830	−0.137465
$690$	0	0
$691$	9.96916	0.379245	0.189622	−	0.981857i	$-0.439274\pi$
$691$	9.96916	0.379245	0.189622	+	0.981857i	$0.439274\pi$
$692$	0	0
$693$	−0.760073	−0.0288728
$694$	0	0
$695$	6.28426	0.238376
$696$	0	0
$697$	−10.7958	−0.408919
$698$	0	0
$699$	−5.78218	−0.218702
$700$	0	0
$701$	−22.5315	−0.851003	−0.425502	−	0.904958i	$-0.639902\pi$
$701$	−22.5315	−0.851003	−0.425502	+	0.904958i	$0.639902\pi$
$702$	0	0
$703$	12.5547	0.473511
$704$	0	0
$705$	19.8576	0.747879
$706$	0	0
$707$	−8.66409	−0.325847
$708$	0	0
$709$	−40.2273	−1.51077	−0.755384	−	0.655283i	$-0.772550\pi$
$709$	−40.2273	−1.51077	−0.755384	+	0.655283i	$0.772550\pi$
$710$	0	0
$711$	−8.10633	−0.304011
$712$	0	0
$713$	−4.24402	−0.158940
$714$	0	0
$715$	−2.25153	−0.0842024
$716$	0	0
$717$	40.3622	1.50736
$718$	0	0
$719$	−19.5147	−0.727777	−0.363889	−	0.931442i	$-0.618551\pi$
$719$	−19.5147	−0.727777	−0.363889	+	0.931442i	$0.618551\pi$
$720$	0	0
$721$	7.57112	0.281963
$722$	0	0
$723$	−15.0225	−0.558694
$724$	0	0
$725$	−0.662312	−0.0245976
$726$	0	0
$727$	−16.9417	−0.628331	−0.314166	−	0.949368i	$-0.601725\pi$
$727$	−16.9417	−0.628331	−0.314166	+	0.949368i	$0.601725\pi$
$728$	0	0
$729$	17.1322	0.634525
$730$	0	0
$731$	15.8545	0.586401
$732$	0	0
$733$	11.8813	0.438846	0.219423	−	0.975630i	$-0.429583\pi$
$733$	11.8813	0.438846	0.219423	+	0.975630i	$0.429583\pi$
$734$	0	0
$735$	−4.36592	−0.161039
$736$	0	0
$737$	−8.06636	−0.297128
$738$	0	0
$739$	−26.4901	−0.974454	−0.487227	−	0.873275i	$-0.661992\pi$
$739$	−26.4901	−0.974454	−0.487227	+	0.873275i	$0.661992\pi$
$740$	0	0
$741$	2.68769	0.0987348
$742$	0	0
$743$	33.8717	1.24263	0.621316	−	0.783560i	$-0.286598\pi$
$743$	33.8717	1.24263	0.621316	+	0.783560i	$0.286598\pi$
$744$	0	0
$745$	21.0172	0.770012
$746$	0	0
$747$	5.23302	0.191466
$748$	0	0
$749$	7.76240	0.283632
$750$	0	0
$751$	32.4054	1.18249	0.591245	−	0.806492i	$-0.298636\pi$
$751$	32.4054	1.18249	0.591245	+	0.806492i	$0.298636\pi$
$752$	0	0
$753$	−16.9202	−0.616606
$754$	0	0
$755$	19.7853	0.720061
$756$	0	0
$757$	−1.02725	−0.0373359	−0.0186680	−	0.999826i	$-0.505943\pi$
$757$	−1.02725	−0.0373359	−0.0186680	+	0.999826i	$0.505943\pi$
$758$	0	0
$759$	11.0273	0.400264
$760$	0	0
$761$	3.72430	0.135006	0.0675030	−	0.997719i	$-0.478497\pi$
$761$	3.72430	0.135006	0.0675030	+	0.997719i	$0.478497\pi$
$762$	0	0
$763$	−19.0796	−0.690727
$764$	0	0
$765$	2.83125	0.102364
$766$	0	0
$767$	0.294575	0.0106365
$768$	0	0
$769$	−36.6933	−1.32319	−0.661597	−	0.749860i	$-0.730121\pi$
$769$	−36.6933	−1.32319	−0.661597	+	0.749860i	$0.730121\pi$
$770$	0	0
$771$	10.2096	0.367690
$772$	0	0
$773$	−36.4706	−1.31176	−0.655878	−	0.754867i	$-0.727701\pi$
$773$	−36.4706	−1.31176	−0.655878	+	0.754867i	$0.727701\pi$
$774$	0	0
$775$	−0.0517757	−0.00185984
$776$	0	0
$777$	17.5641	0.630107
$778$	0	0
$779$	9.04460	0.324056
$780$	0	0
$781$	11.0316	0.394742
$782$	0	0
$783$	−41.4644	−1.48182
$784$	0	0
$785$	−16.1089	−0.574950
$786$	0	0
$787$	37.7914	1.34712	0.673559	−	0.739133i	$-0.264765\pi$
