LMFDB - Embedded newform 8016.2.a.v.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 20x^{5} + 2x^{4} + 87x^{3} + 46x^{2} - 48x - 2 $ x^7 - 2x^6 - 20x^5 + 2x^4 + 87x^3 + 46x^2 - 48x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.09849$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.09849 q^{5} +2.06938 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.09849 q^{5} +2.06938 q^{7} +1.00000 q^{9} +4.75971 q^{11} +0.499475 q^{13} -3.09849 q^{15} +6.36159 q^{17} -6.44057 q^{19} -2.06938 q^{21} -0.293477 q^{23} +4.60061 q^{25} -1.00000 q^{27} +7.05737 q^{29} +1.77590 q^{31} -4.75971 q^{33} +6.41193 q^{35} -3.97068 q^{37} -0.499475 q^{39} -2.87219 q^{41} +11.0906 q^{43} +3.09849 q^{45} +1.77172 q^{47} -2.71768 q^{49} -6.36159 q^{51} +1.45018 q^{53} +14.7479 q^{55} +6.44057 q^{57} +11.4926 q^{59} -7.98819 q^{61} +2.06938 q^{63} +1.54762 q^{65} -12.0378 q^{67} +0.293477 q^{69} +4.64935 q^{71} -9.16773 q^{73} -4.60061 q^{75} +9.84964 q^{77} +1.42547 q^{79} +1.00000 q^{81} +10.1399 q^{83} +19.7113 q^{85} -7.05737 q^{87} +4.77024 q^{89} +1.03360 q^{91} -1.77590 q^{93} -19.9560 q^{95} +8.47739 q^{97} +4.75971 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{3} + 5 q^{5} - 7 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^3 + 5 * q^5 - 7 * q^7 + 7 * q^9	$ 7 q - 7 q^{3} + 5 q^{5} - 7 q^{7} + 7 q^{9} + 6 q^{13} - 5 q^{15} + 6 q^{17} + 2 q^{19} + 7 q^{21} + 12 q^{25} - 7 q^{27} - 4 q^{29} - 7 q^{31} + 13 q^{35} - 3 q^{37} - 6 q^{39} - 12 q^{41} + 2 q^{43} + 5 q^{45} + 11 q^{47} + 10 q^{49} - 6 q^{51} + q^{53} + 2 q^{55} - 2 q^{57} + 19 q^{59} + 12 q^{61} - 7 q^{63} - 10 q^{65} + 17 q^{67} + 20 q^{71} + 10 q^{73} - 12 q^{75} - 24 q^{77} - 2 q^{79} + 7 q^{81} + 7 q^{83} - 18 q^{85} + 4 q^{87} - 3 q^{89} - 4 q^{91} + 7 q^{93} + 24 q^{95} - 3 q^{97}+O(q^{100}) $ 7 * q - 7 * q^3 + 5 * q^5 - 7 * q^7 + 7 * q^9 + 6 * q^13 - 5 * q^15 + 6 * q^17 + 2 * q^19 + 7 * q^21 + 12 * q^25 - 7 * q^27 - 4 * q^29 - 7 * q^31 + 13 * q^35 - 3 * q^37 - 6 * q^39 - 12 * q^41 + 2 * q^43 + 5 * q^45 + 11 * q^47 + 10 * q^49 - 6 * q^51 + q^53 + 2 * q^55 - 2 * q^57 + 19 * q^59 + 12 * q^61 - 7 * q^63 - 10 * q^65 + 17 * q^67 + 20 * q^71 + 10 * q^73 - 12 * q^75 - 24 * q^77 - 2 * q^79 + 7 * q^81 + 7 * q^83 - 18 * q^85 + 4 * q^87 - 3 * q^89 - 4 * q^91 + 7 * q^93 + 24 * q^95 - 3 * q^97

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.09849	1.38568	0.692842	−	0.721089i	$-0.256358\pi$
$5$	3.09849	1.38568	0.692842	+	0.721089i	$0.256358\pi$
$6$	0	0
$7$	2.06938	0.782151	0.391075	−	0.920359i	$-0.372103\pi$
$7$	2.06938	0.782151	0.391075	+	0.920359i	$0.372103\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.75971	1.43511	0.717554	−	0.696503i	$-0.245262\pi$
$11$	4.75971	1.43511	0.717554	+	0.696503i	$0.245262\pi$
$12$	0	0
$13$	0.499475	0.138530	0.0692648	−	0.997598i	$-0.477935\pi$
$13$	0.499475	0.138530	0.0692648	+	0.997598i	$0.477935\pi$
$14$	0	0
$15$	−3.09849	−0.800026
$16$	0	0
$17$	6.36159	1.54291	0.771456	−	0.636283i	$-0.219529\pi$
$17$	6.36159	1.54291	0.771456	+	0.636283i	$0.219529\pi$
$18$	0	0
$19$	−6.44057	−1.47757	−0.738784	−	0.673942i	$-0.764600\pi$
$19$	−6.44057	−1.47757	−0.738784	+	0.673942i	$0.764600\pi$
$20$	0	0
$21$	−2.06938	−0.451575
$22$	0	0
$23$	−0.293477	−0.0611942	−0.0305971	−	0.999532i	$-0.509741\pi$
$23$	−0.293477	−0.0611942	−0.0305971	+	0.999532i	$0.509741\pi$
$24$	0	0
$25$	4.60061	0.920123
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.05737	1.31052	0.655260	−	0.755403i	$-0.272559\pi$
$29$	7.05737	1.31052	0.655260	+	0.755403i	$0.272559\pi$
$30$	0	0
$31$	1.77590	0.318961	0.159480	−	0.987201i	$-0.449018\pi$
$31$	1.77590	0.318961	0.159480	+	0.987201i	$0.449018\pi$
$32$	0	0
$33$	−4.75971	−0.828560
$34$	0	0
$35$	6.41193	1.08381
$36$	0	0
$37$	−3.97068	−0.652775	−0.326388	−	0.945236i	$-0.605831\pi$
$37$	−3.97068	−0.652775	−0.326388	+	0.945236i	$0.605831\pi$
$38$	0	0
$39$	−0.499475	−0.0799801
$40$	0	0
$41$	−2.87219	−0.448561	−0.224280	−	0.974525i	$-0.572003\pi$
$41$	−2.87219	−0.448561	−0.224280	+	0.974525i	$0.572003\pi$
$42$	0	0
$43$	11.0906	1.69130	0.845650	−	0.533738i	$-0.179213\pi$
$43$	11.0906	1.69130	0.845650	+	0.533738i	$0.179213\pi$
$44$	0	0
$45$	3.09849	0.461895
$46$	0	0
$47$	1.77172	0.258432	0.129216	−	0.991616i	$-0.458754\pi$
$47$	1.77172	0.258432	0.129216	+	0.991616i	$0.458754\pi$
$48$	0	0
$49$	−2.71768	−0.388240
$50$	0	0
$51$	−6.36159	−0.890801
$52$	0	0
$53$	1.45018	0.199198	0.0995988	−	0.995028i	$-0.468244\pi$
$53$	1.45018	0.199198	0.0995988	+	0.995028i	$0.468244\pi$
$54$	0	0
$55$	14.7479	1.98861
$56$	0	0
$57$	6.44057	0.853074
$58$	0	0
$59$	11.4926	1.49621	0.748106	−	0.663579i	$-0.230963\pi$
$59$	11.4926	1.49621	0.748106	+	0.663579i	$0.230963\pi$
$60$	0	0
