LMFDB - Embedded newform 2320.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,2,Mod(1,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, 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("2320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.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:	$18.5252932689$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	2320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.90321 q^{3} -1.00000 q^{5} -1.52543 q^{7} +5.42864 q^{9} -4.90321 q^{11} -6.42864 q^{13} -2.90321 q^{15} +2.14764 q^{17} -2.28100 q^{19} -4.42864 q^{21} -6.90321 q^{23} +1.00000 q^{25} +7.05086 q^{27} +1.00000 q^{29} -1.71900 q^{31} -14.2351 q^{33} +1.52543 q^{35} +7.95407 q^{37} -18.6637 q^{39} -3.37778 q^{41} +1.09679 q^{43} -5.42864 q^{45} -12.7096 q^{47} -4.67307 q^{49} +6.23506 q^{51} +3.37778 q^{53} +4.90321 q^{55} -6.62222 q^{57} +3.18421 q^{59} -2.42864 q^{61} -8.28100 q^{63} +6.42864 q^{65} +1.09679 q^{67} -20.0415 q^{69} -3.57136 q^{71} +14.1891 q^{73} +2.90321 q^{75} +7.47949 q^{77} -0.341219 q^{79} +4.18421 q^{81} +7.33185 q^{83} -2.14764 q^{85} +2.90321 q^{87} +2.94914 q^{89} +9.80642 q^{91} -4.99063 q^{93} +2.28100 q^{95} -18.5763 q^{97} -26.6178 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 8 q^{11} - 6 q^{13} - 2 q^{15} - 14 q^{23} + 3 q^{25} + 8 q^{27} + 3 q^{29} - 12 q^{31} - 16 q^{33} - 2 q^{35} + 4 q^{37} - 16 q^{39} - 10 q^{41} + 10 q^{43}+ \cdots - 20 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 8 * q^11 - 6 * q^13 - 2 * q^15 - 14 * q^23 + 3 * q^25 + 8 * q^27 + 3 * q^29 - 12 * q^31 - 16 * q^33 - 2 * q^35 + 4 * q^37 - 16 * q^39 - 10 * q^41 + 10 * q^43 - 3 * q^45 - 18 * q^47 - q^49 - 8 * q^51 + 10 * q^53 + 8 * q^55 - 20 * q^57 - 4 * q^59 + 6 * q^61 - 18 * q^63 + 6 * q^65 + 10 * q^67 - 20 * q^69 - 24 * q^71 - 4 * q^73 + 2 * q^75 - 4 * q^77 - 8 * q^79 - q^81 + 2 * q^83 + 2 * q^87 + 22 * q^89 + 16 * q^91 + 12 * q^93 - 36 * q^97 - 20 * q^99

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
$3$	2.90321	1.67617	0.838085	−	0.545540i	$-0.183675\pi$
$3$	2.90321	1.67617	0.838085	+	0.545540i	$0.183675\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.52543	−0.576557	−0.288279	−	0.957547i	$-0.593083\pi$
$7$	−1.52543	−0.576557	−0.288279	+	0.957547i	$0.593083\pi$
$8$	0	0
$9$	5.42864	1.80955
$10$	0	0
$11$	−4.90321	−1.47837	−0.739187	−	0.673500i	$-0.764790\pi$
$11$	−4.90321	−1.47837	−0.739187	+	0.673500i	$0.764790\pi$
$12$	0	0
$13$	−6.42864	−1.78298	−0.891492	−	0.453037i	$-0.850341\pi$
$13$	−6.42864	−1.78298	−0.891492	+	0.453037i	$0.850341\pi$
$14$	0	0
$15$	−2.90321	−0.749606
$16$	0	0
$17$	2.14764	0.520880	0.260440	−	0.965490i	$-0.416132\pi$
$17$	2.14764	0.520880	0.260440	+	0.965490i	$0.416132\pi$
$18$	0	0
$19$	−2.28100	−0.523296	−0.261648	−	0.965163i	$-0.584266\pi$
$19$	−2.28100	−0.523296	−0.261648	+	0.965163i	$0.584266\pi$
$20$	0	0
$21$	−4.42864	−0.966408
$22$	0	0
$23$	−6.90321	−1.43942	−0.719710	−	0.694275i	$-0.755725\pi$
$23$	−6.90321	−1.43942	−0.719710	+	0.694275i	$0.755725\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	7.05086	1.35694
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−1.71900	−0.308742	−0.154371	−	0.988013i	$-0.549335\pi$
$31$	−1.71900	−0.308742	−0.154371	+	0.988013i	$0.549335\pi$
$32$	0	0
$33$	−14.2351	−2.47801
$34$	0	0
$35$	1.52543	0.257844
$36$	0	0
$37$	7.95407	1.30764	0.653820	−	0.756650i	$-0.273165\pi$
$37$	7.95407	1.30764	0.653820	+	0.756650i	$0.273165\pi$
$38$	0	0
$39$	−18.6637	−2.98858
$40$	0	0
$41$	−3.37778	−0.527521	−0.263761	−	0.964588i	$-0.584963\pi$
$41$	−3.37778	−0.527521	−0.263761	+	0.964588i	$0.584963\pi$
$42$	0	0
$43$	1.09679	0.167259	0.0836293	−	0.996497i	$-0.473349\pi$
$43$	1.09679	0.167259	0.0836293	+	0.996497i	$0.473349\pi$
$44$	0	0
$45$	−5.42864	−0.809254
$46$	0	0
$47$	−12.7096	−1.85389	−0.926945	−	0.375196i	$-0.877575\pi$
$47$	−12.7096	−1.85389	−0.926945	+	0.375196i	$0.877575\pi$
$48$	0	0
$49$	−4.67307	−0.667582
$50$	0	0
$51$	6.23506	0.873084
$52$	0	0
$53$	3.37778	0.463974	0.231987	−	0.972719i	$-0.425477\pi$
$53$	3.37778	0.463974	0.231987	+	0.972719i	$0.425477\pi$
$54$	0	0
$55$	4.90321	0.661149
$56$	0	0
$57$	−6.62222	−0.877134
$58$	0	0
$59$	3.18421	0.414549	0.207274	−	0.978283i	$-0.433541\pi$
$59$	3.18421	0.414549	0.207274	+	0.978283i	$0.433541\pi$
$60$	0	0
$61$	−2.42864	−0.310955	−0.155478	−	0.987839i	$-0.549692\pi$
$61$	−2.42864	−0.310955	−0.155478	+	0.987839i	$0.549692\pi$
$62$	0	0
$63$	−8.28100	−1.04331
$64$	0	0
$65$	6.42864	0.797375
$66$	0	0
$67$	1.09679	0.133994	0.0669970	−	0.997753i	$-0.478658\pi$
$67$	1.09679	0.133994	0.0669970	+	0.997753i	$0.478658\pi$
$68$	0	0
$69$	−20.0415	−2.41271
$70$	0	0
$71$	−3.57136	−0.423843	−0.211921	−	0.977287i	$-0.567972\pi$
