LMFDB - Embedded newform 9680.2.a.cy.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, 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("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.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:	$77.2951891566$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.45753625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20 $ x^6 - x^5 - 13x^4 + 11x^3 + 41x^2 - 30x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.04842$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.31441 q^{3} -1.00000 q^{5} +0.185958 q^{7} +7.98529 q^{9} -3.01352 q^{13} +3.31441 q^{15} -5.11840 q^{17} -4.86246 q^{19} -0.616340 q^{21} -6.20391 q^{23} +1.00000 q^{25} -16.5233 q^{27} +0.190393 q^{29} -8.06089 q^{31} -0.185958 q^{35} -2.55988 q^{37} +9.98803 q^{39} +8.95655 q^{41} -7.83103 q^{43} -7.98529 q^{45} +5.30075 q^{47} -6.96542 q^{49} +16.9645 q^{51} +1.30089 q^{53} +16.1162 q^{57} +0.593594 q^{59} +1.13498 q^{61} +1.48493 q^{63} +3.01352 q^{65} +1.73474 q^{67} +20.5623 q^{69} -13.0710 q^{71} -0.894075 q^{73} -3.31441 q^{75} +13.3047 q^{79} +30.8090 q^{81} -16.9499 q^{83} +5.11840 q^{85} -0.631040 q^{87} -7.22907 q^{89} -0.560388 q^{91} +26.7171 q^{93} +4.86246 q^{95} +3.39538 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} + 6 q^{13} + 2 q^{15} - 11 q^{17} - 11 q^{19} - 2 q^{21} - 18 q^{23} + 6 q^{25} + q^{27} + 6 q^{29} - q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} + 4 q^{41} + 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) $ 6 * q - 2 * q^3 - 6 * q^5 + 6 * q^7 + 10 * q^9 + 6 * q^13 + 2 * q^15 - 11 * q^17 - 11 * q^19 - 2 * q^21 - 18 * q^23 + 6 * q^25 + q^27 + 6 * q^29 - q^31 - 6 * q^35 + 4 * q^37 + 27 * q^39 + 4 * q^41 + 3 * q^43 - 10 * q^45 - 14 * q^47 + 8 * q^49 + 31 * q^51 + 14 * q^53 + 5 * q^57 - 2 * q^59 + 4 * q^61 - 16 * q^63 - 6 * q^65 - 11 * q^67 + 8 * q^69 - 7 * q^71 + 9 * q^73 - 2 * q^75 + 36 * q^79 + 30 * q^81 - 45 * q^83 + 11 * q^85 + 25 * q^87 + q^89 + 8 * q^91 + 55 * q^93 + 11 * q^95 + 20 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.31441	−1.91357	−0.956787	−	0.290790i	$-0.906082\pi$
$3$	−3.31441	−1.91357	−0.956787	+	0.290790i	$0.906082\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.185958	0.0702854	0.0351427	−	0.999382i	$-0.488811\pi$
$7$	0.185958	0.0702854	0.0351427	+	0.999382i	$0.488811\pi$
$8$	0	0
$9$	7.98529	2.66176
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.01352	−0.835800	−0.417900	−	0.908493i	$-0.637234\pi$
$13$	−3.01352	−0.835800	−0.417900	+	0.908493i	$0.637234\pi$
$14$	0	0
$15$	3.31441	0.855776
$16$	0	0
$17$	−5.11840	−1.24139	−0.620697	−	0.784051i	$-0.713150\pi$
$17$	−5.11840	−1.24139	−0.620697	+	0.784051i	$0.713150\pi$
$18$	0	0
$19$	−4.86246	−1.11552	−0.557762	−	0.830001i	$-0.688340\pi$
$19$	−4.86246	−1.11552	−0.557762	+	0.830001i	$0.688340\pi$
$20$	0	0
$21$	−0.616340	−0.134496
$22$	0	0