$61$	−7.98819	−1.02278	−0.511391	−	0.859348i	$-0.670870\pi$
$61$	−7.98819	−1.02278	−0.511391	+	0.859348i	$0.670870\pi$
$62$	0	0
$63$	2.06938	0.260717
$64$	0	0
$65$	1.54762	0.191958
$66$	0	0
$67$	−12.0378	−1.47066	−0.735329	−	0.677711i	$-0.762972\pi$
$67$	−12.0378	−1.47066	−0.735329	+	0.677711i	$0.762972\pi$
$68$	0	0
$69$	0.293477	0.0353305
$70$	0	0
$71$	4.64935	0.551777	0.275888	−	0.961190i	$-0.411028\pi$
$71$	4.64935	0.551777	0.275888	+	0.961190i	$0.411028\pi$
$72$	0	0
$73$	−9.16773	−1.07300	−0.536501	−	0.843900i	$-0.680254\pi$
$73$	−9.16773	−1.07300	−0.536501	+	0.843900i	$0.680254\pi$
$74$	0	0
$75$	−4.60061	−0.531233
$76$	0	0
$77$	9.84964	1.12247
$78$	0	0
$79$	1.42547	0.160378	0.0801888	−	0.996780i	$-0.474448\pi$
$79$	1.42547	0.160378	0.0801888	+	0.996780i	$0.474448\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.1399	1.11300	0.556498	−	0.830849i	$-0.312145\pi$
$83$	10.1399	1.11300	0.556498	+	0.830849i	$0.312145\pi$
$84$	0	0
$85$	19.7113	2.13799
$86$	0	0
$87$	−7.05737	−0.756630
$88$	0	0
$89$	4.77024	0.505644	0.252822	−	0.967513i	$-0.418641\pi$
$89$	4.77024	0.505644	0.252822	+	0.967513i	$0.418641\pi$
$90$	0	0
$91$	1.03360	0.108351
$92$	0	0
$93$	−1.77590	−0.184152
$94$	0	0
$95$	−19.9560	−2.04744
$96$	0	0
$97$	8.47739	0.860749	0.430374	−	0.902650i	$-0.358382\pi$
$97$	8.47739	0.860749	0.430374	+	0.902650i	$0.358382\pi$
$98$	0	0
$99$	4.75971	0.478369
$100$	0	0
$101$	−10.0110	−0.996134	−0.498067	−	0.867138i	$-0.665957\pi$
$101$	−10.0110	−0.996134	−0.498067	+	0.867138i	$0.665957\pi$
$102$	0	0
$103$	−7.38830	−0.727991	−0.363996	−	0.931401i	$-0.618588\pi$
$103$	−7.38830	−0.727991	−0.363996	+	0.931401i	$0.618588\pi$
$104$	0	0
$105$	−6.41193	−0.625741
$106$	0	0
$107$	14.9310	1.44344	0.721718	−	0.692187i	$-0.243353\pi$
$107$	14.9310	1.44344	0.721718	+	0.692187i	$0.243353\pi$
$108$	0	0
$109$	10.4793	1.00374	0.501869	−	0.864944i	$-0.332646\pi$
$109$	10.4793	1.00374	0.501869	+	0.864944i	$0.332646\pi$
$110$	0	0
$111$	3.97068	0.376880
$112$	0	0
$113$	3.10380	0.291981	0.145990	−	0.989286i	$-0.453363\pi$
$113$	3.10380	0.291981	0.145990	+	0.989286i	$0.453363\pi$
$114$	0	0
$115$	−0.909334	−0.0847959
$116$	0	0
$117$	0.499475	0.0461765
$118$	0	0
$119$	13.1645	1.20679
$120$	0	0
$121$	11.6549	1.05953
$122$	0	0
$123$	2.87219	0.258977
$124$	0	0
$125$	−1.23749	−0.110685
$126$	0	0
$127$	3.84421	0.341119	0.170559	−	0.985347i	$-0.445443\pi$
$127$	3.84421	0.341119	0.170559	+	0.985347i	$0.445443\pi$
$128$	0	0
$129$	−11.0906	−0.976472
$130$	0	0
$131$	15.3758	1.34339	0.671696	−	0.740827i	$-0.265566\pi$
$131$	15.3758	1.34339	0.671696	+	0.740827i	$0.265566\pi$
$132$	0	0
$133$	−13.3280	−1.15568
$134$	0	0
$135$	−3.09849	−0.266675
$136$	0	0
$137$	−8.16754	−0.697800	−0.348900	−	0.937160i	$-0.613445\pi$
$137$	−8.16754	−0.697800	−0.348900	+	0.937160i	$0.613445\pi$
$138$	0	0
$139$	−14.5251	−1.23200	−0.616000	−	0.787746i	$-0.711248\pi$
$139$	−14.5251	−1.23200	−0.616000	+	0.787746i	$0.711248\pi$
$140$	0	0
$141$	−1.77172	−0.149206
$142$	0	0
$143$	2.37736	0.198805
$144$	0	0
$145$	21.8672	1.81597
$146$	0	0
$147$	2.71768	0.224151
$148$	0	0
$149$	−10.0660	−0.824642	−0.412321	−	0.911039i	$-0.635282\pi$
$149$	−10.0660	−0.824642	−0.412321	+	0.911039i	$0.635282\pi$
$150$	0	0
$151$	−21.4071	−1.74209	−0.871045	−	0.491204i	$-0.836557\pi$
$151$	−21.4071	−1.74209	−0.871045	+	0.491204i	$0.836557\pi$
$152$	0	0
$153$	6.36159	0.514304
$154$	0	0
$155$	5.50260	0.441979
$156$	0	0
$157$	−18.7779	−1.49864	−0.749321	−	0.662207i	$-0.769620\pi$
$157$	−18.7779	−1.49864	−0.749321	+	0.662207i	$0.769620\pi$
$158$	0	0
$159$	−1.45018	−0.115007
$160$	0	0
$161$	−0.607315	−0.0478631
$162$	0	0
$163$	−16.2023	−1.26906	−0.634530	−	0.772899i	$-0.718806\pi$
$163$	−16.2023	−1.26906	−0.634530	+	0.772899i	$0.718806\pi$
$164$	0	0
$165$	−14.7479	−1.14812
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−12.7505	−0.980810
$170$	0	0
$171$	−6.44057	−0.492523
$172$	0	0
$173$	1.10174	0.0837634	0.0418817	−	0.999123i	$-0.486665\pi$
$173$	1.10174	0.0837634	0.0418817	+	0.999123i	$0.486665\pi$
$174$	0	0
$175$	9.52040	0.719675
$176$	0	0
$177$	−11.4926	−0.863839
$178$	0	0
$179$	−17.6300	−1.31773	−0.658864	−	0.752262i	$-0.728963\pi$
$179$	−17.6300	−1.31773	−0.658864	+	0.752262i	$0.728963\pi$
$180$	0	0
$181$	25.5175	1.89670	0.948352	−	0.317221i	$-0.102750\pi$
$181$	25.5175	1.89670	0.948352	+	0.317221i	$0.102750\pi$
$182$	0	0
$183$	7.98819	0.590504
$184$	0	0
$185$	−12.3031	−0.904541
$186$	0	0
$187$	30.2793	2.21424
$188$	0	0
$189$	−2.06938	−0.150525
$190$	0	0
$191$	2.26824	0.164124	0.0820622	−	0.996627i	$-0.473849\pi$
$191$	2.26824	0.164124	0.0820622	+	0.996627i	$0.473849\pi$
$192$	0	0
$193$	8.82627	0.635329	0.317664	−	0.948203i	$-0.397101\pi$