$71$	−3.57136	−0.423843	−0.211921	+	0.977287i	$0.567972\pi$
$72$	0	0
$73$	14.1891	1.66071	0.830356	−	0.557233i	$-0.188137\pi$
$73$	14.1891	1.66071	0.830356	+	0.557233i	$0.188137\pi$
$74$	0	0
$75$	2.90321	0.335234
$76$	0	0
$77$	7.47949	0.852368
$78$	0	0
$79$	−0.341219	−0.0383902	−0.0191951	−	0.999816i	$-0.506110\pi$
$79$	−0.341219	−0.0383902	−0.0191951	+	0.999816i	$0.506110\pi$
$80$	0	0
$81$	4.18421	0.464912
$82$	0	0
$83$	7.33185	0.804775	0.402388	−	0.915469i	$-0.368180\pi$
$83$	7.33185	0.804775	0.402388	+	0.915469i	$0.368180\pi$
$84$	0	0
$85$	−2.14764	−0.232945
$86$	0	0
$87$	2.90321	0.311257
$88$	0	0
$89$	2.94914	0.312609	0.156304	−	0.987709i	$-0.450042\pi$
$89$	2.94914	0.312609	0.156304	+	0.987709i	$0.450042\pi$
$90$	0	0
$91$	9.80642	1.02799
$92$	0	0
$93$	−4.99063	−0.517504
$94$	0	0
$95$	2.28100	0.234025
$96$	0	0
$97$	−18.5763	−1.88614	−0.943068	−	0.332600i	$-0.892074\pi$
$97$	−18.5763	−1.88614	−0.943068	+	0.332600i	$0.892074\pi$
$98$	0	0
$99$	−26.6178	−2.67519
$100$	0	0
$101$	15.4193	1.53427	0.767137	−	0.641483i	$-0.221680\pi$
$101$	15.4193	1.53427	0.767137	+	0.641483i	$0.221680\pi$
$102$	0	0
$103$	−7.76049	−0.764664	−0.382332	−	0.924025i	$-0.624879\pi$
$103$	−7.76049	−0.764664	−0.382332	+	0.924025i	$0.624879\pi$
$104$	0	0
$105$	4.42864	0.432191
$106$	0	0
$107$	3.03657	0.293556	0.146778	−	0.989169i	$-0.453110\pi$
$107$	3.03657	0.293556	0.146778	+	0.989169i	$0.453110\pi$
$108$	0	0
$109$	−7.93978	−0.760493	−0.380246	−	0.924885i	$-0.624161\pi$
$109$	−7.93978	−0.760493	−0.380246	+	0.924885i	$0.624161\pi$
$110$	0	0
$111$	23.0923	2.19183
$112$	0	0
$113$	7.82071	0.735711	0.367855	−	0.929883i	$-0.380092\pi$
$113$	7.82071	0.735711	0.367855	+	0.929883i	$0.380092\pi$
$114$	0	0
$115$	6.90321	0.643728
$116$	0	0
$117$	−34.8988	−3.22639
$118$	0	0
$119$	−3.27607	−0.300317
$120$	0	0
$121$	13.0415	1.18559
$122$	0	0
$123$	−9.80642	−0.884215
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.4429	1.10413	0.552066	−	0.833801i	$-0.313840\pi$
$127$	12.4429	1.10413	0.552066	+	0.833801i	$0.313840\pi$
$128$	0	0
$129$	3.18421	0.280354
$130$	0	0
$131$	−4.08742	−0.357120	−0.178560	−	0.983929i	$-0.557144\pi$
$131$	−4.08742	−0.357120	−0.178560	+	0.983929i	$0.557144\pi$
$132$	0	0
$133$	3.47949	0.301710
$134$	0	0
$135$	−7.05086	−0.606841
$136$	0	0
$137$	−19.9541	−1.70479	−0.852395	−	0.522898i	$-0.824851\pi$
$137$	−19.9541	−1.70479	−0.852395	+	0.522898i	$0.824851\pi$
$138$	0	0
$139$	−7.90813	−0.670759	−0.335380	−	0.942083i	$-0.608865\pi$
$139$	−7.90813	−0.670759	−0.335380	+	0.942083i	$0.608865\pi$
$140$	0	0
$141$	−36.8988	−3.10744
$142$	0	0
$143$	31.5210	2.63592
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−13.5669	−1.11898
$148$	0	0
$149$	−16.1017	−1.31910	−0.659552	−	0.751659i	$-0.729254\pi$
$149$	−16.1017	−1.31910	−0.659552	+	0.751659i	$0.729254\pi$
$150$	0	0
$151$	−5.67307	−0.461668	−0.230834	−	0.972993i	$-0.574145\pi$
$151$	−5.67307	−0.461668	−0.230834	+	0.972993i	$0.574145\pi$
$152$	0	0
$153$	11.6588	0.942557
$154$	0	0
$155$	1.71900	0.138074
$156$	0	0
$157$	1.89384	0.151145	0.0755726	−	0.997140i	$-0.475922\pi$
$157$	1.89384	0.151145	0.0755726	+	0.997140i	$0.475922\pi$
$158$	0	0
$159$	9.80642	0.777700
$160$	0	0
$161$	10.5303	0.829908
$162$	0	0
$163$	5.95407	0.466359	0.233179	−	0.972434i	$-0.425087\pi$
$163$	5.95407	0.466359	0.233179	+	0.972434i	$0.425087\pi$
$164$	0	0
$165$	14.2351	1.10820
$166$	0	0
$167$	−4.23951	−0.328063	−0.164032	−	0.986455i	$-0.552450\pi$
$167$	−4.23951	−0.328063	−0.164032	+	0.986455i	$0.552450\pi$
$168$	0	0
$169$	28.3274	2.17903
$170$	0	0
$171$	−12.3827	−0.946929
$172$	0	0
$173$	24.4099	1.85585	0.927925	−	0.372766i	$-0.121591\pi$
$173$	24.4099	1.85585	0.927925	+	0.372766i	$0.121591\pi$
$174$	0	0
$175$	−1.52543	−0.115311
$176$	0	0
$177$	9.24443	0.694854
$178$	0	0
$179$	3.61285	0.270037	0.135018	−	0.990843i	$-0.456891\pi$
$179$	3.61285	0.270037	0.135018	+	0.990843i	$0.456891\pi$
$180$	0	0
$181$	18.0415	1.34101	0.670507	−	0.741904i	$-0.266077\pi$
$181$	18.0415	1.34101	0.670507	+	0.741904i	$0.266077\pi$
$182$	0	0
$183$	−7.05086	−0.521214
$184$	0	0
$185$	−7.95407	−0.584795
$186$	0	0
$187$	−10.5303	−0.770055
$188$	0	0
$189$	−10.7556	−0.782353
$190$	0	0
$191$	−9.85236	−0.712892	−0.356446	−	0.934316i	$-0.616012\pi$
$191$	−9.85236	−0.712892	−0.356446	+	0.934316i	$0.616012\pi$
$192$	0	0
$193$	2.23951	0.161203	0.0806017	−	0.996746i	$-0.474316\pi$
$193$	2.23951	0.161203	0.0806017	+	0.996746i	$0.474316\pi$
$194$	0	0
$195$	18.6637	1.33654
$196$	0	0
$197$	−10.5620	−0.752511	−0.376255	−	0.926516i	$-0.622788\pi$
$197$	−10.5620	−0.752511	−0.376255	+	0.926516i	$0.622788\pi$
$198$	0	0
$199$	−3.18421	−0.225723	−0.112861	−	0.993611i	$-0.536002\pi$