$23$	−6.20391	−1.29361	−0.646803	−	0.762657i	$-0.723894\pi$
$23$	−6.20391	−1.29361	−0.646803	+	0.762657i	$0.723894\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−16.5233	−3.17991
$28$	0	0
$29$	0.190393	0.0353551	0.0176776	−	0.999844i	$-0.494373\pi$
$29$	0.190393	0.0353551	0.0176776	+	0.999844i	$0.494373\pi$
$30$	0	0
$31$	−8.06089	−1.44778	−0.723889	−	0.689916i	$-0.757647\pi$
$31$	−8.06089	−1.44778	−0.723889	+	0.689916i	$0.757647\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.185958	−0.0314326
$36$	0	0
$37$	−2.55988	−0.420841	−0.210421	−	0.977611i	$-0.567483\pi$
$37$	−2.55988	−0.420841	−0.210421	+	0.977611i	$0.567483\pi$
$38$	0	0
$39$	9.98803	1.59937
$40$	0	0
$41$	8.95655	1.39878	0.699389	−	0.714741i	$-0.253456\pi$
$41$	8.95655	1.39878	0.699389	+	0.714741i	$0.253456\pi$
$42$	0	0
$43$	−7.83103	−1.19422	−0.597111	−	0.802159i	$-0.703685\pi$
$43$	−7.83103	−1.19422	−0.597111	+	0.802159i	$0.703685\pi$
$44$	0	0
$45$	−7.98529	−1.19038
$46$	0	0
$47$	5.30075	0.773193	0.386597	−	0.922249i	$-0.373651\pi$
$47$	5.30075	0.773193	0.386597	+	0.922249i	$0.373651\pi$
$48$	0	0
$49$	−6.96542	−0.995060
$50$	0	0
$51$	16.9645	2.37550
$52$	0	0
$53$	1.30089	0.178690	0.0893452	−	0.996001i	$-0.471523\pi$
$53$	1.30089	0.178690	0.0893452	+	0.996001i	$0.471523\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	16.1162	2.13464
$58$	0	0
$59$	0.593594	0.0772793	0.0386397	−	0.999253i	$-0.487698\pi$
$59$	0.593594	0.0772793	0.0386397	+	0.999253i	$0.487698\pi$
$60$	0	0
$61$	1.13498	0.145319	0.0726595	−	0.997357i	$-0.476851\pi$
$61$	1.13498	0.145319	0.0726595	+	0.997357i	$0.476851\pi$
$62$	0	0
$63$	1.48493	0.187083
$64$	0	0
$65$	3.01352	0.373781
$66$	0	0
$67$	1.73474	0.211932	0.105966	−	0.994370i	$-0.466207\pi$
$67$	1.73474	0.211932	0.105966	+	0.994370i	$0.466207\pi$
$68$	0	0
$69$	20.5623	2.47541
$70$	0	0
$71$	−13.0710	−1.55125	−0.775623	−	0.631197i	$-0.782564\pi$
$71$	−13.0710	−1.55125	−0.775623	+	0.631197i	$0.782564\pi$
$72$	0	0
$73$	−0.894075	−0.104644	−0.0523218	−	0.998630i	$-0.516662\pi$
$73$	−0.894075	−0.104644	−0.0523218	+	0.998630i	$0.516662\pi$
$74$	0	0
$75$	−3.31441	−0.382715
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.3047	1.49689	0.748446	−	0.663196i	$-0.230800\pi$
$79$	13.3047	1.49689	0.748446	+	0.663196i	$0.230800\pi$
$80$	0	0
$81$	30.8090	3.42322
$82$	0	0
$83$	−16.9499	−1.86049	−0.930246	−	0.366937i	$-0.880407\pi$
$83$	−16.9499	−1.86049	−0.930246	+	0.366937i	$0.880407\pi$
$84$	0	0
$85$	5.11840	0.555168
$86$	0	0
$87$	−0.631040	−0.0676546
$88$	0	0
$89$	−7.22907	−0.766280	−0.383140	−	0.923690i	$-0.625157\pi$
$89$	−7.22907	−0.766280	−0.383140	+	0.923690i	$0.625157\pi$
$90$	0	0
$91$	−0.560388	−0.0587446
$92$	0	0
$93$	26.7171	2.77043
$94$	0	0
$95$	4.86246	0.498878
$96$	0	0
$97$	3.39538	0.344748	0.172374	−	0.985032i	$-0.444856\pi$
$97$	3.39538	0.344748	0.172374	+	0.985032i	$0.444856\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.289237	0.0287802	0.0143901	−	0.999896i	$-0.495419\pi$