$193$	8.82627	0.635329	0.317664	+	0.948203i	$0.397101\pi$
$194$	0	0
$195$	−1.54762	−0.110827
$196$	0	0
$197$	7.67066	0.546512	0.273256	−	0.961941i	$-0.411899\pi$
$197$	7.67066	0.546512	0.273256	+	0.961941i	$0.411899\pi$
$198$	0	0
$199$	16.6020	1.17688	0.588441	−	0.808540i	$-0.299742\pi$
$199$	16.6020	1.17688	0.588441	+	0.808540i	$0.299742\pi$
$200$	0	0
$201$	12.0378	0.849084
$202$	0	0
$203$	14.6044	1.02502
$204$	0	0
$205$	−8.89944	−0.621564
$206$	0	0
$207$	−0.293477	−0.0203981
$208$	0	0
$209$	−30.6553	−2.12047
$210$	0	0
$211$	−18.0639	−1.24357	−0.621786	−	0.783187i	$-0.713593\pi$
$211$	−18.0639	−1.24357	−0.621786	+	0.783187i	$0.713593\pi$
$212$	0	0
$213$	−4.64935	−0.318568
$214$	0	0
$215$	34.3640	2.34361
$216$	0	0
$217$	3.67500	0.249476
$218$	0	0
$219$	9.16773	0.619498
$220$	0	0
$221$	3.17746	0.213739
$222$	0	0
$223$	16.8630	1.12923	0.564615	−	0.825355i	$-0.309025\pi$
$223$	16.8630	1.12923	0.564615	+	0.825355i	$0.309025\pi$
$224$	0	0
$225$	4.60061	0.306708
$226$	0	0
$227$	−17.5706	−1.16620	−0.583102	−	0.812399i	$-0.698161\pi$
$227$	−17.5706	−1.16620	−0.583102	+	0.812399i	$0.698161\pi$
$228$	0	0
$229$	−13.0305	−0.861080	−0.430540	−	0.902571i	$-0.641677\pi$
$229$	−13.0305	−0.861080	−0.430540	+	0.902571i	$0.641677\pi$
$230$	0	0
$231$	−9.84964	−0.648059
$232$	0	0
$233$	−25.9165	−1.69785	−0.848923	−	0.528517i	$-0.822748\pi$
$233$	−25.9165	−1.69785	−0.848923	+	0.528517i	$0.822748\pi$
$234$	0	0
$235$	5.48964	0.358105
$236$	0	0
$237$	−1.42547	−0.0925941
$238$	0	0
$239$	−0.800608	−0.0517870	−0.0258935	−	0.999665i	$-0.508243\pi$
$239$	−0.800608	−0.0517870	−0.0258935	+	0.999665i	$0.508243\pi$
$240$	0	0
$241$	−26.0249	−1.67641	−0.838205	−	0.545356i	$-0.816395\pi$
$241$	−26.0249	−1.67641	−0.838205	+	0.545356i	$0.816395\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−8.42070	−0.537979
$246$	0	0
$247$	−3.21691	−0.204687
$248$	0	0
$249$	−10.1399	−0.642589
$250$	0	0
$251$	−0.646826	−0.0408273	−0.0204136	−	0.999792i	$-0.506498\pi$
$251$	−0.646826	−0.0408273	−0.0204136	+	0.999792i	$0.506498\pi$
$252$	0	0
$253$	−1.39687	−0.0878202
$254$	0	0
$255$	−19.7113	−1.23437
$256$	0	0
$257$	20.9626	1.30761	0.653805	−	0.756663i	$-0.273171\pi$
$257$	20.9626	1.30761	0.653805	+	0.756663i	$0.273171\pi$
$258$	0	0
$259$	−8.21682	−0.510569
$260$	0	0
$261$	7.05737	0.436840
$262$	0	0
$263$	−20.2077	−1.24606	−0.623030	−	0.782198i	$-0.714099\pi$
$263$	−20.2077	−1.24606	−0.623030	+	0.782198i	$0.714099\pi$
$264$	0	0
$265$	4.49337	0.276025
$266$	0	0
$267$	−4.77024	−0.291934
$268$	0	0
$269$	20.9835	1.27939	0.639693	−	0.768631i	$-0.279062\pi$
$269$	20.9835	1.27939	0.639693	+	0.768631i	$0.279062\pi$
$270$	0	0
$271$	−32.5957	−1.98005	−0.990023	−	0.140903i	$-0.954999\pi$
$271$	−32.5957	−1.98005	−0.990023	+	0.140903i	$0.954999\pi$
$272$	0	0
$273$	−1.03360	−0.0625565
$274$	0	0
$275$	21.8976	1.32047
$276$	0	0
$277$	13.9249	0.836664	0.418332	−	0.908294i	$-0.362615\pi$
$277$	13.9249	0.836664	0.418332	+	0.908294i	$0.362615\pi$
$278$	0	0
$279$	1.77590	0.106320
$280$	0	0
$281$	2.95960	0.176555	0.0882775	−	0.996096i	$-0.471864\pi$
$281$	2.95960	0.176555	0.0882775	+	0.996096i	$0.471864\pi$
$282$	0	0
$283$	−18.6587	−1.10915	−0.554573	−	0.832135i	$-0.687118\pi$
$283$	−18.6587	−1.10915	−0.554573	+	0.832135i	$0.687118\pi$
$284$	0	0
$285$	19.9560	1.18209
$286$	0	0
$287$	−5.94364	−0.350842
$288$	0	0
$289$	23.4698	1.38058
$290$	0	0
$291$	−8.47739	−0.496954
$292$	0	0
$293$	8.70037	0.508281	0.254141	−	0.967167i	$-0.418207\pi$
$293$	8.70037	0.508281	0.254141	+	0.967167i	$0.418207\pi$
$294$	0	0
$295$	35.6097	2.07328
$296$	0	0
$297$	−4.75971	−0.276187
$298$	0	0
$299$	−0.146585	−0.00847721
$300$	0	0
$301$	22.9506	1.32285
$302$	0	0
$303$	10.0110	0.575118
$304$	0	0
$305$	−24.7513	−1.41725
$306$	0	0
$307$	2.29116	0.130763	0.0653817	−	0.997860i	$-0.479174\pi$
$307$	2.29116	0.130763	0.0653817	+	0.997860i	$0.479174\pi$
$308$	0	0
$309$	7.38830	0.420306
$310$	0	0
$311$	19.4482	1.10280	0.551402	−	0.834240i	$-0.314093\pi$
$311$	19.4482	1.10280	0.551402	+	0.834240i	$0.314093\pi$
$312$	0	0
$313$	−3.97617	−0.224746	−0.112373	−	0.993666i	$-0.535845\pi$
$313$	−3.97617	−0.224746	−0.112373	+	0.993666i	$0.535845\pi$
$314$	0	0
$315$	6.41193	0.361272
$316$	0	0
$317$	−14.5851	−0.819183	−0.409591	−	0.912269i	$-0.634329\pi$
$317$	−14.5851	−0.819183	−0.409591	+	0.912269i	$0.634329\pi$
$318$	0	0
$319$	33.5911	1.88074
$320$	0	0
$321$	−14.9310	−0.833368
$322$	0	0
$323$	−40.9723	−2.27976
$324$	0	0
$325$	2.29789	0.127464
$326$	0	0
$327$	−10.4793	−0.579508
$328$	0	0
$329$	3.66635	0.202133
$330$	0	0
$331$	18.7063	1.02819	0.514097	−	0.857732i	$-0.328127\pi$
$331$	18.7063	1.02819	0.514097	+	0.857732i	$0.328127\pi$
$332$	0	0
$333$	−3.97068	−0.217592
$334$	0	0
$335$	−37.2991	−2.03787
$336$	0	0