$199$	−3.18421	−0.225723	−0.112861	+	0.993611i	$0.536002\pi$
$200$	0	0
$201$	3.18421	0.224597
$202$	0	0
$203$	−1.52543	−0.107064
$204$	0	0
$205$	3.37778	0.235915
$206$	0	0
$207$	−37.4750	−2.60470
$208$	0	0
$209$	11.1842	0.773628
$210$	0	0
$211$	5.76049	0.396569	0.198284	−	0.980145i	$-0.436463\pi$
$211$	5.76049	0.396569	0.198284	+	0.980145i	$0.436463\pi$
$212$	0	0
$213$	−10.3684	−0.710432
$214$	0	0
$215$	−1.09679	−0.0748003
$216$	0	0
$217$	2.62222	0.178008
$218$	0	0
$219$	41.1941	2.78364
$220$	0	0
$221$	−13.8064	−0.928721
$222$	0	0
$223$	−8.14764	−0.545607	−0.272803	−	0.962070i	$-0.587951\pi$
$223$	−8.14764	−0.545607	−0.272803	+	0.962070i	$0.587951\pi$
$224$	0	0
$225$	5.42864	0.361909
$226$	0	0
$227$	−10.5161	−0.697975	−0.348988	−	0.937127i	$-0.613475\pi$
$227$	−10.5161	−0.697975	−0.348988	+	0.937127i	$0.613475\pi$
$228$	0	0
$229$	8.48886	0.560960	0.280480	−	0.959860i	$-0.409506\pi$
$229$	8.48886	0.560960	0.280480	+	0.959860i	$0.409506\pi$
$230$	0	0
$231$	21.7146	1.42871
$232$	0	0
$233$	−20.5718	−1.34771	−0.673853	−	0.738866i	$-0.735362\pi$
$233$	−20.5718	−1.34771	−0.673853	+	0.738866i	$0.735362\pi$
$234$	0	0
$235$	12.7096	0.829085
$236$	0	0
$237$	−0.990632	−0.0643485
$238$	0	0
$239$	−0.815792	−0.0527692	−0.0263846	−	0.999652i	$-0.508399\pi$
$239$	−0.815792	−0.0527692	−0.0263846	+	0.999652i	$0.508399\pi$
$240$	0	0
$241$	−7.24443	−0.466655	−0.233327	−	0.972398i	$-0.574961\pi$
$241$	−7.24443	−0.466655	−0.233327	+	0.972398i	$0.574961\pi$
$242$	0	0
$243$	−9.00492	−0.577666
$244$	0	0
$245$	4.67307	0.298552
$246$	0	0
$247$	14.6637	0.933029
$248$	0	0
$249$	21.2859	1.34894
$250$	0	0
$251$	20.4242	1.28916	0.644582	−	0.764535i	$-0.277031\pi$
$251$	20.4242	1.28916	0.644582	+	0.764535i	$0.277031\pi$
$252$	0	0
$253$	33.8479	2.12800
$254$	0	0
$255$	−6.23506	−0.390455
$256$	0	0
$257$	3.08250	0.192281	0.0961405	−	0.995368i	$-0.469350\pi$
$257$	3.08250	0.192281	0.0961405	+	0.995368i	$0.469350\pi$
$258$	0	0
$259$	−12.1334	−0.753930
$260$	0	0
$261$	5.42864	0.336024
$262$	0	0
$263$	−20.0558	−1.23669	−0.618346	−	0.785906i	$-0.712197\pi$
$263$	−20.0558	−1.23669	−0.618346	+	0.785906i	$0.712197\pi$
$264$	0	0
$265$	−3.37778	−0.207496
$266$	0	0
$267$	8.56199	0.523985
$268$	0	0
$269$	−23.4608	−1.43043	−0.715214	−	0.698906i	$-0.753671\pi$
$269$	−23.4608	−1.43043	−0.715214	+	0.698906i	$0.753671\pi$
$270$	0	0
$271$	21.9353	1.33248	0.666238	−	0.745739i	$-0.267903\pi$
$271$	21.9353	1.33248	0.666238	+	0.745739i	$0.267903\pi$
$272$	0	0
$273$	28.4701	1.72309
$274$	0	0
$275$	−4.90321	−0.295675
$276$	0	0
$277$	18.3368	1.10175	0.550875	−	0.834588i	$-0.314294\pi$
$277$	18.3368	1.10175	0.550875	+	0.834588i	$0.314294\pi$
$278$	0	0
$279$	−9.33185	−0.558683
$280$	0	0
$281$	6.89877	0.411546	0.205773	−	0.978600i	$-0.434029\pi$
$281$	6.89877	0.411546	0.205773	+	0.978600i	$0.434029\pi$
$282$	0	0
$283$	29.0049	1.72416	0.862082	−	0.506769i	$-0.169160\pi$
$283$	29.0049	1.72416	0.862082	+	0.506769i	$0.169160\pi$
$284$	0	0
$285$	6.62222	0.392266
$286$	0	0
$287$	5.15257	0.304146
$288$	0	0
$289$	−12.3876	−0.728684
$290$	0	0
$291$	−53.9309	−3.16148
$292$	0	0
$293$	7.79213	0.455221	0.227611	−	0.973752i	$-0.426909\pi$
$293$	7.79213	0.455221	0.227611	+	0.973752i	$0.426909\pi$
$294$	0	0
$295$	−3.18421	−0.185392
$296$	0	0
$297$	−34.5718	−2.00606
$298$	0	0
$299$	44.3783	2.56646
$300$	0	0
$301$	−1.67307	−0.0964342
$302$	0	0
$303$	44.7654	2.57171
$304$	0	0
$305$	2.42864	0.139063
$306$	0	0
$307$	−5.92549	−0.338185	−0.169093	−	0.985600i	$-0.554084\pi$
$307$	−5.92549	−0.338185	−0.169093	+	0.985600i	$0.554084\pi$
$308$	0	0
$309$	−22.5303	−1.28171
$310$	0	0
$311$	−8.94470	−0.507207	−0.253604	−	0.967308i	$-0.581616\pi$
$311$	−8.94470	−0.507207	−0.253604	+	0.967308i	$0.581616\pi$
$312$	0	0
$313$	−30.5116	−1.72462	−0.862309	−	0.506382i	$-0.830983\pi$
$313$	−30.5116	−1.72462	−0.862309	+	0.506382i	$0.830983\pi$
$314$	0	0
$315$	8.28100	0.466581
$316$	0	0
$317$	−22.2306	−1.24860	−0.624298	−	0.781186i	$-0.714615\pi$
$317$	−22.2306	−1.24860	−0.624298	+	0.781186i	$0.714615\pi$
$318$	0	0
$319$	−4.90321	−0.274527
$320$	0	0
$321$	8.81579	0.492050
$322$	0	0
$323$	−4.89877	−0.272575
$324$	0	0
$325$	−6.42864	−0.356597
$326$	0	0
$327$	−23.0509	−1.27472
$328$	0	0
$329$	19.3876	1.06887
$330$	0	0
$331$	6.54770	0.359894	0.179947	−	0.983676i	$-0.442407\pi$
$331$	6.54770	0.359894	0.179947	+	0.983676i	$0.442407\pi$
$332$	0	0
$333$	43.1798	2.36624
$334$	0	0
$335$	−1.09679	−0.0599239
$336$	0	0
$337$	2.66815	0.145343	0.0726717	−	0.997356i	$-0.476847\pi$
$337$	2.66815	0.145343	0.0726717	+	0.997356i	$0.476847\pi$
$338$	0	0
$339$	22.7052	1.23318
$340$	0	0
$341$	8.42864	0.456436
$342$	0	0
$343$	17.8064	0.961457
$344$	0	0
$345$	20.0415	1.07900