$101$	0.289237	0.0287802	0.0143901	+	0.999896i	$0.495419\pi$
$102$	0	0
$103$	−17.3893	−1.71342	−0.856711	−	0.515797i	$-0.827496\pi$
$103$	−17.3893	−1.71342	−0.856711	+	0.515797i	$0.827496\pi$
$104$	0	0
$105$	0.616340	0.0601486
$106$	0	0
$107$	−12.1746	−1.17697	−0.588483	−	0.808510i	$-0.700275\pi$
$107$	−12.1746	−1.17697	−0.588483	+	0.808510i	$0.700275\pi$
$108$	0	0
$109$	−14.0940	−1.34996	−0.674978	−	0.737838i	$-0.735847\pi$
$109$	−14.0940	−1.34996	−0.674978	+	0.737838i	$0.735847\pi$
$110$	0	0
$111$	8.48448	0.805311
$112$	0	0
$113$	10.4723	0.985148	0.492574	−	0.870271i	$-0.336056\pi$
$113$	10.4723	0.985148	0.492574	+	0.870271i	$0.336056\pi$
$114$	0	0
$115$	6.20391	0.578518
$116$	0	0
$117$	−24.0638	−2.22470
$118$	0	0
$119$	−0.951806	−0.0872519
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−29.6856	−2.67666
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.09867	0.0974909	0.0487454	−	0.998811i	$-0.484478\pi$
$127$	1.09867	0.0974909	0.0487454	+	0.998811i	$0.484478\pi$
$128$	0	0
$129$	25.9552	2.28523
$130$	0	0
$131$	7.88959	0.689317	0.344659	−	0.938728i	$-0.387995\pi$
$131$	7.88959	0.689317	0.344659	+	0.938728i	$0.387995\pi$
$132$	0	0
$133$	−0.904212	−0.0784051
$134$	0	0
$135$	16.5233	1.42210
$136$	0	0
$137$	5.06861	0.433040	0.216520	−	0.976278i	$-0.430529\pi$
$137$	5.06861	0.433040	0.216520	+	0.976278i	$0.430529\pi$
$138$	0	0
$139$	−8.35531	−0.708689	−0.354344	−	0.935115i	$-0.615296\pi$
$139$	−8.35531	−0.708689	−0.354344	+	0.935115i	$0.615296\pi$
$140$	0	0
$141$	−17.5688	−1.47956
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.190393	−0.0158113
$146$	0	0
$147$	23.0862	1.90412
$148$	0	0
$149$	−20.3880	−1.67025	−0.835126	−	0.550058i	$-0.814606\pi$
$149$	−20.3880	−1.67025	−0.835126	+	0.550058i	$0.814606\pi$
$150$	0	0
$151$	13.1942	1.07373	0.536864	−	0.843669i	$-0.319609\pi$
$151$	13.1942	1.07373	0.536864	+	0.843669i	$0.319609\pi$
$152$	0	0
$153$	−40.8719	−3.30430
$154$	0	0
$155$	8.06089	0.647466
$156$	0	0
$157$	−18.2689	−1.45801	−0.729007	−	0.684506i	$-0.760018\pi$
$157$	−18.2689	−1.45801	−0.729007	+	0.684506i	$0.760018\pi$
$158$	0	0
$159$	−4.31167	−0.341937
$160$	0	0
$161$	−1.15367	−0.0909216
$162$	0	0
$163$	−13.5139	−1.05849	−0.529245	−	0.848469i	$-0.677525\pi$
$163$	−13.5139	−1.05849	−0.529245	+	0.848469i	$0.677525\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.74944	−0.290141	−0.145070	−	0.989421i	$-0.546341\pi$
$167$	−3.74944	−0.290141	−0.145070	+	0.989421i	$0.546341\pi$
$168$	0	0
$169$	−3.91869	−0.301438
$170$	0	0
$171$	−38.8281	−2.96926
$172$	0	0
$173$	−9.68177	−0.736092	−0.368046	−	0.929808i	$-0.619973\pi$
$173$	−9.68177	−0.736092	−0.368046	+	0.929808i	$0.619973\pi$
$174$	0	0
$175$	0.185958	0.0140571
$176$	0	0
$177$	−1.96741	−0.147880
$178$	0	0
$179$	−10.9136	−0.815722	−0.407861	−	0.913044i	$-0.633725\pi$
$179$	−10.9136	−0.815722	−0.407861	+	0.913044i	$0.633725\pi$
$180$	0	0