$337$	17.3101	0.942940	0.471470	−	0.881882i	$-0.343724\pi$
$337$	17.3101	0.942940	0.471470	+	0.881882i	$0.343724\pi$
$338$	0	0
$339$	−3.10380	−0.168575
$340$	0	0
$341$	8.45277	0.457743
$342$	0	0
$343$	−20.1095	−1.08581
$344$	0	0
$345$	0.909334	0.0489569
$346$	0	0
$347$	14.1957	0.762065	0.381032	−	0.924562i	$-0.375569\pi$
$347$	14.1957	0.762065	0.381032	+	0.924562i	$0.375569\pi$
$348$	0	0
$349$	17.4568	0.934443	0.467221	−	0.884140i	$-0.345255\pi$
$349$	17.4568	0.934443	0.467221	+	0.884140i	$0.345255\pi$
$350$	0	0
$351$	−0.499475	−0.0266600
$352$	0	0
$353$	−9.27980	−0.493914	−0.246957	−	0.969026i	$-0.579431\pi$
$353$	−9.27980	−0.493914	−0.246957	+	0.969026i	$0.579431\pi$
$354$	0	0
$355$	14.4060	0.764589
$356$	0	0
$357$	−13.1645	−0.696740
$358$	0	0
$359$	15.9284	0.840668	0.420334	−	0.907369i	$-0.361913\pi$
$359$	15.9284	0.840668	0.420334	+	0.907369i	$0.361913\pi$
$360$	0	0
$361$	22.4810	1.18321
$362$	0	0
$363$	−11.6549	−0.611722
$364$	0	0
$365$	−28.4061	−1.48684
$366$	0	0
$367$	−16.4588	−0.859142	−0.429571	−	0.903033i	$-0.641335\pi$
$367$	−16.4588	−0.859142	−0.429571	+	0.903033i	$0.641335\pi$
$368$	0	0
$369$	−2.87219	−0.149520
$370$	0	0
$371$	3.00097	0.155803
$372$	0	0
$373$	19.6427	1.01706	0.508531	−	0.861044i	$-0.330189\pi$
$373$	19.6427	1.01706	0.508531	+	0.861044i	$0.330189\pi$
$374$	0	0
$375$	1.23749	0.0639039
$376$	0	0
$377$	3.52498	0.181546
$378$	0	0
$379$	5.84243	0.300105	0.150053	−	0.988678i	$-0.452056\pi$
$379$	5.84243	0.300105	0.150053	+	0.988678i	$0.452056\pi$
$380$	0	0
$381$	−3.84421	−0.196945
$382$	0	0
$383$	9.39101	0.479858	0.239929	−	0.970790i	$-0.422876\pi$
$383$	9.39101	0.479858	0.239929	+	0.970790i	$0.422876\pi$
$384$	0	0
$385$	30.5190	1.55539
$386$	0	0
$387$	11.0906	0.563766
$388$	0	0
$389$	−6.47379	−0.328234	−0.164117	−	0.986441i	$-0.552477\pi$
$389$	−6.47379	−0.328234	−0.164117	+	0.986441i	$0.552477\pi$
$390$	0	0
$391$	−1.86698	−0.0944173
$392$	0	0
$393$	−15.3758	−0.775608
$394$	0	0
$395$	4.41679	0.222233
$396$	0	0
$397$	−4.11840	−0.206697	−0.103348	−	0.994645i	$-0.532956\pi$
$397$	−4.11840	−0.206697	−0.103348	+	0.994645i	$0.532956\pi$
$398$	0	0
$399$	13.3280	0.667233
$400$	0	0
$401$	−33.6255	−1.67918	−0.839589	−	0.543222i	$-0.817204\pi$
$401$	−33.6255	−1.67918	−0.839589	+	0.543222i	$0.817204\pi$
$402$	0	0
$403$	0.887018	0.0441855
$404$	0	0
$405$	3.09849	0.153965
$406$	0	0
$407$	−18.8993	−0.936802
$408$	0	0
$409$	25.1679	1.24447	0.622237	−	0.782829i	$-0.286224\pi$
$409$	25.1679	1.24447	0.622237	+	0.782829i	$0.286224\pi$
$410$	0	0
$411$	8.16754	0.402875
$412$	0	0
$413$	23.7826	1.17026
$414$	0	0
$415$	31.4183	1.54226
$416$	0	0
$417$	14.5251	0.711295
$418$	0	0
$419$	5.42044	0.264806	0.132403	−	0.991196i	$-0.457731\pi$
$419$	5.42044	0.264806	0.132403	+	0.991196i	$0.457731\pi$
$420$	0	0
$421$	−9.46445	−0.461269	−0.230634	−	0.973040i	$-0.574080\pi$
$421$	−9.46445	−0.461269	−0.230634	+	0.973040i	$0.574080\pi$
$422$	0	0
$423$	1.77172	0.0861439
$424$	0	0
$425$	29.2672	1.41967
$426$	0	0
$427$	−16.5306	−0.799970
$428$	0	0
$429$	−2.37736	−0.114780
$430$	0	0
$431$	11.4748	0.552721	0.276361	−	0.961054i	$-0.410872\pi$
$431$	11.4748	0.552721	0.276361	+	0.961054i	$0.410872\pi$
$432$	0	0
$433$	27.9160	1.34156	0.670778	−	0.741658i	$-0.265960\pi$
$433$	27.9160	1.34156	0.670778	+	0.741658i	$0.265960\pi$
$434$	0	0
$435$	−21.8672	−1.04845
$436$	0	0
$437$	1.89016	0.0904186
$438$	0	0
$439$	15.0998	0.720674	0.360337	−	0.932822i	$-0.382662\pi$
$439$	15.0998	0.720674	0.360337	+	0.932822i	$0.382662\pi$
$440$	0	0
$441$	−2.71768	−0.129413
$442$	0	0
$443$	17.0042	0.807895	0.403948	−	0.914782i	$-0.367638\pi$
$443$	17.0042	0.807895	0.403948	+	0.914782i	$0.367638\pi$
$444$	0	0
$445$	14.7805	0.700664
$446$	0	0
$447$	10.0660	0.476107
$448$	0	0
$449$	1.49847	0.0707174	0.0353587	−	0.999375i	$-0.488743\pi$
$449$	1.49847	0.0707174	0.0353587	+	0.999375i	$0.488743\pi$
$450$	0	0
$451$	−13.6708	−0.643733
$452$	0	0
$453$	21.4071	1.00580
$454$	0	0
$455$	3.20260	0.150140
$456$	0	0
$457$	−22.7176	−1.06269	−0.531343	−	0.847157i	$-0.678312\pi$
$457$	−22.7176	−1.06269	−0.531343	+	0.847157i	$0.678312\pi$
$458$	0	0
$459$	−6.36159	−0.296934
$460$	0	0
$461$	4.08508	0.190261	0.0951306	−	0.995465i	$-0.469673\pi$
$461$	4.08508	0.190261	0.0951306	+	0.995465i	$0.469673\pi$
$462$	0	0
$463$	−5.70881	−0.265311	−0.132655	−	0.991162i	$-0.542350\pi$
$463$	−5.70881	−0.265311	−0.132655	+	0.991162i	$0.542350\pi$
$464$	0	0
$465$	−5.50260	−0.255177
$466$	0	0
$467$	32.2320	1.49152	0.745759	−	0.666216i	$-0.232087\pi$
$467$	32.2320	1.49152	0.745759	+	0.666216i	$0.232087\pi$
$468$	0	0
$469$	−24.9108	−1.15028
$470$	0	0
$471$	18.7779	0.865241
$472$	0	0
$473$	52.7880	2.42720
$474$	0	0
$475$	−29.6306	−1.35954
$476$	0	0