$346$	0	0
$347$	−14.6780	−0.787956	−0.393978	−	0.919120i	$-0.628901\pi$
$347$	−14.6780	−0.787956	−0.393978	+	0.919120i	$0.628901\pi$
$348$	0	0
$349$	−11.1240	−0.595453	−0.297727	−	0.954651i	$-0.596228\pi$
$349$	−11.1240	−0.595453	−0.297727	+	0.954651i	$0.596228\pi$
$350$	0	0
$351$	−45.3274	−2.41940
$352$	0	0
$353$	−13.4795	−0.717441	−0.358721	−	0.933445i	$-0.616787\pi$
$353$	−13.4795	−0.717441	−0.358721	+	0.933445i	$0.616787\pi$
$354$	0	0
$355$	3.57136	0.189548
$356$	0	0
$357$	−9.51114	−0.503383
$358$	0	0
$359$	−26.1891	−1.38221	−0.691105	−	0.722755i	$-0.742876\pi$
$359$	−26.1891	−1.38221	−0.691105	+	0.722755i	$0.742876\pi$
$360$	0	0
$361$	−13.7971	−0.726161
$362$	0	0
$363$	37.8622	1.98725
$364$	0	0
$365$	−14.1891	−0.742693
$366$	0	0
$367$	22.9862	1.19987	0.599935	−	0.800049i	$-0.295193\pi$
$367$	22.9862	1.19987	0.599935	+	0.800049i	$0.295193\pi$
$368$	0	0
$369$	−18.3368	−0.954574
$370$	0	0
$371$	−5.15257	−0.267508
$372$	0	0
$373$	−24.2766	−1.25699	−0.628496	−	0.777813i	$-0.716329\pi$
$373$	−24.2766	−1.25699	−0.628496	+	0.777813i	$0.716329\pi$
$374$	0	0
$375$	−2.90321	−0.149921
$376$	0	0
$377$	−6.42864	−0.331092
$378$	0	0
$379$	−29.7605	−1.52869	−0.764347	−	0.644805i	$-0.776938\pi$
$379$	−29.7605	−1.52869	−0.764347	+	0.644805i	$0.776938\pi$
$380$	0	0
$381$	36.1245	1.85071
$382$	0	0
$383$	−23.8020	−1.21622	−0.608112	−	0.793851i	$-0.708073\pi$
$383$	−23.8020	−1.21622	−0.608112	+	0.793851i	$0.708073\pi$
$384$	0	0
$385$	−7.47949	−0.381190
$386$	0	0
$387$	5.95407	0.302662
$388$	0	0
$389$	−6.52051	−0.330603	−0.165301	−	0.986243i	$-0.552860\pi$
$389$	−6.52051	−0.330603	−0.165301	+	0.986243i	$0.552860\pi$
$390$	0	0
$391$	−14.8256	−0.749765
$392$	0	0
$393$	−11.8666	−0.598593
$394$	0	0
$395$	0.341219	0.0171686
$396$	0	0
$397$	−14.7654	−0.741055	−0.370527	−	0.928822i	$-0.620823\pi$
$397$	−14.7654	−0.741055	−0.370527	+	0.928822i	$0.620823\pi$
$398$	0	0
$399$	10.1017	0.505718
$400$	0	0
$401$	6.81579	0.340364	0.170182	−	0.985413i	$-0.445564\pi$
$401$	6.81579	0.340364	0.170182	+	0.985413i	$0.445564\pi$
$402$	0	0
$403$	11.0509	0.550482
$404$	0	0
$405$	−4.18421	−0.207915
$406$	0	0
$407$	−39.0005	−1.93318
$408$	0	0
$409$	−11.0825	−0.547994	−0.273997	−	0.961731i	$-0.588346\pi$
$409$	−11.0825	−0.547994	−0.273997	+	0.961731i	$0.588346\pi$
$410$	0	0
$411$	−57.9309	−2.85752
$412$	0	0
$413$	−4.85728	−0.239011
$414$	0	0
$415$	−7.33185	−0.359906
$416$	0	0
$417$	−22.9590	−1.12431
$418$	0	0
$419$	−30.9719	−1.51308	−0.756538	−	0.653950i	$-0.773111\pi$
$419$	−30.9719	−1.51308	−0.756538	+	0.653950i	$0.773111\pi$
$420$	0	0
$421$	22.8988	1.11602	0.558009	−	0.829835i	$-0.311566\pi$
$421$	22.8988	1.11602	0.558009	+	0.829835i	$0.311566\pi$
$422$	0	0
$423$	−68.9960	−3.35470
$424$	0	0
$425$	2.14764	0.104176
$426$	0	0
$427$	3.70471	0.179284
$428$	0	0
$429$	91.5121	4.41825
$430$	0	0
$431$	28.0830	1.35271	0.676355	−	0.736576i	$-0.263559\pi$
$431$	28.0830	1.35271	0.676355	+	0.736576i	$0.263559\pi$
$432$	0	0
$433$	23.0049	1.10555	0.552773	−	0.833332i	$-0.313570\pi$
$433$	23.0049	1.10555	0.552773	+	0.833332i	$0.313570\pi$
$434$	0	0
$435$	−2.90321	−0.139198
$436$	0	0
$437$	15.7462	0.753243
$438$	0	0
$439$	−11.7462	−0.560616	−0.280308	−	0.959910i	$-0.590437\pi$
$439$	−11.7462	−0.560616	−0.280308	+	0.959910i	$0.590437\pi$
$440$	0	0
$441$	−25.3684	−1.20802
$442$	0	0
$443$	−17.6874	−0.840352	−0.420176	−	0.907443i	$-0.638032\pi$
$443$	−17.6874	−0.840352	−0.420176	+	0.907443i	$0.638032\pi$
$444$	0	0
$445$	−2.94914	−0.139803
$446$	0	0
$447$	−46.7467	−2.21104
$448$	0	0
$449$	1.57136	0.0741571	0.0370785	−	0.999312i	$-0.488195\pi$
$449$	1.57136	0.0741571	0.0370785	+	0.999312i	$0.488195\pi$
$450$	0	0
$451$	16.5620	0.779874
$452$	0	0
$453$	−16.4701	−0.773834
$454$	0	0
$455$	−9.80642	−0.459732
$456$	0	0
$457$	1.47949	0.0692078	0.0346039	−	0.999401i	$-0.488983\pi$
$457$	1.47949	0.0692078	0.0346039	+	0.999401i	$0.488983\pi$
$458$	0	0
$459$	15.1427	0.706802
$460$	0	0
$461$	−41.2543	−1.92140	−0.960702	−	0.277583i	$-0.910467\pi$
$461$	−41.2543	−1.92140	−0.960702	+	0.277583i	$0.910467\pi$
$462$	0	0
$463$	34.4242	1.59983	0.799914	−	0.600115i	$-0.204878\pi$
$463$	34.4242	1.59983	0.799914	+	0.600115i	$0.204878\pi$
$464$	0	0
$465$	4.99063	0.231435
$466$	0	0
$467$	−15.1699	−0.701980	−0.350990	−	0.936379i	$-0.614155\pi$
$467$	−15.1699	−0.701980	−0.350990	+	0.936379i	$0.614155\pi$
$468$	0	0
$469$	−1.67307	−0.0772552
$470$	0	0
$471$	5.49823	0.253345
$472$	0	0
$473$	−5.37778	−0.247271
$474$	0	0
$475$	−2.28100	−0.104659
$476$	0	0
$477$	18.3368	0.839583
$478$	0	0
$479$	18.9763	0.867051	0.433526	−	0.901141i	$-0.357269\pi$
$479$	18.9763	0.867051	0.433526	+	0.901141i	$0.357269\pi$
$480$	0	0
$481$	−51.1338	−2.33150
$482$	0	0