$787$	37.7914	1.34712	0.673559	+	0.739133i	$0.264765\pi$
$788$	0	0
$789$	22.2025	0.790431
$790$	0	0
$791$	−14.7967	−0.526111
$792$	0	0
$793$	7.21914	0.256359
$794$	0	0
$795$	15.7535	0.558720
$796$	0	0
$797$	−7.20761	−0.255306	−0.127653	−	0.991819i	$-0.540744\pi$
$797$	−7.20761	−0.255306	−0.127653	+	0.991819i	$0.540744\pi$
$798$	0	0
$799$	−7.52481	−0.266209
$800$	0	0
$801$	4.42356	0.156299
$802$	0	0
$803$	3.69721	0.130472
$804$	0	0
$805$	12.8040	0.451283
$806$	0	0
$807$	−12.5301	−0.441080
$808$	0	0
$809$	39.1459	1.37630	0.688148	−	0.725570i	$-0.258424\pi$
$809$	39.1459	1.37630	0.688148	+	0.725570i	$0.258424\pi$
$810$	0	0
$811$	24.5510	0.862104	0.431052	−	0.902327i	$-0.358143\pi$
$811$	24.5510	0.862104	0.431052	+	0.902327i	$0.358143\pi$
$812$	0	0
$813$	−30.2616	−1.06132
$814$	0	0
$815$	7.62703	0.267163
$816$	0	0
$817$	−13.2828	−0.464706
$818$	0	0
$819$	0.760073	0.0265591
$820$	0	0
$821$	34.8150	1.21505	0.607527	−	0.794299i	$-0.292162\pi$
$821$	34.8150	1.21505	0.607527	+	0.794299i	$0.292162\pi$
$822$	0	0
$823$	42.3558	1.47643	0.738215	−	0.674565i	$-0.235669\pi$
$823$	42.3558	1.47643	0.738215	+	0.674565i	$0.235669\pi$
$824$	0	0
$825$	0.134529	0.00468371
$826$	0	0
$827$	−10.8452	−0.377126	−0.188563	−	0.982061i	$-0.560383\pi$
$827$	−10.8452	−0.377126	−0.188563	+	0.982061i	$0.560383\pi$
$828$	0	0
$829$	41.1646	1.42970	0.714852	−	0.699275i	$-0.246494\pi$
$829$	41.1646	1.42970	0.714852	+	0.699275i	$0.246494\pi$
$830$	0	0
$831$	−47.4717	−1.64678
$832$	0	0
$833$	1.65442	0.0573222
$834$	0	0
$835$	14.3948	0.498153
$836$	0	0
$837$	−3.24145	−0.112041
$838$	0	0
$839$	0.302891	0.0104569	0.00522847	−	0.999986i	$-0.498336\pi$
$839$	0.302891	0.0104569	0.00522847	+	0.999986i	$0.498336\pi$
$840$	0	0
$841$	62.1354	2.14260
$842$	0	0
$843$	52.5436	1.80970
$844$	0	0
$845$	2.25153	0.0774549
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−57.2466	−1.96470
$850$	0	0
$851$	−51.5106	−1.76576
$852$	0	0
$853$	42.5382	1.45648	0.728239	−	0.685323i	$-0.240339\pi$
$853$	42.5382	1.45648	0.728239	+	0.685323i	$0.240339\pi$
$854$	0	0
$855$	−2.37199	−0.0811205
$856$	0	0
$857$	−8.41571	−0.287475	−0.143738	−	0.989616i	$-0.545912\pi$
$857$	−8.41571	−0.287475	−0.143738	+	0.989616i	$0.545912\pi$
$858$	0	0
$859$	−2.00761	−0.0684986	−0.0342493	−	0.999413i	$-0.510904\pi$
$859$	−2.00761	−0.0684986	−0.0342493	+	0.999413i	$0.510904\pi$
$860$	0	0
$861$	12.6534	0.431226
$862$	0	0
$863$	35.8782	1.22131	0.610655	−	0.791897i	$-0.290906\pi$
$863$	35.8782	1.22131	0.610655	+	0.791897i	$0.290906\pi$
$864$	0	0
$865$	−29.6670	−1.00871
$866$	0	0
$867$	27.6571	0.939283
$868$	0	0
$869$	10.6652	0.361792
$870$	0	0
$871$	8.06636	0.273318
$872$	0	0
$873$	−1.14930	−0.0388979
$874$	0	0
$875$	−11.1014	−0.375297
$876$	0	0
$877$	−34.9263	−1.17938	−0.589689	−	0.807630i	$-0.700750\pi$
$877$	−34.9263	−1.17938	−0.589689	+	0.807630i	$0.700750\pi$
$878$	0	0
$879$	−63.9018	−2.15535
$880$	0	0
$881$	−44.6617	−1.50469	−0.752345	−	0.658770i	$-0.771077\pi$
$881$	−44.6617	−1.50469	−0.752345	+	0.658770i	$0.771077\pi$