$477$	1.45018	0.0663992
$478$	0	0
$479$	−24.4320	−1.11633	−0.558163	−	0.829731i	$-0.688494\pi$
$479$	−24.4320	−1.11633	−0.558163	+	0.829731i	$0.688494\pi$
$480$	0	0
$481$	−1.98326	−0.0904287
$482$	0	0
$483$	0.607315	0.0276338
$484$	0	0
$485$	26.2671	1.19273
$486$	0	0
$487$	11.8753	0.538122	0.269061	−	0.963123i	$-0.413287\pi$
$487$	11.8753	0.538122	0.269061	+	0.963123i	$0.413287\pi$
$488$	0	0
$489$	16.2023	0.732692
$490$	0	0
$491$	−7.98913	−0.360544	−0.180272	−	0.983617i	$-0.557698\pi$
$491$	−7.98913	−0.360544	−0.180272	+	0.983617i	$0.557698\pi$
$492$	0	0
$493$	44.8961	2.02202
$494$	0	0
$495$	14.7479	0.662869
$496$	0	0
$497$	9.62126	0.431573
$498$	0	0
$499$	−0.412366	−0.0184600	−0.00923002	−	0.999957i	$-0.502938\pi$
$499$	−0.412366	−0.0184600	−0.00923002	+	0.999957i	$0.502938\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	27.2489	1.21497	0.607483	−	0.794333i	$-0.292179\pi$
$503$	27.2489	1.21497	0.607483	+	0.794333i	$0.292179\pi$
$504$	0	0
$505$	−31.0190	−1.38033
$506$	0	0
$507$	12.7505	0.566271
$508$	0	0
$509$	−0.338765	−0.0150155	−0.00750774	−	0.999972i	$-0.502390\pi$
$509$	−0.338765	−0.0150155	−0.00750774	+	0.999972i	$0.502390\pi$
$510$	0	0
$511$	−18.9715	−0.839249
$512$	0	0
$513$	6.44057	0.284358
$514$	0	0
$515$	−22.8926	−1.00877
$516$	0	0
$517$	8.43287	0.370877
$518$	0	0
$519$	−1.10174	−0.0483608
$520$	0	0
$521$	7.08251	0.310290	0.155145	−	0.987892i	$-0.450415\pi$
$521$	7.08251	0.310290	0.155145	+	0.987892i	$0.450415\pi$
$522$	0	0
$523$	−4.00855	−0.175282	−0.0876408	−	0.996152i	$-0.527933\pi$
$523$	−4.00855	−0.175282	−0.0876408	+	0.996152i	$0.527933\pi$
$524$	0	0
$525$	−9.52040	−0.415504
$526$	0	0
$527$	11.2975	0.492129
$528$	0	0
$529$	−22.9139	−0.996255
$530$	0	0
$531$	11.4926	0.498738
$532$	0	0
$533$	−1.43459	−0.0621389
$534$	0	0
$535$	46.2635	2.00015
$536$	0	0
$537$	17.6300	0.760790
$538$	0	0
$539$	−12.9354	−0.557166
$540$	0	0
$541$	−3.36638	−0.144732	−0.0723659	−	0.997378i	$-0.523055\pi$
$541$	−3.36638	−0.144732	−0.0723659	+	0.997378i	$0.523055\pi$
$542$	0	0
$543$	−25.5175	−1.09506
$544$	0	0
$545$	32.4700	1.39086
$546$	0	0
$547$	−16.3176	−0.697691	−0.348845	−	0.937180i	$-0.613426\pi$
$547$	−16.3176	−0.697691	−0.348845	+	0.937180i	$0.613426\pi$
$548$	0	0
$549$	−7.98819	−0.340928
$550$	0	0
$551$	−45.4535	−1.93638
$552$	0	0
$553$	2.94983	0.125440
$554$	0	0
$555$	12.3031	0.522237
$556$	0	0
$557$	−4.56364	−0.193368	−0.0966839	−	0.995315i	$-0.530824\pi$
$557$	−4.56364	−0.193368	−0.0966839	+	0.995315i	$0.530824\pi$
$558$	0	0
$559$	5.53948	0.234295
$560$	0	0
$561$	−30.2793	−1.27839
$562$	0	0
$563$	−29.2483	−1.23267	−0.616334	−	0.787485i	$-0.711383\pi$
$563$	−29.2483	−1.23267	−0.616334	+	0.787485i	$0.711383\pi$
$564$	0	0
$565$	9.61708	0.404593
$566$	0	0
$567$	2.06938	0.0869056
$568$	0	0
$569$	38.8732	1.62965	0.814824	−	0.579708i	$-0.196833\pi$
$569$	38.8732	1.62965	0.814824	+	0.579708i	$0.196833\pi$
$570$	0	0
$571$	−42.4636	−1.77705	−0.888523	−	0.458832i	$-0.848268\pi$
$571$	−42.4636	−1.77705	−0.888523	+	0.458832i	$0.848268\pi$
$572$	0	0
$573$	−2.26824	−0.0947573
$574$	0	0
$575$	−1.35017	−0.0563062
$576$	0	0
$577$	−14.6306	−0.609078	−0.304539	−	0.952500i	$-0.598502\pi$
$577$	−14.6306	−0.609078	−0.304539	+	0.952500i	$0.598502\pi$
$578$	0	0
$579$	−8.82627	−0.366807
$580$	0	0
$581$	20.9832	0.870531
$582$	0	0
$583$	6.90245	0.285870
$584$	0	0
$585$	1.54762	0.0639861
$586$	0	0
$587$	−26.6740	−1.10095	−0.550477	−	0.834850i	$-0.685554\pi$
$587$	−26.6740	−1.10095	−0.550477	+	0.834850i	$0.685554\pi$
$588$	0	0
$589$	−11.4378	−0.471287
$590$	0	0
$591$	−7.67066	−0.315529
$592$	0	0
$593$	27.8274	1.14274	0.571368	−	0.820694i	$-0.306413\pi$
$593$	27.8274	1.14274	0.571368	+	0.820694i	$0.306413\pi$
$594$	0	0
$595$	40.7901	1.67223
$596$	0	0
$597$	−16.6020	−0.679473
$598$	0	0
$599$	−45.6903	−1.86685	−0.933427	−	0.358767i	$-0.883197\pi$
$599$	−45.6903	−1.86685	−0.933427	+	0.358767i	$0.883197\pi$
$600$	0	0
$601$	−17.6426	−0.719657	−0.359829	−	0.933018i	$-0.617165\pi$
$601$	−17.6426	−0.719657	−0.359829	+	0.933018i	$0.617165\pi$
$602$	0	0
$603$	−12.0378	−0.490219
$604$	0	0
$605$	36.1124	1.46818
$606$	0	0
$607$	−0.216382	−0.00878269	−0.00439134	−	0.999990i	$-0.501398\pi$
$607$	−0.216382	−0.00878269	−0.00439134	+	0.999990i	$0.501398\pi$
$608$	0	0
$609$	−14.6044	−0.591798
$610$	0	0
$611$	0.884930	0.0358004
$612$	0	0
$613$	34.8578	1.40789	0.703947	−	0.710252i	$-0.251419\pi$
$613$	34.8578	1.40789	0.703947	+	0.710252i	$0.251419\pi$
$614$	0	0
$615$	8.89944	0.358860
$616$	0	0
$617$	−29.5545	−1.18982	−0.594910	−	0.803792i	$-0.702812\pi$
$617$	−29.5545	−1.18982	−0.594910	+	0.803792i	$0.702812\pi$
$618$	0	0
$619$	34.5417	1.38835	0.694174	−	0.719807i	$-0.255770\pi$
$619$	34.5417	1.38835	0.694174	+	0.719807i	$0.255770\pi$