$483$	30.5718	1.39107
$484$	0	0
$485$	18.5763	0.843506
$486$	0	0
$487$	32.3926	1.46785	0.733923	−	0.679232i	$-0.237687\pi$
$487$	32.3926	1.46785	0.733923	+	0.679232i	$0.237687\pi$
$488$	0	0
$489$	17.2859	0.781696
$490$	0	0
$491$	−2.69673	−0.121702	−0.0608508	−	0.998147i	$-0.519381\pi$
$491$	−2.69673	−0.121702	−0.0608508	+	0.998147i	$0.519381\pi$
$492$	0	0
$493$	2.14764	0.0967250
$494$	0	0
$495$	26.6178	1.19638
$496$	0	0
$497$	5.44785	0.244370
$498$	0	0
$499$	−14.5718	−0.652325	−0.326163	−	0.945314i	$-0.605756\pi$
$499$	−14.5718	−0.652325	−0.326163	+	0.945314i	$0.605756\pi$
$500$	0	0
$501$	−12.3082	−0.549890
$502$	0	0
$503$	22.2494	0.992050	0.496025	−	0.868308i	$-0.334792\pi$
$503$	22.2494	0.992050	0.496025	+	0.868308i	$0.334792\pi$
$504$	0	0
$505$	−15.4193	−0.686149
$506$	0	0
$507$	82.2405	3.65243
$508$	0	0
$509$	−9.18421	−0.407083	−0.203541	−	0.979066i	$-0.565245\pi$
$509$	−9.18421	−0.407083	−0.203541	+	0.979066i	$0.565245\pi$
$510$	0	0
$511$	−21.6445	−0.957496
$512$	0	0
$513$	−16.0830	−0.710081
$514$	0	0
$515$	7.76049	0.341968
$516$	0	0
$517$	62.3180	2.74074
$518$	0	0
$519$	70.8671	3.11072
$520$	0	0
$521$	7.01921	0.307517	0.153759	−	0.988108i	$-0.450862\pi$
$521$	7.01921	0.307517	0.153759	+	0.988108i	$0.450862\pi$
$522$	0	0
$523$	7.29036	0.318785	0.159393	−	0.987215i	$-0.449046\pi$
$523$	7.29036	0.318785	0.159393	+	0.987215i	$0.449046\pi$
$524$	0	0
$525$	−4.42864	−0.193282
$526$	0	0
$527$	−3.69181	−0.160818
$528$	0	0
$529$	24.6543	1.07193
$530$	0	0
$531$	17.2859	0.750145
$532$	0	0
$533$	21.7146	0.940562
$534$	0	0
$535$	−3.03657	−0.131282
$536$	0	0
$537$	10.4889	0.452628
$538$	0	0
$539$	22.9131	0.986935
$540$	0	0
$541$	30.9491	1.33061	0.665304	−	0.746573i	$-0.268302\pi$
$541$	30.9491	1.33061	0.665304	+	0.746573i	$0.268302\pi$
$542$	0	0
$543$	52.3783	2.24777
$544$	0	0
$545$	7.93978	0.340103
$546$	0	0
$547$	19.4237	0.830498	0.415249	−	0.909708i	$-0.363694\pi$
$547$	19.4237	0.830498	0.415249	+	0.909708i	$0.363694\pi$
$548$	0	0
$549$	−13.1842	−0.562688
$550$	0	0
$551$	−2.28100	−0.0971737
$552$	0	0
$553$	0.520505	0.0221341
$554$	0	0
$555$	−23.0923	−0.980215
$556$	0	0
$557$	30.3497	1.28596	0.642979	−	0.765884i	$-0.277698\pi$
$557$	30.3497	1.28596	0.642979	+	0.765884i	$0.277698\pi$
$558$	0	0
$559$	−7.05086	−0.298219
$560$	0	0
$561$	−30.5718	−1.29074
$562$	0	0
$563$	33.1798	1.39836	0.699180	−	0.714946i	$-0.253549\pi$
$563$	33.1798	1.39836	0.699180	+	0.714946i	$0.253549\pi$
$564$	0	0
$565$	−7.82071	−0.329020
$566$	0	0
$567$	−6.38271	−0.268048
$568$	0	0
$569$	−4.06022	−0.170213	−0.0851067	−	0.996372i	$-0.527123\pi$
$569$	−4.06022	−0.170213	−0.0851067	+	0.996372i	$0.527123\pi$
$570$	0	0
$571$	31.5496	1.32031	0.660154	−	0.751130i	$-0.270491\pi$
$571$	31.5496	1.32031	0.660154	+	0.751130i	$0.270491\pi$
$572$	0	0
$573$	−28.6035	−1.19493
$574$	0	0
$575$	−6.90321	−0.287884
$576$	0	0
$577$	33.7891	1.40666	0.703329	−	0.710865i	$-0.251696\pi$
$577$	33.7891	1.40666	0.703329	+	0.710865i	$0.251696\pi$
$578$	0	0
$579$	6.50177	0.270204
$580$	0	0
$581$	−11.1842	−0.463999
$582$	0	0
$583$	−16.5620	−0.685928
$584$	0	0
$585$	34.8988	1.44289
$586$	0	0
$587$	25.5669	1.05526	0.527630	−	0.849474i	$-0.323081\pi$
$587$	25.5669	1.05526	0.527630	+	0.849474i	$0.323081\pi$
$588$	0	0
$589$	3.92104	0.161564
$590$	0	0
$591$	−30.6637	−1.26134
$592$	0	0
$593$	7.96836	0.327221	0.163611	−	0.986525i	$-0.447686\pi$
$593$	7.96836	0.327221	0.163611	+	0.986525i	$0.447686\pi$
$594$	0	0
$595$	3.27607	0.134306
$596$	0	0
$597$	−9.24443	−0.378349
$598$	0	0
$599$	−37.2815	−1.52328	−0.761640	−	0.648001i	$-0.775605\pi$
$599$	−37.2815	−1.52328	−0.761640	+	0.648001i	$0.775605\pi$
$600$	0	0
$601$	−29.9496	−1.22167	−0.610835	−	0.791758i	$-0.709166\pi$
$601$	−29.9496	−1.22167	−0.610835	+	0.791758i	$0.709166\pi$
$602$	0	0
$603$	5.95407	0.242468
$604$	0	0
$605$	−13.0415	−0.530212
$606$	0	0
$607$	31.8435	1.29249	0.646243	−	0.763132i	$-0.276339\pi$
$607$	31.8435	1.29249	0.646243	+	0.763132i	$0.276339\pi$
$608$	0	0
$609$	−4.42864	−0.179458
$610$	0	0
$611$	81.7057	3.30546
$612$	0	0
$613$	−2.65386	−0.107188	−0.0535942	−	0.998563i	$-0.517068\pi$
$613$	−2.65386	−0.107188	−0.0535942	+	0.998563i	$0.517068\pi$
$614$	0	0
$615$	9.80642	0.395433
$616$	0	0
$617$	−18.3096	−0.737116	−0.368558	−	0.929605i	$-0.620148\pi$
$617$	−18.3096	−0.737116	−0.368558	+	0.929605i	$0.620148\pi$
$618$	0	0
$619$	−12.8113	−0.514931	−0.257466	−	0.966287i	$-0.582887\pi$
$619$	−12.8113	−0.514931	−0.257466	+	0.966287i	$0.582887\pi$
$620$	0	0
$621$	−48.6735	−1.95320
$622$	0	0
$623$	−4.49871	−0.180237
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	32.4701	1.29673
$628$	0	0
$629$	17.0825	0.681124
$630$	0	0
$631$	−30.2766	−1.20529	−0.602645	−	0.798009i	$-0.705887\pi$
