LMFDB - Embedded newform 920.2.a.i.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [920,2,Mod(1,920)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("920.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$7.34623698596$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	920.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93543 q^{3} +1.00000 q^{5} +4.93543 q^{7} +0.745898 q^{9} +0.745898 q^{11} +1.74590 q^{13} -1.93543 q^{15} -6.10856 q^{17} +5.44364 q^{19} -9.55220 q^{21} -1.00000 q^{23} +1.00000 q^{25} +4.36266 q^{27} -1.66492 q^{29} -1.61676 q^{31} -1.44364 q^{33} +4.93543 q^{35} +4.34625 q^{37} -3.37907 q^{39} -6.95184 q^{41} +5.01641 q^{43} +0.745898 q^{45} +2.68133 q^{47} +17.3585 q^{49} +11.8227 q^{51} +13.7417 q^{53} +0.745898 q^{55} -10.5358 q^{57} +12.2171 q^{59} -13.9794 q^{61} +3.68133 q^{63} +1.74590 q^{65} +13.1044 q^{67} +1.93543 q^{69} -9.67716 q^{71} +5.69774 q^{73} -1.93543 q^{75} +3.68133 q^{77} +10.3791 q^{79} -10.6813 q^{81} +0.637339 q^{83} -6.10856 q^{85} +3.22235 q^{87} +2.72532 q^{89} +8.61676 q^{91} +3.12914 q^{93} +5.44364 q^{95} +7.12497 q^{97} +0.556364 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 3 q^{5} + 7 q^{7} + 3 q^{9} + 3 q^{11} + 6 q^{13} + 2 q^{15} - 5 q^{17} + 7 q^{19} - 6 q^{21} - 3 q^{23} + 3 q^{25} - q^{27} - q^{29} + 10 q^{31} + 5 q^{33} + 7 q^{35} + 2 q^{37} + 7 q^{39}+ \cdots + 11 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 3 * q^5 + 7 * q^7 + 3 * q^9 + 3 * q^11 + 6 * q^13 + 2 * q^15 - 5 * q^17 + 7 * q^19 - 6 * q^21 - 3 * q^23 + 3 * q^25 - q^27 - q^29 + 10 * q^31 + 5 * q^33 + 7 * q^35 + 2 * q^37 + 7 * q^39 - 10 * q^41 + 12 * q^43 + 3 * q^45 + q^47 + 6 * q^49 + 9 * q^51 + 10 * q^53 + 3 * q^55 - 12 * q^57 + 10 * q^59 - 13 * q^61 + 4 * q^63 + 6 * q^65 - 6 * q^67 - 2 * q^69 + 10 * q^71 + 7 * q^73 + 2 * q^75 + 4 * q^77 + 14 * q^79 - 25 * q^81 + 16 * q^83 - 5 * q^85 - 5 * q^87 - 20 * q^89 + 11 * q^91 + 25 * q^93 + 7 * q^95 + 5 * q^97 + 11 * 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
$3$	−1.93543	−1.11742	−0.558711	−	0.829362i	$-0.688704\pi$
$3$	−1.93543	−1.11742	−0.558711	+	0.829362i	$0.688704\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.93543	1.86542	0.932709	−	0.360630i	$-0.117438\pi$
$7$	4.93543	1.86542	0.932709	+	0.360630i	$0.117438\pi$
$8$	0	0
$9$	0.745898	0.248633
$10$	0	0
$11$	0.745898	0.224897	0.112448	−	0.993658i	$-0.464131\pi$
$11$	0.745898	0.224897	0.112448	+	0.993658i	$0.464131\pi$
$12$	0	0
$13$	1.74590	0.484225	0.242113	−	0.970248i	$-0.422160\pi$
$13$	1.74590	0.484225	0.242113	+	0.970248i	$0.422160\pi$
$14$	0	0
$15$	−1.93543	−0.499726
$16$	0	0
$17$	−6.10856	−1.48154	−0.740772	−	0.671757i	$-0.765540\pi$
$17$	−6.10856	−1.48154	−0.740772	+	0.671757i	$0.765540\pi$
$18$	0	0
$19$	5.44364	1.24886	0.624428	−	0.781083i	$-0.285332\pi$
$19$	5.44364	1.24886	0.624428	+	0.781083i	$0.285332\pi$
$20$	0	0
$21$	−9.55220	−2.08446
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.36266	0.839595
$28$	0	0
$29$	−1.66492	−0.309169	−0.154584	−	0.987980i	$-0.549404\pi$
$29$	−1.66492	−0.309169	−0.154584	+	0.987980i	$0.549404\pi$
$30$	0	0
$31$	−1.61676	−0.290379	−0.145190	−	0.989404i	$-0.546379\pi$
$31$	−1.61676	−0.290379	−0.145190	+	0.989404i	$0.546379\pi$
$32$	0	0
$33$	−1.44364	−0.251305
$34$	0	0
$35$	4.93543	0.834240
$36$	0	0
$37$	4.34625	0.714520	0.357260	−	0.934005i	$-0.383711\pi$
$37$	4.34625	0.714520	0.357260	+	0.934005i	$0.383711\pi$
$38$	0	0
$39$	−3.37907	−0.541084
$40$	0	0
$41$	−6.95184	−1.08569	−0.542847	−	0.839831i	$-0.682654\pi$
$41$	−6.95184	−1.08569	−0.542847	+	0.839831i	$0.682654\pi$
$42$	0	0
$43$	5.01641	0.764995	0.382497	−	0.923957i	$-0.375064\pi$
$43$	5.01641	0.764995	0.382497	+	0.923957i	$0.375064\pi$
$44$	0	0
$45$	0.745898	0.111192
$46$	0	0
$47$	2.68133	0.391112	0.195556	−	0.980693i	$-0.437349\pi$
$47$	2.68133	0.391112	0.195556	+	0.980693i	$0.437349\pi$
$48$	0	0
$49$	17.3585	2.47978
$50$	0	0
$51$	11.8227	1.65551
$52$	0	0
$53$	13.7417	1.88757	0.943786	−	0.330558i	$-0.107237\pi$
$53$	13.7417	1.88757	0.943786	+	0.330558i	$0.107237\pi$
$54$	0	0
$55$	0.745898	0.100577
$56$	0	0
$57$	−10.5358	−1.39550
$58$	0	0
$59$	12.2171	1.59053	0.795267	−	0.606260i	$-0.207331\pi$
$59$	12.2171	1.59053	0.795267	+	0.606260i	$0.207331\pi$
$60$	0	0
$61$	−13.9794	−1.78988	−0.894941	−	0.446185i	$-0.852782\pi$
$61$	−13.9794	−1.78988	−0.894941	+	0.446185i	$0.852782\pi$
$62$	0	0
$63$	3.68133	0.463804
$64$	0	0
$65$	1.74590	0.216552
$66$	0	0
$67$	13.1044	1.60096	0.800478	−	0.599362i	$-0.204579\pi$
$67$	13.1044	1.60096	0.800478	+	0.599362i	$0.204579\pi$
$68$	0	0
$69$	1.93543	0.232999
$70$	0	0
$71$	−9.67716	−1.14847	−0.574234	−	0.818691i	$-0.694700\pi$