$620$	0	0
$621$	0.293477	0.0117768
$622$	0	0
$623$	9.87142	0.395490
$624$	0	0
$625$	−26.8374	−1.07350
$626$	0	0
$627$	30.6553	1.22425
$628$	0	0
$629$	−25.2598	−1.00717
$630$	0	0
$631$	−45.5913	−1.81496	−0.907480	−	0.420096i	$-0.861996\pi$
$631$	−45.5913	−1.81496	−0.907480	+	0.420096i	$0.861996\pi$
$632$	0	0
$633$	18.0639	0.717977
$634$	0	0
$635$	11.9112	0.472683
$636$	0	0
$637$	−1.35742	−0.0537827
$638$	0	0
$639$	4.64935	0.183926
$640$	0	0
$641$	−34.2370	−1.35228	−0.676140	−	0.736774i	$-0.736348\pi$
$641$	−34.2370	−1.35228	−0.676140	+	0.736774i	$0.736348\pi$
$642$	0	0
$643$	6.49446	0.256116	0.128058	−	0.991767i	$-0.459126\pi$
$643$	6.49446	0.256116	0.128058	+	0.991767i	$0.459126\pi$
$644$	0	0
$645$	−34.3640	−1.35308
$646$	0	0
$647$	−34.1427	−1.34229	−0.671144	−	0.741327i	$-0.734197\pi$
$647$	−34.1427	−1.34229	−0.671144	+	0.741327i	$0.734197\pi$
$648$	0	0
$649$	54.7016	2.14723
$650$	0	0
$651$	−3.67500	−0.144035
$652$	0	0
$653$	14.9336	0.584398	0.292199	−	0.956358i	$-0.405613\pi$
$653$	14.9336	0.584398	0.292199	+	0.956358i	$0.405613\pi$
$654$	0	0
$655$	47.6418	1.86152
$656$	0	0
$657$	−9.16773	−0.357667
$658$	0	0
$659$	24.8384	0.967566	0.483783	−	0.875188i	$-0.339262\pi$
$659$	24.8384	0.967566	0.483783	+	0.875188i	$0.339262\pi$
$660$	0	0
$661$	−19.4186	−0.755295	−0.377648	−	0.925949i	$-0.623267\pi$
$661$	−19.4186	−0.755295	−0.377648	+	0.925949i	$0.623267\pi$
$662$	0	0
$663$	−3.17746	−0.123402
$664$	0	0
$665$	−41.2965	−1.60141
$666$	0	0
$667$	−2.07118	−0.0801963
$668$	0	0
$669$	−16.8630	−0.651961
$670$	0	0
$671$	−38.0215	−1.46780
$672$	0	0
$673$	10.3390	0.398538	0.199269	−	0.979945i	$-0.436143\pi$
$673$	10.3390	0.398538	0.199269	+	0.979945i	$0.436143\pi$
$674$	0	0
$675$	−4.60061	−0.177078
$676$	0	0
$677$	35.3102	1.35708	0.678540	−	0.734563i	$-0.262613\pi$
$677$	35.3102	1.35708	0.678540	+	0.734563i	$0.262613\pi$
$678$	0	0
$679$	17.5429	0.673235
$680$	0	0
$681$	17.5706	0.673308
$682$	0	0
$683$	25.2542	0.966326	0.483163	−	0.875530i	$-0.339488\pi$
$683$	25.2542	0.966326	0.483163	+	0.875530i	$0.339488\pi$
$684$	0	0
$685$	−25.3070	−0.966930
$686$	0	0
$687$	13.0305	0.497145
$688$	0	0
$689$	0.724330	0.0275948
$690$	0	0
$691$	16.7374	0.636721	0.318360	−	0.947970i	$-0.396868\pi$
$691$	16.7374	0.636721	0.318360	+	0.947970i	$0.396868\pi$
$692$	0	0
$693$	9.84964	0.374157
$694$	0	0
$695$	−45.0057	−1.70716
$696$	0	0
$697$	−18.2717	−0.692090
$698$	0	0
$699$	25.9165	0.980252
$700$	0	0
$701$	45.1045	1.70357	0.851787	−	0.523888i	$-0.175519\pi$
$701$	45.1045	1.70357	0.851787	+	0.523888i	$0.175519\pi$
$702$	0	0
$703$	25.5734	0.964520
$704$	0	0
$705$	−5.48964	−0.206752
$706$	0	0
$707$	−20.7166	−0.779127
$708$	0	0
$709$	32.4749	1.21962	0.609811	−	0.792547i	$-0.291246\pi$
$709$	32.4749	1.21962	0.609811	+	0.792547i	$0.291246\pi$
$710$	0	0
$711$	1.42547	0.0534592
$712$	0	0
$713$	−0.521186	−0.0195186
$714$	0	0
$715$	7.36622	0.275481
$716$	0	0
$717$	0.800608	0.0298992
$718$	0	0
$719$	33.5295	1.25044	0.625219	−	0.780449i	$-0.285010\pi$
$719$	33.5295	1.25044	0.625219	+	0.780449i	$0.285010\pi$
$720$	0	0
$721$	−15.2892	−0.569399
$722$	0	0
$723$	26.0249	0.967875
$724$	0	0
$725$	32.4682	1.20584
$726$	0	0
$727$	−11.1937	−0.415150	−0.207575	−	0.978219i	$-0.566557\pi$
$727$	−11.1937	−0.415150	−0.207575	+	0.978219i	$0.566557\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	70.5538	2.60953
$732$	0	0
$733$	29.6881	1.09656	0.548278	−	0.836296i	$-0.315284\pi$
$733$	29.6881	1.09656	0.548278	+	0.836296i	$0.315284\pi$
$734$	0	0
$735$	8.42070	0.310602
$736$	0	0
$737$	−57.2967	−2.11055
$738$	0	0
$739$	31.1744	1.14677	0.573385	−	0.819286i	$-0.305630\pi$
$739$	31.1744	1.14677	0.573385	+	0.819286i	$0.305630\pi$
$740$	0	0
$741$	3.21691	0.118176
$742$	0	0
$743$	4.46999	0.163988	0.0819939	−	0.996633i	$-0.473871\pi$
$743$	4.46999	0.163988	0.0819939	+	0.996633i	$0.473871\pi$
$744$	0	0
$745$	−31.1895	−1.14269
$746$	0	0
$747$	10.1399	0.370999
$748$	0	0
$749$	30.8979	1.12898
$750$	0	0
$751$	−16.3874	−0.597986	−0.298993	−	0.954255i	$-0.596651\pi$
$751$	−16.3874	−0.597986	−0.298993	+	0.954255i	$0.596651\pi$
$752$	0	0
$753$	0.646826	0.0235717
$754$	0	0
$755$	−66.3297	−2.41399
$756$	0	0
$757$	16.3823	0.595426	0.297713	−	0.954655i	$-0.403776\pi$
$757$	16.3823	0.595426	0.297713	+	0.954655i	$0.403776\pi$
$758$	0	0
$759$	1.39687	0.0507030
$760$	0	0
$761$	20.2313	0.733383	0.366691	−	0.930343i	$-0.380490\pi$
$761$	20.2313	0.733383	0.366691	+	0.930343i	$0.380490\pi$
$762$	0	0
$763$	21.6857	0.785074
$764$	0	0
$765$	19.7113	0.712663
$766$	0	0
$767$	5.74029	0.207270
$768$	0	0
$769$	38.4297	1.38581	0.692905	−	0.721029i	$-0.256331\pi$
$769$	38.4297	1.38581	0.692905	+	0.721029i	$0.256331\pi$
$770$	0	0
$771$	−20.9626	−0.754950
$772$	0	0