$631$	−30.2766	−1.20529	−0.602645	+	0.798009i	$0.705887\pi$
$632$	0	0
$633$	16.7239	0.664716
$634$	0	0
$635$	−12.4429	−0.493783
$636$	0	0
$637$	30.0415	1.19029
$638$	0	0
$639$	−19.3876	−0.766963
$640$	0	0
$641$	−4.50177	−0.177809	−0.0889046	−	0.996040i	$-0.528337\pi$
$641$	−4.50177	−0.177809	−0.0889046	+	0.996040i	$0.528337\pi$
$642$	0	0
$643$	40.0272	1.57852	0.789259	−	0.614060i	$-0.210465\pi$
$643$	40.0272	1.57852	0.789259	+	0.614060i	$0.210465\pi$
$644$	0	0
$645$	−3.18421	−0.125378
$646$	0	0
$647$	−27.3604	−1.07565	−0.537825	−	0.843057i	$-0.680754\pi$
$647$	−27.3604	−1.07565	−0.537825	+	0.843057i	$0.680754\pi$
$648$	0	0
$649$	−15.6128	−0.612858
$650$	0	0
$651$	7.61285	0.298371
$652$	0	0
$653$	22.8430	0.893915	0.446958	−	0.894555i	$-0.352507\pi$
$653$	22.8430	0.893915	0.446958	+	0.894555i	$0.352507\pi$
$654$	0	0
$655$	4.08742	0.159709
$656$	0	0
$657$	77.0277	3.00514
$658$	0	0
$659$	−15.0178	−0.585012	−0.292506	−	0.956264i	$-0.594489\pi$
$659$	−15.0178	−0.585012	−0.292506	+	0.956264i	$0.594489\pi$
$660$	0	0
$661$	−4.65080	−0.180895	−0.0904475	−	0.995901i	$-0.528830\pi$
$661$	−4.65080	−0.180895	−0.0904475	+	0.995901i	$0.528830\pi$
$662$	0	0
$663$	−40.0830	−1.55669
$664$	0	0
$665$	−3.47949	−0.134929
$666$	0	0
$667$	−6.90321	−0.267293
$668$	0	0
$669$	−23.6543	−0.914529
$670$	0	0
$671$	11.9081	0.459708
$672$	0	0
$673$	−44.8671	−1.72950	−0.864750	−	0.502202i	$-0.832523\pi$
$673$	−44.8671	−1.72950	−0.864750	+	0.502202i	$0.832523\pi$
$674$	0	0
$675$	7.05086	0.271388
$676$	0	0
$677$	27.2212	1.04620	0.523099	−	0.852272i	$-0.324776\pi$
$677$	27.2212	1.04620	0.523099	+	0.852272i	$0.324776\pi$
$678$	0	0
$679$	28.3368	1.08747
$680$	0	0
$681$	−30.5303	−1.16993
$682$	0	0
$683$	−12.0558	−0.461301	−0.230651	−	0.973037i	$-0.574085\pi$
$683$	−12.0558	−0.461301	−0.230651	+	0.973037i	$0.574085\pi$
$684$	0	0
$685$	19.9541	0.762406
$686$	0	0
$687$	24.6450	0.940264
$688$	0	0
$689$	−21.7146	−0.827259
$690$	0	0
$691$	−37.5812	−1.42966	−0.714828	−	0.699300i	$-0.753495\pi$
$691$	−37.5812	−1.42966	−0.714828	+	0.699300i	$0.753495\pi$
$692$	0	0
$693$	40.6035	1.54240
$694$	0	0
$695$	7.90813	0.299973
$696$	0	0
$697$	−7.25428	−0.274775
$698$	0	0
$699$	−59.7244	−2.25898
$700$	0	0
$701$	2.04149	0.0771059	0.0385530	−	0.999257i	$-0.487725\pi$
$701$	2.04149	0.0771059	0.0385530	+	0.999257i	$0.487725\pi$
$702$	0	0
$703$	−18.1432	−0.684284
$704$	0	0
$705$	36.8988	1.38969
$706$	0	0
$707$	−23.5210	−0.884598
$708$	0	0
$709$	−32.1432	−1.20716	−0.603582	−	0.797301i	$-0.706260\pi$
$709$	−32.1432	−1.20716	−0.603582	+	0.797301i	$0.706260\pi$
$710$	0	0
$711$	−1.85236	−0.0694688
$712$	0	0
$713$	11.8666	0.444409
$714$	0	0
$715$	−31.5210	−1.17882
$716$	0	0
$717$	−2.36842	−0.0884501
$718$	0	0
$719$	−1.01921	−0.0380102	−0.0190051	−	0.999819i	$-0.506050\pi$
$719$	−1.01921	−0.0380102	−0.0190051	+	0.999819i	$0.506050\pi$
$720$	0	0
$721$	11.8381	0.440873
$722$	0	0
$723$	−21.0321	−0.782193
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	24.1476	0.895587	0.447793	−	0.894137i	$-0.352210\pi$
$727$	24.1476	0.895587	0.447793	+	0.894137i	$0.352210\pi$
$728$	0	0
$729$	−38.6958	−1.43318
$730$	0	0
$731$	2.35551	0.0871217
$732$	0	0
$733$	18.8845	0.697514	0.348757	−	0.937213i	$-0.386604\pi$
$733$	18.8845	0.697514	0.348757	+	0.937213i	$0.386604\pi$
$734$	0	0
$735$	13.5669	0.500423
$736$	0	0
$737$	−5.37778	−0.198093
$738$	0	0
$739$	3.31312	0.121875	0.0609375	−	0.998142i	$-0.480591\pi$
$739$	3.31312	0.121875	0.0609375	+	0.998142i	$0.480591\pi$
$740$	0	0
$741$	42.5718	1.56392
$742$	0	0
$743$	−2.99508	−0.109879	−0.0549394	−	0.998490i	$-0.517497\pi$
$743$	−2.99508	−0.109879	−0.0549394	+	0.998490i	$0.517497\pi$
$744$	0	0
$745$	16.1017	0.589921
$746$	0	0
$747$	39.8020	1.45628
$748$	0	0
$749$	−4.63206	−0.169252
$750$	0	0
$751$	−11.9956	−0.437724	−0.218862	−	0.975756i	$-0.570234\pi$
$751$	−11.9956	−0.437724	−0.218862	+	0.975756i	$0.570234\pi$
$752$	0	0
$753$	59.2958	2.16086
$754$	0	0
$755$	5.67307	0.206464
$756$	0	0
$757$	18.5763	0.675166	0.337583	−	0.941296i	$-0.390391\pi$
$757$	18.5763	0.675166	0.337583	+	0.941296i	$0.390391\pi$
$758$	0	0
$759$	98.2677	3.56689
$760$	0	0
$761$	2.59057	0.0939082	0.0469541	−	0.998897i	$-0.485049\pi$
$761$	2.59057	0.0939082	0.0469541	+	0.998897i	$0.485049\pi$
$762$	0	0
$763$	12.1116	0.438468
$764$	0	0
$765$	−11.6588	−0.421524
$766$	0	0
$767$	−20.4701	−0.739133
$768$	0	0
$769$	32.7467	1.18088	0.590438	−	0.807083i	$-0.298955\pi$
$769$	32.7467	1.18088	0.590438	+	0.807083i	$0.298955\pi$
$770$	0	0
$771$	8.94914	0.322296
$772$	0	0
$773$	28.2208	1.01503	0.507515	−	0.861643i	$-0.330564\pi$
$773$	28.2208	1.01503	0.507515	+	0.861643i	$0.330564\pi$
$774$	0	0
$775$	−1.71900	−0.0617484