$71$	−9.67716	−1.14847	−0.574234	+	0.818691i	$0.694700\pi$
$72$	0	0
$73$	5.69774	0.666870	0.333435	−	0.942773i	$-0.391792\pi$
$73$	5.69774	0.666870	0.333435	+	0.942773i	$0.391792\pi$
$74$	0	0
$75$	−1.93543	−0.223484
$76$	0	0
$77$	3.68133	0.419527
$78$	0	0
$79$	10.3791	1.16774	0.583868	−	0.811848i	$-0.301538\pi$
$79$	10.3791	1.16774	0.583868	+	0.811848i	$0.301538\pi$
$80$	0	0
$81$	−10.6813	−1.18681
$82$	0	0
$83$	0.637339	0.0699570	0.0349785	−	0.999388i	$-0.488864\pi$
$83$	0.637339	0.0699570	0.0349785	+	0.999388i	$0.488864\pi$
$84$	0	0
$85$	−6.10856	−0.662566
$86$	0	0
$87$	3.22235	0.345472
$88$	0	0
$89$	2.72532	0.288884	0.144442	−	0.989513i	$-0.453861\pi$
$89$	2.72532	0.288884	0.144442	+	0.989513i	$0.453861\pi$
$90$	0	0
$91$	8.61676	0.903282
$92$	0	0
$93$	3.12914	0.324476
$94$	0	0
$95$	5.44364	0.558505
$96$	0	0
$97$	7.12497	0.723431	0.361715	−	0.932289i	$-0.382191\pi$
$97$	7.12497	0.723431	0.361715	+	0.932289i	$0.382191\pi$
$98$	0	0
$99$	0.556364	0.0559167
$100$	0	0
$101$	14.9753	1.49009	0.745047	−	0.667012i	$-0.232427\pi$
$101$	14.9753	1.49009	0.745047	+	0.667012i	$0.232427\pi$
$102$	0	0
$103$	−10.8667	−1.07073	−0.535364	−	0.844622i	$-0.679825\pi$
$103$	−10.8667	−1.07073	−0.535364	+	0.844622i	$0.679825\pi$
$104$	0	0
$105$	−9.55220	−0.932199
$106$	0	0
$107$	−7.49180	−0.724259	−0.362130	−	0.932128i	$-0.617950\pi$
$107$	−7.49180	−0.724259	−0.362130	+	0.932128i	$0.617950\pi$
$108$	0	0
$109$	−18.3309	−1.75578	−0.877891	−	0.478860i	$-0.841050\pi$
$109$	−18.3309	−1.75578	−0.877891	+	0.478860i	$0.841050\pi$
$110$	0	0
$111$	−8.41188	−0.798420
$112$	0	0
$113$	0.379068	0.0356597	0.0178299	−	0.999841i	$-0.494324\pi$
$113$	0.379068	0.0356597	0.0178299	+	0.999841i	$0.494324\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	1.30226	0.120394
$118$	0	0
$119$	−30.1484	−2.76370
$120$	0	0
$121$	−10.4436	−0.949421
$122$	0	0
$123$	13.4548	1.21318
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.64852	0.235018	0.117509	−	0.993072i	$-0.462509\pi$
$127$	2.64852	0.235018	0.117509	+	0.993072i	$0.462509\pi$
$128$	0	0
$129$	−9.70892	−0.854822
$130$	0	0
$131$	−5.53579	−0.483664	−0.241832	−	0.970318i	$-0.577748\pi$
$131$	−5.53579	−0.483664	−0.241832	+	0.970318i	$0.577748\pi$
$132$	0	0
$133$	26.8667	2.32964
$134$	0	0
$135$	4.36266	0.375478
$136$	0	0
$137$	6.16896	0.527050	0.263525	−	0.964653i	$-0.415115\pi$
$137$	6.16896	0.527050	0.263525	+	0.964653i	$0.415115\pi$
$138$	0	0
$139$	−20.7693	−1.76163	−0.880815	−	0.473460i	$-0.843005\pi$
$139$	−20.7693	−1.76163	−0.880815	+	0.473460i	$0.843005\pi$
$140$	0	0
$141$	−5.18953	−0.437038
$142$	0	0
$143$	1.30226	0.108901
$144$	0	0
$145$	−1.66492	−0.138264
$146$	0	0
$147$	−33.5962	−2.77097
$148$	0	0
$149$	−1.06457	−0.0872128	−0.0436064	−	0.999049i	$-0.513885\pi$
$149$	−1.06457	−0.0872128	−0.0436064	+	0.999049i	$0.513885\pi$
$150$	0	0
$151$	−12.6608	−1.03032	−0.515159	−	0.857095i	$-0.672267\pi$
$151$	−12.6608	−1.03032	−0.515159	+	0.857095i	$0.672267\pi$
$152$	0	0
$153$	−4.55636	−0.368360
$154$	0	0
$155$	−1.61676	−0.129862
$156$	0	0
$157$	14.8873	1.18813	0.594067	−	0.804416i	$-0.297521\pi$
$157$	14.8873	1.18813	0.594067	+	0.804416i	$0.297521\pi$
$158$	0	0
$159$	−26.5962	−2.10921
$160$	0	0
$161$	−4.93543	−0.388967
$162$	0	0
$163$	6.72949	0.527094	0.263547	−	0.964646i	$-0.415108\pi$
$163$	6.72949	0.527094	0.263547	+	0.964646i	$0.415108\pi$
$164$	0	0
$165$	−1.44364	−0.112387
$166$	0	0
$167$	−2.98359	−0.230877	−0.115439	−	0.993315i	$-0.536827\pi$
$167$	−2.98359	−0.230877	−0.115439	+	0.993315i	$0.536827\pi$
$168$	0	0
$169$	−9.95184	−0.765526
$170$	0	0
$171$	4.06040	0.310506
$172$	0	0
$173$	−15.8503	−1.20508	−0.602538	−	0.798091i	$-0.705844\pi$
$173$	−15.8503	−1.20508	−0.602538	+	0.798091i	$0.705844\pi$
$174$	0	0
$175$	4.93543	0.373084
$176$	0	0
$177$	−23.6454	−1.77730
$178$	0	0
$179$	3.02759	0.226292	0.113146	−	0.993578i	$-0.463907\pi$
$179$	3.02759	0.226292	0.113146	+	0.993578i	$0.463907\pi$
$180$	0	0
$181$	1.82270	0.135481	0.0677403	−	0.997703i	$-0.478421\pi$
$181$	1.82270	0.135481	0.0677403	+	0.997703i	$0.478421\pi$
$182$	0	0
$183$	27.0562	2.00005
$184$	0	0
$185$	4.34625	0.319543
$186$	0	0
$187$	−4.55636	−0.333194
$188$	0	0
$189$	21.5316	1.56619
$190$	0	0
$191$	19.6126	1.41912	0.709559	−	0.704646i	$-0.248894\pi$
$191$	19.6126	1.41912	0.709559	+	0.704646i	$0.248894\pi$
$192$	0	0
$193$	−0.335076	−0.0241193	−0.0120597	−	0.999927i	$-0.503839\pi$
$193$	−0.335076	−0.0241193	−0.0120597	+	0.999927i	$0.503839\pi$
$194$	0	0
$195$	−3.37907	−0.241980
$196$	0	0
$197$	−20.5316	−1.46282	−0.731409	−	0.681939i	$-0.761137\pi$
$197$	−20.5316	−1.46282	−0.731409	+	0.681939i	$0.761137\pi$
$198$	0	0
$199$	16.7253	1.18563	0.592813	−	0.805340i	$-0.298017\pi$