$181$	−8.06298	−0.599317	−0.299658	−	0.954047i	$-0.596873\pi$
$181$	−8.06298	−0.599317	−0.299658	+	0.954047i	$0.596873\pi$
$182$	0	0
$183$	−3.76178	−0.278079
$184$	0	0
$185$	2.55988	0.188206
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−3.07263	−0.223501
$190$	0	0
$191$	12.7826	0.924917	0.462459	−	0.886641i	$-0.346967\pi$
$191$	12.7826	0.924917	0.462459	+	0.886641i	$0.346967\pi$
$192$	0	0
$193$	−10.4395	−0.751449	−0.375725	−	0.926731i	$-0.622606\pi$
$193$	−10.4395	−0.751449	−0.375725	+	0.926731i	$0.622606\pi$
$194$	0	0
$195$	−9.98803	−0.715258
$196$	0	0
$197$	19.4651	1.38683	0.693417	−	0.720537i	$-0.256105\pi$
$197$	19.4651	1.38683	0.693417	+	0.720537i	$0.256105\pi$
$198$	0	0
$199$	16.0219	1.13576	0.567882	−	0.823110i	$-0.307763\pi$
$199$	16.0219	1.13576	0.567882	+	0.823110i	$0.307763\pi$
$200$	0	0
$201$	−5.74963	−0.405548
$202$	0	0
$203$	0.0354051	0.00248495
$204$	0	0
$205$	−8.95655	−0.625553
$206$	0	0
$207$	−49.5401	−3.44327
$208$	0	0
$209$	0	0
$210$	0	0
$211$	8.78115	0.604519	0.302260	−	0.953226i	$-0.402259\pi$
$211$	8.78115	0.604519	0.302260	+	0.953226i	$0.402259\pi$
$212$	0	0
$213$	43.3227	2.96842
$214$	0	0
$215$	7.83103	0.534072
$216$	0	0
$217$	−1.49899	−0.101758
$218$	0	0
$219$	2.96333	0.200243
$220$	0	0
$221$	15.4244	1.03756
$222$	0	0
$223$	−2.82855	−0.189414	−0.0947069	−	0.995505i	$-0.530191\pi$
$223$	−2.82855	−0.189414	−0.0947069	+	0.995505i	$0.530191\pi$
$224$	0	0
$225$	7.98529	0.532353
$226$	0	0
$227$	−17.2352	−1.14394	−0.571971	−	0.820274i	$-0.693821\pi$
$227$	−17.2352	−1.14394	−0.571971	+	0.820274i	$0.693821\pi$
$228$	0	0
$229$	−11.0399	−0.729540	−0.364770	−	0.931098i	$-0.618852\pi$
$229$	−11.0399	−0.729540	−0.364770	+	0.931098i	$0.618852\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−23.7716	−1.55733	−0.778665	−	0.627440i	$-0.784103\pi$
$233$	−23.7716	−1.55733	−0.778665	+	0.627440i	$0.784103\pi$
$234$	0	0
$235$	−5.30075	−0.345783
$236$	0	0
$237$	−44.0971	−2.86441
$238$	0	0
$239$	21.1341	1.36705	0.683527	−	0.729925i	$-0.260445\pi$
$239$	21.1341	1.36705	0.683527	+	0.729925i	$0.260445\pi$
$240$	0	0
$241$	−2.31116	−0.148875	−0.0744373	−	0.997226i	$-0.523716\pi$
$241$	−2.31116	−0.148875	−0.0744373	+	0.997226i	$0.523716\pi$
$242$	0	0
$243$	−52.5437	−3.37068
$244$	0	0
$245$	6.96542	0.445004
$246$	0	0
$247$	14.6531	0.932356
$248$	0	0
$249$	56.1788	3.56019
$250$	0	0
$251$	−6.10022	−0.385043	−0.192521	−	0.981293i	$-0.561666\pi$
$251$	−6.10022	−0.385043	−0.192521	+	0.981293i	$0.561666\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−16.9645	−1.06236
$256$	0	0
$257$	8.39635	0.523750	0.261875	−	0.965102i	$-0.415659\pi$
$257$	8.39635	0.523750	0.261875	+	0.965102i	$0.415659\pi$
$258$	0	0
$259$	−0.476029	−0.0295790
$260$	0	0
$261$	1.52034	0.0941070
$262$	0	0
$263$	2.04759	0.126260	0.0631299	−	0.998005i	$-0.479892\pi$
$263$	2.04759	0.126260	0.0631299	+	0.998005i	$0.479892\pi$
$264$	0	0
$265$	−1.30089	−0.0799128