$882$	0	0
$883$	21.9976	0.740278	0.370139	−	0.928976i	$-0.379310\pi$
$883$	21.9976	0.740278	0.370139	+	0.928976i	$0.379310\pi$
$884$	0	0
$885$	−1.28609	−0.0432314
$886$	0	0
$887$	−16.8059	−0.564288	−0.282144	−	0.959372i	$-0.591046\pi$
$887$	−16.8059	−0.564288	−0.282144	+	0.959372i	$0.591046\pi$
$888$	0	0
$889$	−15.2957	−0.513000
$890$	0	0
$891$	10.7025	0.358548
$892$	0	0
$893$	6.30423	0.210963
$894$	0	0
$895$	35.7862	1.19620
$896$	0	0
$897$	−11.0273	−0.368190
$898$	0	0
$899$	7.12445	0.237614
$900$	0	0
$901$	−5.96963	−0.198877
$902$	0	0
$903$	−18.5826	−0.618390
$904$	0	0
$905$	−13.4033	−0.445541
$906$	0	0
$907$	0.545998	0.0181296	0.00906479	−	0.999959i	$-0.497115\pi$
$907$	0.545998	0.0181296	0.00906479	+	0.999959i	$0.497115\pi$
$908$	0	0
$909$	−6.58534	−0.218422
$910$	0	0
$911$	−17.5301	−0.580799	−0.290400	−	0.956905i	$-0.593788\pi$
$911$	−17.5301	−0.580799	−0.290400	+	0.956905i	$0.593788\pi$
$912$	0	0
$913$	−6.88489	−0.227857
$914$	0	0
$915$	−31.5182	−1.04196
$916$	0	0
$917$	−11.5725	−0.382157
$918$	0	0
$919$	16.9099	0.557805	0.278903	−	0.960319i	$-0.410029\pi$
$919$	16.9099	0.557805	0.278903	+	0.960319i	$0.410029\pi$
$920$	0	0
$921$	49.4259	1.62864
$922$	0	0
$923$	−11.0316	−0.363110
$924$	0	0
$925$	−0.628413	−0.0206621
$926$	0	0
$927$	5.75460	0.189006
$928$	0	0
$929$	−26.4474	−0.867711	−0.433855	−	0.900983i	$-0.642847\pi$
$929$	−26.4474	−0.867711	−0.433855	+	0.900983i	$0.642847\pi$
$930$	0	0
$931$	−1.38606	−0.0454262
$932$	0	0
$933$	7.10534	0.232618
$934$	0	0
$935$	−3.72497	−0.121819
$936$	0	0
$937$	−40.5559	−1.32490	−0.662452	−	0.749104i	$-0.730484\pi$
$937$	−40.5559	−1.32490	−0.662452	+	0.749104i	$0.730484\pi$
$938$	0	0
$939$	1.33797	0.0436630
$940$	0	0
$941$	−25.7659	−0.839945	−0.419973	−	0.907537i	$-0.637960\pi$
$941$	−25.7659	−0.839945	−0.419973	+	0.907537i	$0.637960\pi$
$942$	0	0
$943$	−37.1089	−1.20843
$944$	0	0
$945$	9.77933	0.318122
$946$	0	0
$947$	7.81450	0.253937	0.126969	−	0.991907i	$-0.459475\pi$
$947$	7.81450	0.253937	0.126969	+	0.991907i	$0.459475\pi$
$948$	0	0
$949$	−3.69721	−0.120017
$950$	0	0
$951$	−50.3770	−1.63359
$952$	0	0
$953$	−13.6985	−0.443738	−0.221869	−	0.975076i	$-0.571216\pi$
$953$	−13.6985	−0.443738	−0.221869	+	0.975076i	$0.571216\pi$
$954$	0	0
$955$	−2.48552	−0.0804296
$956$	0	0
$957$	−18.5115	−0.598392
$958$	0	0
$959$	7.21055	0.232841
$960$	0	0
$961$	−30.4431	−0.982034
$962$	0	0
$963$	5.89999	0.190124
$964$	0	0
$965$	53.7767	1.73113
$966$	0	0
$967$	−28.5507	−0.918127	−0.459064	−	0.888403i	$-0.651815\pi$
$967$	−28.5507	−0.918127	−0.459064	+	0.888403i	$0.651815\pi$
$968$	0	0
$969$	4.44656	0.142844
$970$	0	0
$971$	6.75335	0.216725	0.108363	−	0.994111i	$-0.465439\pi$
$971$	6.75335	0.216725	0.108363	+	0.994111i	$0.465439\pi$
$972$	0	0
$973$	2.79111	0.0894789
$974$	0	0
$975$	−0.134529	−0.00430839
$976$	0	0
$977$	−9.29483	−0.297368	−0.148684	−	0.988885i	$-0.547504\pi$
$977$	−9.29483	−0.297368	−0.148684	+	0.988885i	$0.547504\pi$
$978$	0	0
$979$	−5.81991	−0.186005
$980$	0	0
$981$	−14.5019	−0.463009