$199$	16.7253	1.18563	0.592813	+	0.805340i	$0.298017\pi$
$200$	0	0
$201$	−25.3627	−1.78894
$202$	0	0
$203$	−8.21712	−0.576729
$204$	0	0
$205$	−6.95184	−0.485538
$206$	0	0
$207$	−0.745898	−0.0518435
$208$	0	0
$209$	4.06040	0.280864
$210$	0	0
$211$	−13.3955	−0.922183	−0.461091	−	0.887353i	$-0.652542\pi$
$211$	−13.3955	−0.922183	−0.461091	+	0.887353i	$0.652542\pi$
$212$	0	0
$213$	18.7295	1.28332
$214$	0	0
$215$	5.01641	0.342116
$216$	0	0
$217$	−7.97942	−0.541679
$218$	0	0
$219$	−11.0276	−0.745175
$220$	0	0
$221$	−10.6649	−0.717400
$222$	0	0
$223$	1.27468	0.0853587	0.0426794	−	0.999089i	$-0.486411\pi$
$223$	1.27468	0.0853587	0.0426794	+	0.999089i	$0.486411\pi$
$224$	0	0
$225$	0.745898	0.0497266
$226$	0	0
$227$	3.57978	0.237598	0.118799	−	0.992918i	$-0.462096\pi$
$227$	3.57978	0.237598	0.118799	+	0.992918i	$0.462096\pi$
$228$	0	0
$229$	10.4119	0.688036	0.344018	−	0.938963i	$-0.388212\pi$
$229$	10.4119	0.688036	0.344018	+	0.938963i	$0.388212\pi$
$230$	0	0
$231$	−7.12497	−0.468788
$232$	0	0
$233$	3.06040	0.200493	0.100247	−	0.994963i	$-0.468037\pi$
$233$	3.06040	0.200493	0.100247	+	0.994963i	$0.468037\pi$
$234$	0	0
$235$	2.68133	0.174911
$236$	0	0
$237$	−20.0880	−1.30485
$238$	0	0
$239$	−19.1895	−1.24127	−0.620634	−	0.784100i	$-0.713125\pi$
$239$	−19.1895	−1.24127	−0.620634	+	0.784100i	$0.713125\pi$
$240$	0	0
$241$	0.346255	0.0223042	0.0111521	−	0.999938i	$-0.496450\pi$
$241$	0.346255	0.0223042	0.0111521	+	0.999938i	$0.496450\pi$
$242$	0	0
$243$	7.58501	0.486579
$244$	0	0
$245$	17.3585	1.10899
$246$	0	0
$247$	9.50403	0.604727
$248$	0	0
$249$	−1.23353	−0.0781715
$250$	0	0
$251$	14.3861	0.908041	0.454021	−	0.890991i	$-0.349989\pi$
$251$	14.3861	0.908041	0.454021	+	0.890991i	$0.349989\pi$
$252$	0	0
$253$	−0.745898	−0.0468942
$254$	0	0
$255$	11.8227	0.740366
$256$	0	0
$257$	1.28586	0.0802095	0.0401047	−	0.999195i	$-0.487231\pi$
$257$	1.28586	0.0802095	0.0401047	+	0.999195i	$0.487231\pi$
$258$	0	0
$259$	21.4506	1.33288
$260$	0	0
$261$	−1.24186	−0.0768694
$262$	0	0
$263$	28.2294	1.74070	0.870348	−	0.492437i	$-0.163894\pi$
$263$	28.2294	1.74070	0.870348	+	0.492437i	$0.163894\pi$
$264$	0	0
$265$	13.7417	0.844148
$266$	0	0
$267$	−5.27468	−0.322805
$268$	0	0
$269$	−19.4395	−1.18525	−0.592623	−	0.805480i	$-0.701907\pi$
$269$	−19.4395	−1.18525	−0.592623	+	0.805480i	$0.701907\pi$
$270$	0	0
$271$	−6.42723	−0.390426	−0.195213	−	0.980761i	$-0.562540\pi$
$271$	−6.42723	−0.390426	−0.195213	+	0.980761i	$0.562540\pi$
$272$	0	0
$273$	−16.6772	−1.00935
$274$	0	0
$275$	0.745898	0.0449794
$276$	0	0
$277$	4.08514	0.245452	0.122726	−	0.992441i	$-0.460836\pi$
$277$	4.08514	0.245452	0.122726	+	0.992441i	$0.460836\pi$
$278$	0	0
$279$	−1.20594	−0.0721978
$280$	0	0
$281$	−6.38741	−0.381041	−0.190520	−	0.981683i	$-0.561018\pi$
$281$	−6.38741	−0.381041	−0.190520	+	0.981683i	$0.561018\pi$
$282$	0	0
$283$	−21.4835	−1.27706	−0.638530	−	0.769597i	$-0.720457\pi$
$283$	−21.4835	−1.27706	−0.638530	+	0.769597i	$0.720457\pi$
$284$	0	0
$285$	−10.5358	−0.624086
$286$	0	0
$287$	−34.3103	−2.02527
$288$	0	0
$289$	20.3145	1.19497
$290$	0	0
$291$	−13.7899	−0.808378
$292$	0	0
$293$	−16.1208	−0.941787	−0.470894	−	0.882190i	$-0.656068\pi$
$293$	−16.1208	−0.941787	−0.470894	+	0.882190i	$0.656068\pi$
$294$	0	0
$295$	12.2171	0.711308
$296$	0	0
$297$	3.25410	0.188822
$298$	0	0
$299$	−1.74590	−0.100968
$300$	0	0
$301$	24.7581	1.42704
$302$	0	0
$303$	−28.9836	−1.66506
$304$	0	0
$305$	−13.9794	−0.800460
$306$	0	0
$307$	−6.96302	−0.397400	−0.198700	−	0.980060i	$-0.563672\pi$
$307$	−6.96302	−0.397400	−0.198700	+	0.980060i	$0.563672\pi$
$308$	0	0
$309$	21.0318	1.19645
$310$	0	0
$311$	−4.14031	−0.234776	−0.117388	−	0.993086i	$-0.537452\pi$
$311$	−4.14031	−0.234776	−0.117388	+	0.993086i	$0.537452\pi$
$312$	0	0
$313$	6.26528	0.354135	0.177067	−	0.984199i	$-0.443339\pi$
$313$	6.26528	0.354135	0.177067	+	0.984199i	$0.443339\pi$
$314$	0	0
$315$	3.68133	0.207419
$316$	0	0
$317$	9.53996	0.535817	0.267909	−	0.963444i	$-0.413667\pi$
$317$	9.53996	0.535817	0.267909	+	0.963444i	$0.413667\pi$
$318$	0	0
$319$	−1.24186	−0.0695310
$320$	0	0
$321$	14.4999	0.809304
$322$	0	0
$323$	−33.2528	−1.85023
$324$	0	0
$325$	1.74590	0.0968450
$326$	0	0
$327$	35.4782	1.96195
$328$	0	0
$329$	13.2335	0.729588
$330$	0	0
$331$	−14.3023	−0.786123	−0.393062	−	0.919512i	$-0.628584\pi$
$331$	−14.3023	−0.786123	−0.393062	+	0.919512i	$0.628584\pi$
$332$	0	0
$333$	3.24186	0.177653
$334$	0	0
$335$	13.1044	0.715969
$336$	0	0
$337$	−3.34731	−0.182340	−0.0911699	−	0.995835i	$-0.529061\pi$
$337$	−3.34731	−0.182340	−0.0911699	+	0.995835i	$0.529061\pi$
$338$	0	0
$339$	−0.733661	−0.0398470
$340$	0	0
$341$	−1.20594	−0.0653054
$342$	0	0