$776$	0	0
$777$	−35.2257	−1.26371
$778$	0	0
$779$	7.70471	0.276050
$780$	0	0
$781$	17.5111	0.626598
$782$	0	0
$783$	7.05086	0.251977
$784$	0	0
$785$	−1.89384	−0.0675942
$786$	0	0
$787$	−11.9857	−0.427244	−0.213622	−	0.976916i	$-0.568526\pi$
$787$	−11.9857	−0.427244	−0.213622	+	0.976916i	$0.568526\pi$
$788$	0	0
$789$	−58.2262	−2.07291
$790$	0	0
$791$	−11.9299	−0.424180
$792$	0	0
$793$	15.6128	0.554428
$794$	0	0
$795$	−9.80642	−0.347798
$796$	0	0
$797$	−33.0366	−1.17022	−0.585108	−	0.810956i	$-0.698948\pi$
$797$	−33.0366	−1.17022	−0.585108	+	0.810956i	$0.698948\pi$
$798$	0	0
$799$	−27.2958	−0.965655
$800$	0	0
$801$	16.0098	0.565680
$802$	0	0
$803$	−69.5723	−2.45515
$804$	0	0
$805$	−10.5303	−0.371146
$806$	0	0
$807$	−68.1116	−2.39764
$808$	0	0
$809$	−32.1303	−1.12964	−0.564820	−	0.825214i	$-0.691055\pi$
$809$	−32.1303	−1.12964	−0.564820	+	0.825214i	$0.691055\pi$
$810$	0	0
$811$	−15.3176	−0.537872	−0.268936	−	0.963158i	$-0.586672\pi$
$811$	−15.3176	−0.537872	−0.268936	+	0.963158i	$0.586672\pi$
$812$	0	0
$813$	63.6829	2.23346
$814$	0	0
$815$	−5.95407	−0.208562
$816$	0	0
$817$	−2.50177	−0.0875258
$818$	0	0
$819$	53.2355	1.86020
$820$	0	0
$821$	−17.1427	−0.598285	−0.299143	−	0.954208i	$-0.596701\pi$
$821$	−17.1427	−0.598285	−0.299143	+	0.954208i	$0.596701\pi$
$822$	0	0
$823$	35.6400	1.24233	0.621167	−	0.783678i	$-0.286659\pi$
$823$	35.6400	1.24233	0.621167	+	0.783678i	$0.286659\pi$
$824$	0	0
$825$	−14.2351	−0.495601
$826$	0	0
$827$	−4.70964	−0.163770	−0.0818850	−	0.996642i	$-0.526094\pi$
$827$	−4.70964	−0.163770	−0.0818850	+	0.996642i	$0.526094\pi$
$828$	0	0
$829$	2.25380	0.0782777	0.0391388	−	0.999234i	$-0.487539\pi$
$829$	2.25380	0.0782777	0.0391388	+	0.999234i	$0.487539\pi$
$830$	0	0
$831$	53.2355	1.84672
$832$	0	0
$833$	−10.0361	−0.347730
$834$	0	0
$835$	4.23951	0.146714
$836$	0	0
$837$	−12.1204	−0.418944
$838$	0	0
$839$	8.42419	0.290835	0.145418	−	0.989370i	$-0.453547\pi$
$839$	8.42419	0.290835	0.145418	+	0.989370i	$0.453547\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	20.0286	0.689821
$844$	0	0
$845$	−28.3274	−0.974492
$846$	0	0
$847$	−19.8938	−0.683561
$848$	0	0
$849$	84.2074	2.88999
$850$	0	0
$851$	−54.9086	−1.88224
$852$	0	0
$853$	51.8247	1.77444	0.887222	−	0.461342i	$-0.152632\pi$
$853$	51.8247	1.77444	0.887222	+	0.461342i	$0.152632\pi$
$854$	0	0
$855$	12.3827	0.423480
$856$	0	0
$857$	−30.0415	−1.02620	−0.513099	−	0.858330i	$-0.671503\pi$
$857$	−30.0415	−1.02620	−0.513099	+	0.858330i	$0.671503\pi$
$858$	0	0
$859$	−19.6874	−0.671724	−0.335862	−	0.941911i	$-0.609028\pi$
$859$	−19.6874	−0.671724	−0.335862	+	0.941911i	$0.609028\pi$
$860$	0	0
$861$	14.9590	0.509801
$862$	0	0
$863$	−19.5986	−0.667143	−0.333571	−	0.942725i	$-0.608254\pi$
$863$	−19.5986	−0.667143	−0.333571	+	0.942725i	$0.608254\pi$
$864$	0	0
$865$	−24.4099	−0.829962
$866$	0	0
$867$	−35.9639	−1.22140
$868$	0	0
$869$	1.67307	0.0567550
$870$	0	0
$871$	−7.05086	−0.238909
$872$	0	0
$873$	−100.844	−3.41305
$874$	0	0
$875$	1.52543	0.0515689
$876$	0	0
$877$	−16.2351	−0.548219	−0.274110	−	0.961698i	$-0.588383\pi$
$877$	−16.2351	−0.548219	−0.274110	+	0.961698i	$0.588383\pi$
$878$	0	0
$879$	22.6222	0.763028
$880$	0	0
$881$	−20.7052	−0.697576	−0.348788	−	0.937202i	$-0.613407\pi$
$881$	−20.7052	−0.697576	−0.348788	+	0.937202i	$0.613407\pi$
$882$	0	0
$883$	−8.75112	−0.294499	−0.147249	−	0.989099i	$-0.547042\pi$
$883$	−8.75112	−0.294499	−0.147249	+	0.989099i	$0.547042\pi$
$884$	0	0
$885$	−9.24443	−0.310748
$886$	0	0
$887$	−0.414349	−0.0139125	−0.00695625	−	0.999976i	$-0.502214\pi$
$887$	−0.414349	−0.0139125	−0.00695625	+	0.999976i	$0.502214\pi$
$888$	0	0
$889$	−18.9808	−0.636595
$890$	0	0
$891$	−20.5161	−0.687314
$892$	0	0
$893$	28.9906	0.970135
$894$	0	0
$895$	−3.61285	−0.120764
$896$	0	0
$897$	128.839	4.30183
$898$	0	0
$899$	−1.71900	−0.0573320
$900$	0	0
$901$	7.25428	0.241675
$902$	0	0
$903$	−4.85728	−0.161640
$904$	0	0
$905$	−18.0415	−0.599719
$906$	0	0
$907$	46.9862	1.56015	0.780075	−	0.625686i	$-0.215181\pi$
$907$	46.9862	1.56015	0.780075	+	0.625686i	$0.215181\pi$
$908$	0	0
$909$	83.7057	2.77634
$910$	0	0
$911$	12.1704	0.403223	0.201612	−	0.979466i	$-0.435382\pi$
$911$	12.1704	0.403223	0.201612	+	0.979466i	$0.435382\pi$
$912$	0	0
$913$	−35.9496	−1.18976
$914$	0	0
$915$	7.05086	0.233094
$916$	0	0
$917$	6.23506	0.205900
$918$	0	0
$919$	23.2672	0.767514	0.383757	−	0.923434i	$-0.374630\pi$
$919$	23.2672	0.767514	0.383757	+	0.923434i	$0.374630\pi$
$920$	0	0
$921$	−17.2029	−0.566856
$922$	0	0
$923$	22.9590	0.755704
$924$	0	0
$925$	7.95407	0.261528
$926$	0	0
$927$	−42.1289	−1.38369
$928$	0	0
$929$	44.7556	1.46838	0.734191	−	0.678943i	$-0.237562\pi$
$929$	44.7556	1.46838	0.734191	+	0.678943i	$0.237562\pi$