$982$	0	0
$983$	6.08385	0.194045	0.0970223	−	0.995282i	$-0.469068\pi$
$983$	6.08385	0.194045	0.0970223	+	0.995282i	$0.469068\pi$
$984$	0	0
$985$	36.6861	1.16892
$986$	0	0
$987$	8.81960	0.280731
$988$	0	0
$989$	54.4977	1.73293
$990$	0	0
$991$	2.77514	0.0881551	0.0440776	−	0.999028i	$-0.485965\pi$
$991$	2.77514	0.0881551	0.0440776	+	0.999028i	$0.485965\pi$
$992$	0	0
$993$	−5.99201	−0.190151
$994$	0	0
$995$	−13.4149	−0.425280
$996$	0	0
$997$	46.9897	1.48818	0.744090	−	0.668080i	$-0.232884\pi$
$997$	46.9897	1.48818	0.744090	+	0.668080i	$0.232884\pi$
$998$	0	0
$999$	−39.3422	−1.24473

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.o.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.o.1.3	✓	9	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.93909 q^{3} +2.25153 q^{5} +1.00000 q^{7} +0.760073 q^{9} +O(q^{10})\)	\(q-1.93909 q^{3} +2.25153 q^{5} +1.00000 q^{7} +0.760073 q^{9} -1.00000 q^{11} +1.00000 q^{13} -4.36592 q^{15} +1.65442 q^{17} -1.38606 q^{19} -1.93909 q^{21} +5.68682 q^{23} +0.0693775 q^{25} +4.34342 q^{27} -9.54649 q^{29} -0.746290 q^{31} +1.93909 q^{33} +2.25153 q^{35} -9.05788 q^{37} -1.93909 q^{39} -6.52541 q^{41} +9.58315 q^{43} +1.71133 q^{45} -4.54832 q^{47} +1.00000 q^{49} -3.20807 q^{51} -3.60830 q^{53} -2.25153 q^{55} +2.68769 q^{57} +0.294575 q^{59} +7.21914 q^{61} +0.760073 q^{63} +2.25153 q^{65} +8.06636 q^{67} -11.0273 q^{69} -11.0316 q^{71} -3.69721 q^{73} -0.134529 q^{75} -1.00000 q^{77} -10.6652 q^{79} -10.7025 q^{81} +6.88489 q^{83} +3.72497 q^{85} +18.5115 q^{87} +5.81991 q^{89} +1.00000 q^{91} +1.44712 q^{93} -3.12075 q^{95} -1.51209 q^{97} -0.760073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) \) 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9	\( 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9} - 9 q^{11} + 9 q^{13} - 4 q^{15} - 9 q^{17} + 9 q^{19} - 2 q^{21} - 10 q^{23} - q^{25} - 8 q^{27} - 13 q^{29} + 7 q^{31} + 2 q^{33} - 4 q^{35} - 15 q^{37} - 2 q^{39} - 18 q^{41} + 7 q^{43} + 9 q^{45} - 11 q^{47} + 9 q^{49} + q^{51} - 16 q^{53} + 4 q^{55} - 26 q^{57} - 2 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + 13 q^{67} - 3 q^{69} - q^{71} - 6 q^{73} - 4 q^{75} - 9 q^{77} - 21 q^{79} - 23 q^{81} - 30 q^{83} + q^{85} + q^{87} - 36 q^{89} + 9 q^{91} - 22 q^{93} - 23 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) \) 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9 - 9 * q^11 + 9 * q^13 - 4 * q^15 - 9 * q^17 + 9 * q^19 - 2 * q^21 - 10 * q^23 - q^25 - 8 * q^27 - 13 * q^29 + 7 * q^31 + 2 * q^33 - 4 * q^35 - 15 * q^37 - 2 * q^39 - 18 * q^41 + 7 * q^43 + 9 * q^45 - 11 * q^47 + 9 * q^49 + q^51 - 16 * q^53 + 4 * q^55 - 26 * q^57 - 2 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + 13 * q^67 - 3 * q^69 - q^71 - 6 * q^73 - 4 * q^75 - 9 * q^77 - 21 * q^79 - 23 * q^81 - 30 * q^83 + q^85 + q^87 - 36 * q^89 + 9 * q^91 - 22 * q^93 - 23 * q^95 - 5 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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