$343$	51.1236	2.76042
$344$	0	0
$345$	1.93543	0.104200
$346$	0	0
$347$	−23.9383	−1.28507	−0.642537	−	0.766255i	$-0.722118\pi$
$347$	−23.9383	−1.28507	−0.642537	+	0.766255i	$0.722118\pi$
$348$	0	0
$349$	−11.9149	−0.637788	−0.318894	−	0.947790i	$-0.603311\pi$
$349$	−11.9149	−0.637788	−0.318894	+	0.947790i	$0.603311\pi$
$350$	0	0
$351$	7.61676	0.406553
$352$	0	0
$353$	−25.2447	−1.34364	−0.671820	−	0.740714i	$-0.734487\pi$
$353$	−25.2447	−1.34364	−0.671820	+	0.740714i	$0.734487\pi$
$354$	0	0
$355$	−9.67716	−0.513610
$356$	0	0
$357$	58.3502	3.08822
$358$	0	0
$359$	20.4671	1.08021	0.540105	−	0.841598i	$-0.318385\pi$
$359$	20.4671	1.08021	0.540105	+	0.841598i	$0.318385\pi$
$360$	0	0
$361$	10.6332	0.559641
$362$	0	0
$363$	20.2130	1.06090
$364$	0	0
$365$	5.69774	0.298233
$366$	0	0
$367$	−36.4999	−1.90528	−0.952639	−	0.304104i	$-0.901643\pi$
$367$	−36.4999	−1.90528	−0.952639	+	0.304104i	$0.901643\pi$
$368$	0	0
$369$	−5.18537	−0.269939
$370$	0	0
$371$	67.8214	3.52111
$372$	0	0
$373$	−5.57978	−0.288910	−0.144455	−	0.989511i	$-0.546143\pi$
$373$	−5.57978	−0.288910	−0.144455	+	0.989511i	$0.546143\pi$
$374$	0	0
$375$	−1.93543	−0.0999453
$376$	0	0
$377$	−2.90679	−0.149707
$378$	0	0
$379$	36.6168	1.88088	0.940438	−	0.339964i	$-0.110415\pi$
$379$	36.6168	1.88088	0.940438	+	0.339964i	$0.110415\pi$
$380$	0	0
$381$	−5.12603	−0.262614
$382$	0	0
$383$	−28.7581	−1.46947	−0.734736	−	0.678353i	$-0.762694\pi$
$383$	−28.7581	−1.46947	−0.734736	+	0.678353i	$0.762694\pi$
$384$	0	0
$385$	3.68133	0.187618
$386$	0	0
$387$	3.74173	0.190203
$388$	0	0
$389$	26.3913	1.33809	0.669046	−	0.743221i	$-0.266703\pi$
$389$	26.3913	1.33809	0.669046	+	0.743221i	$0.266703\pi$
$390$	0	0
$391$	6.10856	0.308923
$392$	0	0
$393$	10.7141	0.540457
$394$	0	0
$395$	10.3791	0.522228
$396$	0	0
$397$	1.08931	0.0546710	0.0273355	−	0.999626i	$-0.491298\pi$
$397$	1.08931	0.0546710	0.0273355	+	0.999626i	$0.491298\pi$
$398$	0	0
$399$	−51.9987	−2.60319
$400$	0	0
$401$	−22.5962	−1.12840	−0.564200	−	0.825638i	$-0.690815\pi$
$401$	−22.5962	−1.12840	−0.564200	+	0.825638i	$0.690815\pi$
$402$	0	0
$403$	−2.82270	−0.140609
$404$	0	0
$405$	−10.6813	−0.530760
$406$	0	0
$407$	3.24186	0.160693
$408$	0	0
$409$	−16.0510	−0.793671	−0.396835	−	0.917890i	$-0.629892\pi$
$409$	−16.0510	−0.793671	−0.396835	+	0.917890i	$0.629892\pi$
$410$	0	0
$411$	−11.9396	−0.588937
$412$	0	0
$413$	60.2968	2.96701
$414$	0	0
$415$	0.637339	0.0312857
$416$	0	0
$417$	40.1976	1.96849
$418$	0	0
$419$	−24.4342	−1.19369	−0.596845	−	0.802356i	$-0.703579\pi$
$419$	−24.4342	−1.19369	−0.596845	+	0.802356i	$0.703579\pi$
$420$	0	0
$421$	−11.3473	−0.553034	−0.276517	−	0.961009i	$-0.589180\pi$
$421$	−11.3473	−0.553034	−0.276517	+	0.961009i	$0.589180\pi$
$422$	0	0
$423$	2.00000	0.0972433
$424$	0	0
$425$	−6.10856	−0.296309
$426$	0	0
$427$	−68.9945	−3.33888
$428$	0	0
$429$	−2.52044	−0.121688
$430$	0	0
$431$	7.83805	0.377546	0.188773	−	0.982021i	$-0.439549\pi$
$431$	7.83805	0.377546	0.188773	+	0.982021i	$0.439549\pi$
$432$	0	0
$433$	−37.9023	−1.82147	−0.910735	−	0.412990i	$-0.864484\pi$
$433$	−37.9023	−1.82147	−0.910735	+	0.412990i	$0.864484\pi$
$434$	0	0
$435$	3.22235	0.154500
$436$	0	0
$437$	−5.44364	−0.260404
$438$	0	0
$439$	−17.7704	−0.848134	−0.424067	−	0.905631i	$-0.639398\pi$
$439$	−17.7704	−0.848134	−0.424067	+	0.905631i	$0.639398\pi$
$440$	0	0
$441$	12.9477	0.616556
$442$	0	0
$443$	16.9466	0.805158	0.402579	−	0.915385i	$-0.368114\pi$
$443$	16.9466	0.805158	0.402579	+	0.915385i	$0.368114\pi$
$444$	0	0
$445$	2.72532	0.129193
$446$	0	0
$447$	2.06040	0.0974535
$448$	0	0
$449$	−21.0890	−0.995253	−0.497627	−	0.867391i	$-0.665795\pi$
$449$	−21.0890	−0.995253	−0.497627	+	0.867391i	$0.665795\pi$
$450$	0	0
$451$	−5.18537	−0.244169
$452$	0	0
$453$	24.5040	1.15130
$454$	0	0
$455$	8.61676	0.403960
$456$	0	0
$457$	13.1372	0.614532	0.307266	−	0.951624i	$-0.400586\pi$
$457$	13.1372	0.614532	0.307266	+	0.951624i	$0.400586\pi$
$458$	0	0
$459$	−26.6496	−1.24390
$460$	0	0
$461$	−14.5439	−0.677375	−0.338687	−	0.940899i	$-0.609983\pi$
$461$	−14.5439	−0.677375	−0.338687	+	0.940899i	$0.609983\pi$
$462$	0	0
$463$	19.4283	0.902909	0.451455	−	0.892294i	$-0.350905\pi$
$463$	19.4283	0.902909	0.451455	+	0.892294i	$0.350905\pi$
$464$	0	0
$465$	3.12914	0.145110
$466$	0	0
$467$	−3.61259	−0.167171	−0.0835855	−	0.996501i	$-0.526637\pi$
$467$	−3.61259	−0.167171	−0.0835855	+	0.996501i	$0.526637\pi$
$468$	0	0
$469$	64.6758	2.98645
$470$	0	0
$471$	−28.8133	−1.32765
$472$	0	0
$473$	3.74173	0.172045
$474$	0	0
$475$	5.44364	0.249771
$476$	0	0
$477$	10.2499	0.469312
$478$	0	0
$479$	5.13720	0.234725	0.117362	−	0.993089i	$-0.462556\pi$
$479$	5.13720	0.234725	0.117362	+	0.993089i	$0.462556\pi$