$930$	0	0
$931$	10.6593	0.349343
$932$	0	0
$933$	−25.9684	−0.850166
$934$	0	0
$935$	10.5303	0.344379
$936$	0	0
$937$	−22.7239	−0.742358	−0.371179	−	0.928561i	$-0.621046\pi$
$937$	−22.7239	−0.742358	−0.371179	+	0.928561i	$0.621046\pi$
$938$	0	0
$939$	−88.5817	−2.89075
$940$	0	0
$941$	4.10171	0.133712	0.0668560	−	0.997763i	$-0.478703\pi$
$941$	4.10171	0.133712	0.0668560	+	0.997763i	$0.478703\pi$
$942$	0	0
$943$	23.3176	0.759324
$944$	0	0
$945$	10.7556	0.349879
$946$	0	0
$947$	−16.6178	−0.540005	−0.270002	−	0.962860i	$-0.587025\pi$
$947$	−16.6178	−0.540005	−0.270002	+	0.962860i	$0.587025\pi$
$948$	0	0
$949$	−91.2168	−2.96102
$950$	0	0
$951$	−64.5402	−2.09286
$952$	0	0
$953$	2.85728	0.0925563	0.0462782	−	0.998929i	$-0.485264\pi$
$953$	2.85728	0.0925563	0.0462782	+	0.998929i	$0.485264\pi$
$954$	0	0
$955$	9.85236	0.318815
$956$	0	0
$957$	−14.2351	−0.460154
$958$	0	0
$959$	30.4385	0.982910
$960$	0	0
$961$	−28.0450	−0.904678
$962$	0	0
$963$	16.4844	0.531203
$964$	0	0
$965$	−2.23951	−0.0720923
$966$	0	0
$967$	−16.8015	−0.540300	−0.270150	−	0.962818i	$-0.587073\pi$
$967$	−16.8015	−0.540300	−0.270150	+	0.962818i	$0.587073\pi$
$968$	0	0
$969$	−14.2222	−0.456881
$970$	0	0
$971$	−38.3640	−1.23116	−0.615579	−	0.788075i	$-0.711078\pi$
$971$	−38.3640	−1.23116	−0.615579	+	0.788075i	$0.711078\pi$
$972$	0	0
$973$	12.0633	0.386731
$974$	0	0
$975$	−18.6637	−0.597717
$976$	0	0
$977$	31.6356	1.01211	0.506056	−	0.862500i	$-0.331103\pi$
$977$	31.6356	1.01211	0.506056	+	0.862500i	$0.331103\pi$
$978$	0	0
$979$	−14.4603	−0.462153
$980$	0	0
$981$	−43.1022	−1.37615
$982$	0	0
$983$	35.3733	1.12823	0.564117	−	0.825695i	$-0.309217\pi$
$983$	35.3733	1.12823	0.564117	+	0.825695i	$0.309217\pi$
$984$	0	0
$985$	10.5620	0.336533
$986$	0	0
$987$	56.2864	1.79162
$988$	0	0
$989$	−7.57136	−0.240755
$990$	0	0
$991$	−24.6953	−0.784474	−0.392237	−	0.919864i	$-0.628299\pi$
$991$	−24.6953	−0.784474	−0.392237	+	0.919864i	$0.628299\pi$
$992$	0	0
$993$	19.0094	0.603244
$994$	0	0
$995$	3.18421	0.100946
$996$	0	0
$997$	−11.4050	−0.361199	−0.180600	−	0.983557i	$-0.557804\pi$
$997$	−11.4050	−0.361199	−0.180600	+	0.983557i	$0.557804\pi$
$998$	0	0
$999$	56.0830	1.77439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.a.s.1.3		3
4.3	odd	2		145.2.a.d.1.1	✓	3
8.3	odd	2		9280.2.a.bu.1.3		3
8.5	even	2		9280.2.a.bm.1.1		3
12.11	even	2		1305.2.a.o.1.3		3
20.3	even	4		725.2.b.d.349.4		6
20.7	even	4		725.2.b.d.349.3		6
20.19	odd	2		725.2.a.d.1.3		3
28.27	even	2		7105.2.a.p.1.1		3
60.59	even	2		6525.2.a.bh.1.1		3
116.115	odd	2		4205.2.a.e.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.d.1.1	✓	3	4.3	odd	2
725.2.a.d.1.3		3	20.19	odd	2
725.2.b.d.349.3		6	20.7	even	4
725.2.b.d.349.4		6	20.3	even	4
1305.2.a.o.1.3		3	12.11	even	2
2320.2.a.s.1.3		3	1.1	even	1	trivial
4205.2.a.e.1.3		3	116.115	odd	2
6525.2.a.bh.1.1		3	60.59	even	2
7105.2.a.p.1.1		3	28.27	even	2
9280.2.a.bm.1.1		3	8.5	even	2
9280.2.a.bu.1.3		3	8.3	odd	2

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:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.a (trivial)

\(f(q)\)	\(=\)	\(q+2.90321 q^{3} -1.00000 q^{5} -1.52543 q^{7} +5.42864 q^{9} -4.90321 q^{11} -6.42864 q^{13} -2.90321 q^{15} +2.14764 q^{17} -2.28100 q^{19} -4.42864 q^{21} -6.90321 q^{23} +1.00000 q^{25} +7.05086 q^{27} +1.00000 q^{29} -1.71900 q^{31} -14.2351 q^{33} +1.52543 q^{35} +7.95407 q^{37} -18.6637 q^{39} -3.37778 q^{41} +1.09679 q^{43} -5.42864 q^{45} -12.7096 q^{47} -4.67307 q^{49} +6.23506 q^{51} +3.37778 q^{53} +4.90321 q^{55} -6.62222 q^{57} +3.18421 q^{59} -2.42864 q^{61} -8.28100 q^{63} +6.42864 q^{65} +1.09679 q^{67} -20.0415 q^{69} -3.57136 q^{71} +14.1891 q^{73} +2.90321 q^{75} +7.47949 q^{77} -0.341219 q^{79} +4.18421 q^{81} +7.33185 q^{83} -2.14764 q^{85} +2.90321 q^{87} +2.94914 q^{89} +9.80642 q^{91} -4.99063 q^{93} +2.28100 q^{95} -18.5763 q^{97} -26.6178 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 8 q^{11} - 6 q^{13} - 2 q^{15} - 14 q^{23} + 3 q^{25} + 8 q^{27} + 3 q^{29} - 12 q^{31} - 16 q^{33} - 2 q^{35} + 4 q^{37} - 16 q^{39} - 10 q^{41} + 10 q^{43}+ \cdots - 20 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 8 * q^11 - 6 * q^13 - 2 * q^15 - 14 * q^23 + 3 * q^25 + 8 * q^27 + 3 * q^29 - 12 * q^31 - 16 * q^33 - 2 * q^35 + 4 * q^37 - 16 * q^39 - 10 * q^41 + 10 * q^43 - 3 * q^45 - 18 * q^47 - q^49 - 8 * q^51 + 10 * q^53 + 8 * q^55 - 20 * q^57 - 4 * q^59 + 6 * q^61 - 18 * q^63 + 6 * q^65 + 10 * q^67 - 20 * q^69 - 24 * q^71 - 4 * q^73 + 2 * q^75 - 4 * q^77 - 8 * q^79 - q^81 + 2 * q^83 + 2 * q^87 + 22 * q^89 + 16 * q^91 + 12 * q^93 - 36 * q^97 - 20 * q^99

Introduction

L-functions

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