$773$	20.9232	0.752553	0.376277	−	0.926507i	$-0.377204\pi$
$773$	20.9232	0.752553	0.376277	+	0.926507i	$0.377204\pi$
$774$	0	0
$775$	8.17023	0.293483
$776$	0	0
$777$	8.21682	0.294777
$778$	0	0
$779$	18.4985	0.662779
$780$	0	0
$781$	22.1296	0.791859
$782$	0	0
$783$	−7.05737	−0.252210
$784$	0	0
$785$	−58.1831	−2.07665
$786$	0	0
$787$	26.0386	0.928175	0.464088	−	0.885789i	$-0.346382\pi$
$787$	26.0386	0.928175	0.464088	+	0.885789i	$0.346382\pi$
$788$	0	0
$789$	20.2077	0.719413
$790$	0	0
$791$	6.42293	0.228373
$792$	0	0
$793$	−3.98990	−0.141686
$794$	0	0
$795$	−4.49337	−0.159363
$796$	0	0
$797$	−44.8876	−1.59000	−0.795001	−	0.606609i	$-0.792530\pi$
$797$	−44.8876	−1.59000	−0.795001	+	0.606609i	$0.792530\pi$
$798$	0	0
$799$	11.2709	0.398737
$800$	0	0
$801$	4.77024	0.168548
$802$	0	0
$803$	−43.6358	−1.53987
$804$	0	0
$805$	−1.88176	−0.0663232
$806$	0	0
$807$	−20.9835	−0.738654
$808$	0	0
$809$	−46.6766	−1.64106	−0.820531	−	0.571603i	$-0.806322\pi$
$809$	−46.6766	−1.64106	−0.820531	+	0.571603i	$0.806322\pi$
$810$	0	0
$811$	47.7251	1.67585	0.837927	−	0.545782i	$-0.183767\pi$
$811$	47.7251	1.67585	0.837927	+	0.545782i	$0.183767\pi$
$812$	0	0
$813$	32.5957	1.14318
$814$	0	0
$815$	−50.2025	−1.75852
$816$	0	0
$817$	−71.4297	−2.49901
$818$	0	0
$819$	1.03360	0.0361170
$820$	0	0
$821$	17.7600	0.619830	0.309915	−	0.950764i	$-0.399699\pi$
$821$	17.7600	0.619830	0.309915	+	0.950764i	$0.399699\pi$
$822$	0	0
$823$	−48.6430	−1.69559	−0.847794	−	0.530326i	$-0.822070\pi$
$823$	−48.6430	−1.69559	−0.847794	+	0.530326i	$0.822070\pi$
$824$	0	0
$825$	−21.8976	−0.762376
$826$	0	0
$827$	36.8651	1.28192	0.640962	−	0.767572i	$-0.278535\pi$
$827$	36.8651	1.28192	0.640962	+	0.767572i	$0.278535\pi$
$828$	0	0
$829$	21.1205	0.733546	0.366773	−	0.930310i	$-0.380463\pi$
$829$	21.1205	0.733546	0.366773	+	0.930310i	$0.380463\pi$
$830$	0	0
$831$	−13.9249	−0.483048
$832$	0	0
$833$	−17.2888	−0.599020
$834$	0	0
$835$	−3.09849	−0.107228
$836$	0	0
$837$	−1.77590	−0.0613841
$838$	0	0
$839$	38.7535	1.33792	0.668959	−	0.743299i	$-0.266740\pi$
$839$	38.7535	1.33792	0.668959	+	0.743299i	$0.266740\pi$
$840$	0	0
$841$	20.8065	0.717465
$842$	0	0
$843$	−2.95960	−0.101934
$844$	0	0
$845$	−39.5073	−1.35909
$846$	0	0
$847$	24.1183	0.828715
$848$	0	0
$849$	18.6587	0.640366
$850$	0	0
$851$	1.16530	0.0399461
$852$	0	0
$853$	−39.8301	−1.36376	−0.681878	−	0.731466i	$-0.738836\pi$
$853$	−39.8301	−1.36376	−0.681878	+	0.731466i	$0.738836\pi$
$854$	0	0
$855$	−19.9560	−0.682481
$856$	0	0
$857$	36.6824	1.25305	0.626523	−	0.779403i	$-0.284478\pi$
$857$	36.6824	1.25305	0.626523	+	0.779403i	$0.284478\pi$
$858$	0	0
$859$	44.0642	1.50345	0.751726	−	0.659476i	$-0.229222\pi$
$859$	44.0642	1.50345	0.751726	+	0.659476i	$0.229222\pi$
$860$	0	0
$861$	5.94364	0.202559
$862$	0	0
$863$	−56.1682	−1.91199	−0.955994	−	0.293387i	$-0.905217\pi$
$863$	−56.1682	−1.91199	−0.955994	+	0.293387i	$0.905217\pi$
$864$	0	0
$865$	3.41371	0.116070
$866$	0	0
$867$	−23.4698	−0.797077
$868$	0	0
$869$	6.78482	0.230159
$870$	0	0
$871$	−6.01261	−0.203729
$872$	0	0
$873$	8.47739	0.286916
$874$	0	0
$875$	−2.56084	−0.0865722
$876$	0	0
$877$	−22.4650	−0.758589	−0.379294	−	0.925276i	$-0.623833\pi$
$877$	−22.4650	−0.758589	−0.379294	+	0.925276i	$0.623833\pi$
$878$	0	0
$879$	−8.70037	−0.293456
$880$	0	0
$881$	−9.96465	−0.335718	−0.167859	−	0.985811i	$-0.553685\pi$
$881$	−9.96465	−0.335718	−0.167859	+	0.985811i	$0.553685\pi$
$882$	0	0
$883$	−11.4787	−0.386290	−0.193145	−	0.981170i	$-0.561869\pi$
$883$	−11.4787	−0.386290	−0.193145	+	0.981170i	$0.561869\pi$
$884$	0	0
$885$	−35.6097	−1.19701
$886$	0	0
$887$	27.9739	0.939272	0.469636	−	0.882860i	$-0.344385\pi$
$887$	27.9739	0.939272	0.469636	+	0.882860i	$0.344385\pi$
$888$	0	0
$889$	7.95512	0.266806
$890$	0	0
$891$	4.75971	0.159456
$892$	0	0
$893$	−11.4109	−0.381850
$894$	0	0
$895$	−54.6263	−1.82595
$896$	0	0
$897$	0.146585	0.00489432
$898$	0	0
$899$	12.5332	0.418005
$900$	0	0
$901$	9.22546	0.307345
$902$	0	0
$903$	−22.9506	−0.763748
$904$	0	0
$905$	79.0657	2.62823
$906$	0	0
$907$	−16.4613	−0.546587	−0.273294	−	0.961931i	$-0.588113\pi$
$907$	−16.4613	−0.546587	−0.273294	+	0.961931i	$0.588113\pi$
$908$	0	0
$909$	−10.0110	−0.332045
$910$	0	0
$911$	−11.9620	−0.396318	−0.198159	−	0.980170i	$-0.563496\pi$
$911$	−11.9620	−0.396318	−0.198159	+	0.980170i	$0.563496\pi$
$912$	0	0
$913$	48.2629	1.59727
$914$	0	0
$915$	24.7513	0.818252
$916$	0	0
$917$	31.8184	1.05074
$918$	0	0
$919$	2.24583	0.0740832	0.0370416	−	0.999314i	$-0.488207\pi$
$919$	2.24583	0.0740832	0.0370416	+	0.999314i	$0.488207\pi$
$920$	0	0
$921$	−2.29116	−0.0754963
$922$	0	0
$923$	2.32224	0.0764374
$924$	0	0
$925$	−18.2675	−0.600633
$926$	0	0
$927$	−7.38830	−0.242664
$928$	0	0