$480$	0	0
$481$	7.58812	0.345988
$482$	0	0
$483$	9.55220	0.434640
$484$	0	0
$485$	7.12497	0.323528
$486$	0	0
$487$	−24.7693	−1.12240	−0.561202	−	0.827679i	$-0.689661\pi$
$487$	−24.7693	−1.12240	−0.561202	+	0.827679i	$0.689661\pi$
$488$	0	0
$489$	−13.0245	−0.588987
$490$	0	0
$491$	3.54413	0.159944	0.0799721	−	0.996797i	$-0.474517\pi$
$491$	3.54413	0.159944	0.0799721	+	0.996797i	$0.474517\pi$
$492$	0	0
$493$	10.1703	0.458047
$494$	0	0
$495$	0.556364	0.0250067
$496$	0	0
$497$	−47.7610	−2.14237
$498$	0	0
$499$	31.5358	1.41174	0.705868	−	0.708344i	$-0.250557\pi$
$499$	31.5358	1.41174	0.705868	+	0.708344i	$0.250557\pi$
$500$	0	0
$501$	5.77454	0.257988
$502$	0	0
$503$	−32.2346	−1.43727	−0.718635	−	0.695388i	$-0.755233\pi$
$503$	−32.2346	−1.43727	−0.718635	+	0.695388i	$0.755233\pi$
$504$	0	0
$505$	14.9753	0.666390
$506$	0	0
$507$	19.2611	0.855416
$508$	0	0
$509$	0.302263	0.0133976	0.00669878	−	0.999978i	$-0.497868\pi$
$509$	0.302263	0.0133976	0.00669878	+	0.999978i	$0.497868\pi$
$510$	0	0
$511$	28.1208	1.24399
$512$	0	0
$513$	23.7487	1.04853
$514$	0	0
$515$	−10.8667	−0.478844
$516$	0	0
$517$	2.00000	0.0879599
$518$	0	0
$519$	30.6772	1.34658
$520$	0	0
$521$	−30.3463	−1.32949	−0.664747	−	0.747069i	$-0.731461\pi$
$521$	−30.3463	−1.32949	−0.664747	+	0.747069i	$0.731461\pi$
$522$	0	0
$523$	−42.5878	−1.86224	−0.931118	−	0.364717i	$-0.881166\pi$
$523$	−42.5878	−1.86224	−0.931118	+	0.364717i	$0.881166\pi$
$524$	0	0
$525$	−9.55220	−0.416892
$526$	0	0
$527$	9.87609	0.430209
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	9.11273	0.395459
$532$	0	0
$533$	−12.1372	−0.525721
$534$	0	0
$535$	−7.49180	−0.323899
$536$	0	0
$537$	−5.85969	−0.252864
$538$	0	0
$539$	12.9477	0.557696
$540$	0	0
$541$	38.9313	1.67379	0.836893	−	0.547367i	$-0.184370\pi$
$541$	38.9313	1.67379	0.836893	+	0.547367i	$0.184370\pi$
$542$	0	0
$543$	−3.52772	−0.151389
$544$	0	0
$545$	−18.3309	−0.785210
$546$	0	0
$547$	16.6004	0.709780	0.354890	−	0.934908i	$-0.384518\pi$
$547$	16.6004	0.709780	0.354890	+	0.934908i	$0.384518\pi$
$548$	0	0
$549$	−10.4272	−0.445023
$550$	0	0
$551$	−9.06324	−0.386107
$552$	0	0
$553$	51.2252	2.17832
$554$	0	0
$555$	−8.41188	−0.357064
$556$	0	0
$557$	17.4506	0.739408	0.369704	−	0.929150i	$-0.379459\pi$
$557$	17.4506	0.739408	0.369704	+	0.929150i	$0.379459\pi$
$558$	0	0
$559$	8.75814	0.370430
$560$	0	0
$561$	8.81853	0.372319
$562$	0	0
$563$	8.25827	0.348045	0.174022	−	0.984742i	$-0.444324\pi$
$563$	8.25827	0.348045	0.174022	+	0.984742i	$0.444324\pi$
$564$	0	0
$565$	0.379068	0.0159475
$566$	0	0
$567$	−52.7170	−2.21391
$568$	0	0
$569$	−41.4178	−1.73633	−0.868163	−	0.496279i	$-0.834699\pi$
$569$	−41.4178	−1.73633	−0.868163	+	0.496279i	$0.834699\pi$
$570$	0	0
$571$	7.19059	0.300917	0.150458	−	0.988616i	$-0.451925\pi$
$571$	7.19059	0.300917	0.150458	+	0.988616i	$0.451925\pi$
$572$	0	0
$573$	−37.9588	−1.58575
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	39.3103	1.63651	0.818255	−	0.574855i	$-0.194942\pi$
$577$	39.3103	1.63651	0.818255	+	0.574855i	$0.194942\pi$
$578$	0	0
$579$	0.648517	0.0269515
$580$	0	0
$581$	3.14554	0.130499
$582$	0	0
$583$	10.2499	0.424509
$584$	0	0
$585$	1.30226	0.0538419
$586$	0	0
$587$	28.7899	1.18829	0.594143	−	0.804359i	$-0.297491\pi$
$587$	28.7899	1.18829	0.594143	+	0.804359i	$0.297491\pi$
$588$	0	0
$589$	−8.80107	−0.362642
$590$	0	0
$591$	39.7376	1.63458
$592$	0	0
$593$	−40.8873	−1.67904	−0.839519	−	0.543330i	$-0.817163\pi$
$593$	−40.8873	−1.67904	−0.839519	+	0.543330i	$0.817163\pi$
$594$	0	0
$595$	−30.1484	−1.23596
$596$	0	0
$597$	−32.3707	−1.32485
$598$	0	0
$599$	19.1801	0.783679	0.391840	−	0.920034i	$-0.371839\pi$
$599$	19.1801	0.783679	0.391840	+	0.920034i	$0.371839\pi$
$600$	0	0
$601$	3.36683	0.137336	0.0686679	−	0.997640i	$-0.478125\pi$
$601$	3.36683	0.137336	0.0686679	+	0.997640i	$0.478125\pi$
$602$	0	0
$603$	9.77454	0.398050
$604$	0	0
$605$	−10.4436	−0.424594
$606$	0	0
$607$	13.1924	0.535462	0.267731	−	0.963494i	$-0.413726\pi$
$607$	13.1924	0.535462	0.267731	+	0.963494i	$0.413726\pi$
$608$	0	0
$609$	15.9037	0.644450
$610$	0	0
$611$	4.68133	0.189386
$612$	0	0
$613$	32.6207	1.31754	0.658768	−	0.752346i	$-0.271078\pi$
$613$	32.6207	1.31754	0.658768	+	0.752346i	$0.271078\pi$
$614$	0	0
$615$	13.4548	0.542550
$616$	0	0
$617$	41.7212	1.67963	0.839815	−	0.542872i	$-0.182663\pi$
$617$	41.7212	1.67963	0.839815	+	0.542872i	$0.182663\pi$
$618$	0	0
$619$	−0.339245	−0.0136354	−0.00681771	−	0.999977i	$-0.502170\pi$
$619$	−0.339245	−0.0136354	−0.00681771	+	0.999977i	$0.502170\pi$
$620$	0	0
$621$	−4.36266	−0.175068
$622$	0	0
$623$	13.4506	0.538889
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.85863	−0.313843
$628$	0	0
$629$	−26.5494	−1.05859
$630$	0	0
$631$	3.71725	0.147982	0.0739908	−	0.997259i	$-0.476426\pi$
$631$	3.71725	0.147982	0.0739908	+	0.997259i	$0.476426\pi$
$632$	0	0
$633$	25.9260	1.03047
$634$	0	0
$635$	2.64852	0.105103
$636$	0	0
$637$	30.3062	1.20077
$638$	0	0
$639$	−7.21818	−0.285547
$640$	0	0
$641$	−22.0328	−0.870244	−0.435122	−	0.900372i	$-0.643295\pi$
$641$	−22.0328	−0.870244	−0.435122	+	0.900372i	$0.643295\pi$
$642$	0	0
$643$	−24.6842	−0.973449	−0.486724	−	0.873556i	$-0.661808\pi$
$643$	−24.6842	−0.973449	−0.486724	+	0.873556i	$0.661808\pi$
$644$	0	0
$645$	−9.70892	−0.382288
$646$	0	0
$647$	32.1731	1.26486	0.632428	−	0.774619i	$-0.282058\pi$
$647$	32.1731	1.26486	0.632428	+	0.774619i	$0.282058\pi$
$648$	0	0
$649$	9.11273	0.357706
$650$	0	0
$651$	15.4436	0.605284
$652$	0	0
$653$	−26.9466	−1.05450	−0.527251	−	0.849709i	$-0.676777\pi$
$653$	−26.9466	−1.05450	−0.527251	+	0.849709i	$0.676777\pi$
$654$	0	0
$655$	−5.53579	−0.216301
$656$	0	0
$657$	4.24993	0.165806
$658$	0	0
$659$	−19.7417	−0.769029	−0.384514	−	0.923119i	$-0.625631\pi$
$659$	−19.7417	−0.769029	−0.384514	+	0.923119i	$0.625631\pi$
$660$	0	0
$661$	−1.28692	−0.0500552	−0.0250276	−	0.999687i	$-0.507967\pi$
$661$	−1.28692	−0.0500552	−0.0250276	+	0.999687i	$0.507967\pi$
$662$	0	0
$663$	20.6412	0.801639
$664$	0	0
$665$	26.8667	1.04185
$666$	0	0
$667$	1.66492	0.0644661
$668$	0	0
$669$	−2.46705	−0.0953817
$670$	0	0
$671$	−10.4272	−0.402539
$672$	0	0
$673$	12.4970	0.481725	0.240862	−	0.970559i	$-0.422570\pi$
$673$	12.4970	0.481725	0.240862	+	0.970559i	$0.422570\pi$
$674$	0	0
$675$	4.36266	0.167919
$676$	0	0
$677$	−2.69251	−0.103482	−0.0517408	−	0.998661i	$-0.516477\pi$
$677$	−2.69251	−0.103482	−0.0517408	+	0.998661i	$0.516477\pi$
$678$	0	0
$679$	35.1648	1.34950
$680$	0	0
$681$	−6.92842	−0.265498
$682$	0	0
$683$	−2.72115	−0.104122	−0.0520610	−	0.998644i	$-0.516579\pi$
$683$	−2.72115	−0.104122	−0.0520610	+	0.998644i	$0.516579\pi$
$684$	0	0
$685$	6.16896	0.235704
$686$	0	0
$687$	−20.1515	−0.768827
$688$	0	0
$689$	23.9917	0.914010
$690$	0	0
$691$	38.3051	1.45719	0.728597	−	0.684942i	$-0.240173\pi$
$691$	38.3051	1.45719	0.728597	+	0.684942i	$0.240173\pi$
$692$	0	0
$693$	2.74590	0.104308
$694$	0	0
$695$	−20.7693	−0.787825
$696$	0	0
$697$	42.4657	1.60850
$698$	0	0
$699$	−5.92319	−0.224036
$700$	0	0
$701$	39.8472	1.50501	0.752504	−	0.658588i	$-0.228846\pi$
$701$	39.8472	1.50501	0.752504	+	0.658588i	$0.228846\pi$
$702$	0	0
$703$	23.6594	0.892332
$704$	0	0
$705$	−5.18953	−0.195449
$706$	0	0
$707$	73.9094	2.77965
$708$	0	0
$709$	−44.8995	−1.68624	−0.843118	−	0.537728i	$-0.819283\pi$
$709$	−44.8995	−1.68624	−0.843118	+	0.537728i	$0.819283\pi$
$710$	0	0
$711$	7.74173	0.290338
$712$	0	0
$713$	1.61676	0.0605482
$714$	0	0
$715$	1.30226	0.0487019
$716$	0	0
$717$	37.1400	1.38702
$718$	0	0
$719$	−31.5400	−1.17624	−0.588121	−	0.808773i	$-0.700132\pi$
$719$	−31.5400	−1.17624	−0.588121	+	0.808773i	$0.700132\pi$
$720$	0	0
$721$	−53.6318	−1.99735
$722$	0	0
$723$	−0.670152	−0.0249232
$724$	0	0
$725$	−1.66492	−0.0618337
$726$	0	0
$727$	−48.9219	−1.81441	−0.907206	−	0.420687i	$-0.861789\pi$
$727$	−48.9219	−1.81441	−0.907206	+	0.420687i	$0.861789\pi$
$728$	0	0
$729$	17.3637	0.643101
$730$	0	0
$731$	−30.6430	−1.13337
$732$	0	0
$733$	−14.1536	−0.522776	−0.261388	−	0.965234i	$-0.584180\pi$
$733$	−14.1536	−0.522776	−0.261388	+	0.965234i	$0.584180\pi$
$734$	0	0
$735$	−33.5962	−1.23921
$736$	0	0
$737$	9.77454	0.360050
$738$	0	0
$739$	4.46421	0.164219	0.0821093	−	0.996623i	$-0.473834\pi$
$739$	4.46421	0.164219	0.0821093	+	0.996623i	$0.473834\pi$
$740$	0	0
$741$	−18.3944	−0.675736
$742$	0	0
$743$	−34.2898	−1.25797	−0.628985	−	0.777418i	$-0.716529\pi$
$743$	−34.2898	−1.25797	−0.628985	+	0.777418i	$0.716529\pi$
$744$	0	0
$745$	−1.06457	−0.0390027
$746$	0	0
$747$	0.475390	0.0173936
$748$	0	0
$749$	−36.9753	−1.35105
$750$	0	0
$751$	−8.28275	−0.302242	−0.151121	−	0.988515i	$-0.548288\pi$
$751$	−8.28275	−0.302242	−0.151121	+	0.988515i	$0.548288\pi$
$752$	0	0
$753$	−27.8433	−1.01467
$754$	0	0
$755$	−12.6608	−0.460772
$756$	0	0
$757$	3.32985	0.121025	0.0605127	−	0.998167i	$-0.480726\pi$
$757$	3.32985	0.121025	0.0605127	+	0.998167i	$0.480726\pi$
$758$	0	0
$759$	1.44364	0.0524007
$760$	0	0
$761$	−18.4077	−0.667279	−0.333640	−	0.942701i	$-0.608277\pi$
$761$	−18.4077	−0.667279	−0.333640	+	0.942701i	$0.608277\pi$
$762$	0	0
$763$	−90.4710	−3.27527
$764$	0	0
$765$	−4.55636	−0.164736
$766$	0	0
$767$	21.3298	0.770176
$768$	0	0
$769$	51.4506	1.85536	0.927679	−	0.373379i	$-0.121801\pi$
$769$	51.4506	1.85536	0.927679	+	0.373379i	$0.121801\pi$
$770$	0	0
$771$	−2.48869	−0.0896279
$772$	0	0
$773$	17.4506	0.627656	0.313828	−	0.949480i	$-0.398388\pi$