$929$	−46.0338	−1.51032	−0.755160	−	0.655541i	$-0.772441\pi$
$929$	−46.0338	−1.51032	−0.755160	+	0.655541i	$0.772441\pi$
$930$	0	0
$931$	17.5034	0.573651
$932$	0	0
$933$	−19.4482	−0.636704
$934$	0	0
$935$	93.8201	3.06825
$936$	0	0
$937$	31.7244	1.03639	0.518195	−	0.855262i	$-0.326604\pi$
$937$	31.7244	1.03639	0.518195	+	0.855262i	$0.326604\pi$
$938$	0	0
$939$	3.97617	0.129757
$940$	0	0
$941$	−30.6481	−0.999099	−0.499549	−	0.866285i	$-0.666501\pi$
$941$	−30.6481	−0.999099	−0.499549	+	0.866285i	$0.666501\pi$
$942$	0	0
$943$	0.842922	0.0274493
$944$	0	0
$945$	−6.41193	−0.208580
$946$	0	0
$947$	49.2796	1.60137	0.800686	−	0.599085i	$-0.204469\pi$
$947$	49.2796	1.60137	0.800686	+	0.599085i	$0.204469\pi$
$948$	0	0
$949$	−4.57906	−0.148642
$950$	0	0
$951$	14.5851	0.472955
$952$	0	0
$953$	−18.8597	−0.610927	−0.305464	−	0.952204i	$-0.598811\pi$
$953$	−18.8597	−0.610927	−0.305464	+	0.952204i	$0.598811\pi$
$954$	0	0
$955$	7.02812	0.227425
$956$	0	0
$957$	−33.5911	−1.08584
$958$	0	0
$959$	−16.9017	−0.545785
$960$	0	0
$961$	−27.8462	−0.898264
$962$	0	0
$963$	14.9310	0.481145
$964$	0	0
$965$	27.3481	0.880366
$966$	0	0
$967$	3.37791	0.108626	0.0543131	−	0.998524i	$-0.482703\pi$
$967$	3.37791	0.108626	0.0543131	+	0.998524i	$0.482703\pi$
$968$	0	0
$969$	40.9723	1.31622
$970$	0	0
$971$	39.3570	1.26303	0.631513	−	0.775365i	$-0.282434\pi$
$971$	39.3570	1.26303	0.631513	+	0.775365i	$0.282434\pi$
$972$	0	0
$973$	−30.0578	−0.963609
$974$	0	0
$975$	−2.29789	−0.0735915
$976$	0	0
$977$	23.1672	0.741183	0.370591	−	0.928796i	$-0.379155\pi$
$977$	23.1672	0.741183	0.370591	+	0.928796i	$0.379155\pi$
$978$	0	0
$979$	22.7050	0.725654
$980$	0	0
$981$	10.4793	0.334579
$982$	0	0
$983$	−52.3700	−1.67034	−0.835171	−	0.549990i	$-0.814632\pi$
$983$	−52.3700	−1.67034	−0.835171	+	0.549990i	$0.814632\pi$
$984$	0	0
$985$	23.7674	0.757293
$986$	0	0
$987$	−3.66635	−0.116701
$988$	0	0
$989$	−3.25483	−0.103498
$990$	0	0
$991$	−0.204823	−0.00650643	−0.00325321	−	0.999995i	$-0.501036\pi$
$991$	−0.204823	−0.00650643	−0.00325321	+	0.999995i	$0.501036\pi$
$992$	0	0
$993$	−18.7063	−0.593628
$994$	0	0
$995$	51.4409	1.63079
$996$	0	0
$997$	22.5864	0.715318	0.357659	−	0.933852i	$-0.383575\pi$
$997$	22.5864	0.715318	0.357659	+	0.933852i	$0.383575\pi$
$998$	0	0
$999$	3.97068	0.125627

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.v.1.6		7
4.3	odd	2		1002.2.a.k.1.6	✓	7
12.11	even	2		3006.2.a.u.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.k.1.6	✓	7	4.3	odd	2
3006.2.a.u.1.2		7	12.11	even	2
8016.2.a.v.1.6		7	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.09849 q^{5} +2.06938 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.09849 q^{5} +2.06938 q^{7} +1.00000 q^{9} +4.75971 q^{11} +0.499475 q^{13} -3.09849 q^{15} +6.36159 q^{17} -6.44057 q^{19} -2.06938 q^{21} -0.293477 q^{23} +4.60061 q^{25} -1.00000 q^{27} +7.05737 q^{29} +1.77590 q^{31} -4.75971 q^{33} +6.41193 q^{35} -3.97068 q^{37} -0.499475 q^{39} -2.87219 q^{41} +11.0906 q^{43} +3.09849 q^{45} +1.77172 q^{47} -2.71768 q^{49} -6.36159 q^{51} +1.45018 q^{53} +14.7479 q^{55} +6.44057 q^{57} +11.4926 q^{59} -7.98819 q^{61} +2.06938 q^{63} +1.54762 q^{65} -12.0378 q^{67} +0.293477 q^{69} +4.64935 q^{71} -9.16773 q^{73} -4.60061 q^{75} +9.84964 q^{77} +1.42547 q^{79} +1.00000 q^{81} +10.1399 q^{83} +19.7113 q^{85} -7.05737 q^{87} +4.77024 q^{89} +1.03360 q^{91} -1.77590 q^{93} -19.9560 q^{95} +8.47739 q^{97} +4.75971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{3} + 5 q^{5} - 7 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^3 + 5 * q^5 - 7 * q^7 + 7 * q^9	\( 7 q - 7 q^{3} + 5 q^{5} - 7 q^{7} + 7 q^{9} + 6 q^{13} - 5 q^{15} + 6 q^{17} + 2 q^{19} + 7 q^{21} + 12 q^{25} - 7 q^{27} - 4 q^{29} - 7 q^{31} + 13 q^{35} - 3 q^{37} - 6 q^{39} - 12 q^{41} + 2 q^{43} + 5 q^{45} + 11 q^{47} + 10 q^{49} - 6 q^{51} + q^{53} + 2 q^{55} - 2 q^{57} + 19 q^{59} + 12 q^{61} - 7 q^{63} - 10 q^{65} + 17 q^{67} + 20 q^{71} + 10 q^{73} - 12 q^{75} - 24 q^{77} - 2 q^{79} + 7 q^{81} + 7 q^{83} - 18 q^{85} + 4 q^{87} - 3 q^{89} - 4 q^{91} + 7 q^{93} + 24 q^{95} - 3 q^{97}+O(q^{100}) \) 7 * q - 7 * q^3 + 5 * q^5 - 7 * q^7 + 7 * q^9 + 6 * q^13 - 5 * q^15 + 6 * q^17 + 2 * q^19 + 7 * q^21 + 12 * q^25 - 7 * q^27 - 4 * q^29 - 7 * q^31 + 13 * q^35 - 3 * q^37 - 6 * q^39 - 12 * q^41 + 2 * q^43 + 5 * q^45 + 11 * q^47 + 10 * q^49 - 6 * q^51 + q^53 + 2 * q^55 - 2 * q^57 + 19 * q^59 + 12 * q^61 - 7 * q^63 - 10 * q^65 + 17 * q^67 + 20 * q^71 + 10 * q^73 - 12 * q^75 - 24 * q^77 - 2 * q^79 + 7 * q^81 + 7 * q^83 - 18 * q^85 + 4 * q^87 - 3 * q^89 - 4 * q^91 + 7 * q^93 + 24 * q^95 - 3 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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