$773$	17.4506	0.627656	0.313828	+	0.949480i	$0.398388\pi$
$774$	0	0
$775$	−1.61676	−0.0580758
$776$	0	0
$777$	−41.5163	−1.48939
$778$	0	0
$779$	−37.8433	−1.35588
$780$	0	0
$781$	−7.21818	−0.258287
$782$	0	0
$783$	−7.26350	−0.259576
$784$	0	0
$785$	14.8873	0.531350
$786$	0	0
$787$	−21.2335	−0.756893	−0.378447	−	0.925623i	$-0.623542\pi$
$787$	−21.2335	−0.756893	−0.378447	+	0.925623i	$0.623542\pi$
$788$	0	0
$789$	−54.6360	−1.94509
$790$	0	0
$791$	1.87086	0.0665203
$792$	0	0
$793$	−24.4067	−0.866706
$794$	0	0
$795$	−26.5962	−0.943270
$796$	0	0
$797$	−25.3215	−0.896934	−0.448467	−	0.893799i	$-0.648030\pi$
$797$	−25.3215	−0.896934	−0.448467	+	0.893799i	$0.648030\pi$
$798$	0	0
$799$	−16.3791	−0.579450
$800$	0	0
$801$	2.03281	0.0718259
$802$	0	0
$803$	4.24993	0.149977
$804$	0	0
$805$	−4.93543	−0.173951
$806$	0	0
$807$	37.6238	1.32442
$808$	0	0
$809$	6.14137	0.215919	0.107960	−	0.994155i	$-0.465568\pi$
$809$	6.14137	0.215919	0.107960	+	0.994155i	$0.465568\pi$
$810$	0	0
$811$	15.5051	0.544457	0.272229	−	0.962233i	$-0.412239\pi$
$811$	15.5051	0.544457	0.272229	+	0.962233i	$0.412239\pi$
$812$	0	0
$813$	12.4395	0.436271
$814$	0	0
$815$	6.72949	0.235724
$816$	0	0
$817$	27.3075	0.955368
$818$	0	0
$819$	6.42723	0.224586
$820$	0	0
$821$	−21.4835	−0.749778	−0.374889	−	0.927070i	$-0.622319\pi$
$821$	−21.4835	−0.749778	−0.374889	+	0.927070i	$0.622319\pi$
$822$	0	0
$823$	40.1565	1.39977	0.699883	−	0.714258i	$-0.253235\pi$
$823$	40.1565	1.39977	0.699883	+	0.714258i	$0.253235\pi$
$824$	0	0
$825$	−1.44364	−0.0502609
$826$	0	0
$827$	38.0573	1.32338	0.661691	−	0.749777i	$-0.269839\pi$
$827$	38.0573	1.32338	0.661691	+	0.749777i	$0.269839\pi$
$828$	0	0
$829$	−51.2580	−1.78026	−0.890132	−	0.455703i	$-0.849388\pi$
$829$	−51.2580	−1.78026	−0.890132	+	0.455703i	$0.849388\pi$
$830$	0	0
$831$	−7.90652	−0.274274
$832$	0	0
$833$	−106.035	−3.67391
$834$	0	0
$835$	−2.98359	−0.103252
$836$	0	0
$837$	−7.05339	−0.243801
$838$	0	0
$839$	45.4200	1.56807	0.784035	−	0.620716i	$-0.213158\pi$
$839$	45.4200	1.56807	0.784035	+	0.620716i	$0.213158\pi$
$840$	0	0
$841$	−26.2280	−0.904415
$842$	0	0
$843$	12.3624	0.425783
$844$	0	0
$845$	−9.95184	−0.342354
$846$	0	0
$847$	−51.5439	−1.77107
$848$	0	0
$849$	41.5798	1.42701
$850$	0	0
$851$	−4.34625	−0.148988
$852$	0	0
$853$	40.0867	1.37254	0.686270	−	0.727346i	$-0.259247\pi$
$853$	40.0867	1.37254	0.686270	+	0.727346i	$0.259247\pi$
$854$	0	0
$855$	4.06040	0.138863
$856$	0	0
$857$	−3.63423	−0.124143	−0.0620715	−	0.998072i	$-0.519771\pi$
$857$	−3.63423	−0.124143	−0.0620715	+	0.998072i	$0.519771\pi$
$858$	0	0
$859$	−37.7529	−1.28811	−0.644056	−	0.764978i	$-0.722750\pi$
$859$	−37.7529	−1.28811	−0.644056	+	0.764978i	$0.722750\pi$
$860$	0	0
$861$	66.4053	2.26309
$862$	0	0
$863$	−3.94555	−0.134308	−0.0671541	−	0.997743i	$-0.521392\pi$
$863$	−3.94555	−0.134308	−0.0671541	+	0.997743i	$0.521392\pi$
$864$	0	0
$865$	−15.8503	−0.538926
$866$	0	0
$867$	−39.3173	−1.33529
$868$	0	0
$869$	7.74173	0.262620
$870$	0	0
$871$	22.8789	0.775223
$872$	0	0
$873$	5.31450	0.179869
$874$	0	0
$875$	4.93543	0.166848
$876$	0	0
$877$	−8.64958	−0.292075	−0.146038	−	0.989279i	$-0.546652\pi$
$877$	−8.64958	−0.292075	−0.146038	+	0.989279i	$0.546652\pi$
$878$	0	0
$879$	31.2007	1.05237
$880$	0	0
$881$	−56.1760	−1.89262	−0.946308	−	0.323266i	$-0.895219\pi$
$881$	−56.1760	−1.89262	−0.946308	+	0.323266i	$0.895219\pi$
$882$	0	0
$883$	−12.1054	−0.407381	−0.203690	−	0.979035i	$-0.565294\pi$
$883$	−12.1054	−0.407381	−0.203690	+	0.979035i	$0.565294\pi$
$884$	0	0
$885$	−23.6454	−0.794832
$886$	0	0
$887$	−24.7058	−0.829540	−0.414770	−	0.909926i	$-0.636138\pi$
$887$	−24.7058	−0.829540	−0.414770	+	0.909926i	$0.636138\pi$
$888$	0	0
$889$	13.0716	0.438407
$890$	0	0
$891$	−7.96719	−0.266911
$892$	0	0
$893$	14.5962	0.488443
$894$	0	0
$895$	3.02759	0.101201
$896$	0	0
$897$	3.37907	0.112824
$898$	0	0
$899$	2.69179	0.0897761
$900$	0	0
$901$	−83.9422	−2.79652
$902$	0	0
$903$	−47.9177	−1.59460
$904$	0	0
$905$	1.82270	0.0605887
$906$	0	0
$907$	22.7170	0.754305	0.377153	−	0.926151i	$-0.376903\pi$
$907$	22.7170	0.754305	0.377153	+	0.926151i	$0.376903\pi$
$908$	0	0
$909$	11.1700	0.370486
$910$	0	0
$911$	−12.3156	−0.408033	−0.204016	−	0.978967i	$-0.565400\pi$
$911$	−12.3156	−0.408033	−0.204016	+	0.978967i	$0.565400\pi$
$912$	0	0
$913$	0.475390	0.0157331
$914$	0	0
$915$	27.0562	0.894451
$916$	0	0
$917$	−27.3215	−0.902236
$918$	0	0
$919$	46.2088	1.52429	0.762144	−	0.647408i	$-0.224147\pi$
$919$	46.2088	1.52429	0.762144	+	0.647408i	$0.224147\pi$
$920$	0	0
$921$	13.4764	0.444064
$922$	0	0
$923$	−16.8953	−0.556117
$924$	0	0
$925$	4.34625	0.142904
$926$	0	0
$927$	−8.10545	−0.266218
$928$	0	0
$929$	9.12391	0.299346	0.149673	−	0.988736i	$-0.452178\pi$
$929$	9.12391	0.299346	0.149673	+	0.988736i	$0.452178\pi$
$930$	0	0
$931$	94.4933	3.09689
$932$	0	0
$933$	8.01330	0.262344
$934$	0	0
$935$	−4.55636	−0.149009
$936$	0	0
$937$	−33.9190	−1.10809	−0.554043	−	0.832488i	$-0.686916\pi$
$937$	−33.9190	−1.10809	−0.554043	+	0.832488i	$0.686916\pi$
$938$	0	0
$939$	−12.1260	−0.395718
$940$	0	0
$941$	−37.2926	−1.21570	−0.607852	−	0.794050i	$-0.707969\pi$
$941$	−37.2926	−1.21570	−0.607852	+	0.794050i	$0.707969\pi$
$942$	0	0
$943$	6.95184	0.226383
$944$	0	0
$945$	21.5316	0.700424
$946$	0	0
$947$	33.7047	1.09526	0.547629	−	0.836722i	$-0.315531\pi$
$947$	33.7047	1.09526	0.547629	+	0.836722i	$0.315531\pi$
$948$	0	0
$949$	9.94767	0.322915
$950$	0	0
$951$	−18.4639	−0.598734
$952$	0	0
$953$	−48.4465	−1.56934	−0.784668	−	0.619917i	$-0.787166\pi$
$953$	−48.4465	−1.56934	−0.784668	+	0.619917i	$0.787166\pi$
$954$	0	0
$955$	19.6126	0.634649
$956$	0	0
$957$	2.40354	0.0776955
$958$	0	0
$959$	30.4465	0.983168
$960$	0	0
$961$	−28.3861	−0.915680
$962$	0	0
$963$	−5.58812	−0.180075
$964$	0	0
$965$	−0.335076	−0.0107865
$966$	0	0
$967$	−28.2528	−0.908548	−0.454274	−	0.890862i	$-0.650101\pi$
$967$	−28.2528	−0.908548	−0.454274	+	0.890862i	$0.650101\pi$
$968$	0	0
$969$	64.3585	2.06749
$970$	0	0
$971$	1.65553	0.0531284	0.0265642	−	0.999647i	$-0.491543\pi$
$971$	1.65553	0.0531284	0.0265642	+	0.999647i	$0.491543\pi$
$972$	0	0
$973$	−102.506	−3.28618
$974$	0	0
$975$	−3.37907	−0.108217
$976$	0	0
$977$	−11.1630	−0.357136	−0.178568	−	0.983928i	$-0.557146\pi$
$977$	−11.1630	−0.357136	−0.178568	+	0.983928i	$0.557146\pi$
$978$	0	0
$979$	2.03281	0.0649690
$980$	0	0
$981$	−13.6730	−0.436545
$982$	0	0
$983$	15.0615	0.480386	0.240193	−	0.970725i	$-0.422789\pi$
$983$	15.0615	0.480386	0.240193	+	0.970725i	$0.422789\pi$
$984$	0	0
$985$	−20.5316	−0.654192
$986$	0	0
$987$	−25.6126	−0.815258
$988$	0	0
$989$	−5.01641	−0.159512
$990$	0	0
$991$	−43.8971	−1.39444	−0.697219	−	0.716858i	$-0.745579\pi$
$991$	−43.8971	−1.39444	−0.697219	+	0.716858i	$0.745579\pi$
$992$	0	0
$993$	27.6811	0.878432
$994$	0	0
$995$	16.7253	0.530228
$996$	0	0
$997$	38.5655	1.22138	0.610691	−	0.791869i	$-0.290892\pi$
$997$	38.5655	1.22138	0.610691	+	0.791869i	$0.290892\pi$
$998$	0	0
$999$	18.9612	0.599907

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.a.i.1.1	✓	3
3.2	odd	2		8280.2.a.bl.1.3		3
4.3	odd	2		1840.2.a.q.1.3		3
5.2	odd	4		4600.2.e.q.4049.5		6
5.3	odd	4		4600.2.e.q.4049.2		6
5.4	even	2		4600.2.a.v.1.3		3
8.3	odd	2		7360.2.a.cf.1.1		3
8.5	even	2		7360.2.a.bw.1.3		3
20.19	odd	2		9200.2.a.ci.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.i.1.1	✓	3	1.1	even	1	trivial
1840.2.a.q.1.3		3	4.3	odd	2
4600.2.a.v.1.3		3	5.4	even	2
4600.2.e.q.4049.2		6	5.3	odd	4
4600.2.e.q.4049.5		6	5.2	odd	4
7360.2.a.bw.1.3		3	8.5	even	2
7360.2.a.cf.1.1		3	8.3	odd	2
8280.2.a.bl.1.3		3	3.2	odd	2
9200.2.a.ci.1.1		3	20.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q-1.93543 q^{3} +1.00000 q^{5} +4.93543 q^{7} +0.745898 q^{9} +0.745898 q^{11} +1.74590 q^{13} -1.93543 q^{15} -6.10856 q^{17} +5.44364 q^{19} -9.55220 q^{21} -1.00000 q^{23} +1.00000 q^{25} +4.36266 q^{27} -1.66492 q^{29} -1.61676 q^{31} -1.44364 q^{33} +4.93543 q^{35} +4.34625 q^{37} -3.37907 q^{39} -6.95184 q^{41} +5.01641 q^{43} +0.745898 q^{45} +2.68133 q^{47} +17.3585 q^{49} +11.8227 q^{51} +13.7417 q^{53} +0.745898 q^{55} -10.5358 q^{57} +12.2171 q^{59} -13.9794 q^{61} +3.68133 q^{63} +1.74590 q^{65} +13.1044 q^{67} +1.93543 q^{69} -9.67716 q^{71} +5.69774 q^{73} -1.93543 q^{75} +3.68133 q^{77} +10.3791 q^{79} -10.6813 q^{81} +0.637339 q^{83} -6.10856 q^{85} +3.22235 q^{87} +2.72532 q^{89} +8.61676 q^{91} +3.12914 q^{93} +5.44364 q^{95} +7.12497 q^{97} +0.556364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 3 q^{5} + 7 q^{7} + 3 q^{9} + 3 q^{11} + 6 q^{13} + 2 q^{15} - 5 q^{17} + 7 q^{19} - 6 q^{21} - 3 q^{23} + 3 q^{25} - q^{27} - q^{29} + 10 q^{31} + 5 q^{33} + 7 q^{35} + 2 q^{37} + 7 q^{39}+ \cdots + 11 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 3 * q^5 + 7 * q^7 + 3 * q^9 + 3 * q^11 + 6 * q^13 + 2 * q^15 - 5 * q^17 + 7 * q^19 - 6 * q^21 - 3 * q^23 + 3 * q^25 - q^27 - q^29 + 10 * q^31 + 5 * q^33 + 7 * q^35 + 2 * q^37 + 7 * q^39 - 10 * q^41 + 12 * q^43 + 3 * q^45 + q^47 + 6 * q^49 + 9 * q^51 + 10 * q^53 + 3 * q^55 - 12 * q^57 + 10 * q^59 - 13 * q^61 + 4 * q^63 + 6 * q^65 - 6 * q^67 - 2 * q^69 + 10 * q^71 + 7 * q^73 + 2 * q^75 + 4 * q^77 + 14 * q^79 - 25 * q^81 + 16 * q^83 - 5 * q^85 - 5 * q^87 - 20 * q^89 + 11 * q^91 + 25 * q^93 + 7 * q^95 + 5 * q^97 + 11 * q^99

Introduction

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