$266$	0	0
$267$	23.9601	1.46633
$268$	0	0
$269$	−19.2656	−1.17465	−0.587323	−	0.809353i	$-0.699818\pi$
$269$	−19.2656	−1.17465	−0.587323	+	0.809353i	$0.699818\pi$
$270$	0	0
$271$	−19.3285	−1.17412	−0.587062	−	0.809542i	$-0.699715\pi$
$271$	−19.3285	−1.17412	−0.587062	+	0.809542i	$0.699715\pi$
$272$	0	0
$273$	1.85735	0.112412
$274$	0	0
$275$	0	0
$276$	0	0
$277$	26.2455	1.57694	0.788471	−	0.615072i	$-0.210873\pi$
$277$	26.2455	1.57694	0.788471	+	0.615072i	$0.210873\pi$
$278$	0	0
$279$	−64.3686	−3.85364
$280$	0	0
$281$	−4.01816	−0.239703	−0.119852	−	0.992792i	$-0.538242\pi$
$281$	−4.01816	−0.239703	−0.119852	+	0.992792i	$0.538242\pi$
$282$	0	0
$283$	−6.68519	−0.397393	−0.198697	−	0.980061i	$-0.563671\pi$
$283$	−6.68519	−0.397393	−0.198697	+	0.980061i	$0.563671\pi$
$284$	0	0
$285$	−16.1162	−0.954639
$286$	0	0
$287$	1.66554	0.0983137
$288$	0	0
$289$	9.19800	0.541059
$290$	0	0
$291$	−11.2537	−0.659701
$292$	0	0
$293$	8.38301	0.489741	0.244870	−	0.969556i	$-0.421255\pi$
$293$	8.38301	0.489741	0.244870	+	0.969556i	$0.421255\pi$
$294$	0	0
$295$	−0.593594	−0.0345604
$296$	0	0
$297$	0	0
$298$	0	0
$299$	18.6956	1.08120
$300$	0	0
$301$	−1.45624	−0.0839364
$302$	0	0
$303$	−0.958650	−0.0550730
$304$	0	0
$305$	−1.13498	−0.0649887
$306$	0	0
$307$	5.76000	0.328741	0.164370	−	0.986399i	$-0.447441\pi$
$307$	5.76000	0.328741	0.164370	+	0.986399i	$0.447441\pi$
$308$	0	0
$309$	57.6353	3.27876
$310$	0	0
$311$	1.02983	0.0583965	0.0291982	−	0.999574i	$-0.490705\pi$
$311$	1.02983	0.0583965	0.0291982	+	0.999574i	$0.490705\pi$
$312$	0	0
$313$	−7.93402	−0.448457	−0.224229	−	0.974537i	$-0.571986\pi$
$313$	−7.93402	−0.448457	−0.224229	+	0.974537i	$0.571986\pi$
$314$	0	0
$315$	−1.48493	−0.0836662
$316$	0	0
$317$	−21.3964	−1.20174	−0.600871	−	0.799346i	$-0.705179\pi$
$317$	−21.3964	−1.20174	−0.600871	+	0.799346i	$0.705179\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	40.3517	2.25221
$322$	0	0
$323$	24.8880	1.38481
$324$	0	0
$325$	−3.01352	−0.167160
$326$	0	0
$327$	46.7131	2.58324
$328$	0	0
$329$	0.985715	0.0543442
$330$	0	0
$331$	18.0669	0.993047	0.496523	−	0.868023i	$-0.334610\pi$
$331$	18.0669	0.993047	0.496523	+	0.868023i	$0.334610\pi$
$332$	0	0
$333$	−20.4414	−1.12018
$334$	0	0
$335$	−1.73474	−0.0947789
$336$	0	0
$337$	−2.38869	−0.130120	−0.0650601	−	0.997881i	$-0.520724\pi$
$337$	−2.38869	−0.130120	−0.0650601	+	0.997881i	$0.520724\pi$
$338$	0	0
$339$	−34.7094	−1.88515
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−2.59698	−0.140224
$344$	0	0
$345$	−20.5623	−1.10704
$346$	0	0
$347$	19.3558	1.03908	0.519538	−	0.854447i	$-0.326104\pi$
$347$	19.3558	1.03908	0.519538	+	0.854447i	$0.326104\pi$
$348$	0	0
$349$	30.0338	1.60767	0.803837	−	0.594849i	$-0.202788\pi$
$349$	30.0338	1.60767	0.803837	+	0.594849i	$0.202788\pi$
$350$	0	0
$351$	49.7933	2.65777
$352$	0	0
$353$	−23.8329	−1.26850	−0.634249	−	0.773129i	$-0.718691\pi$
$353$	−23.8329	−1.26850	−0.634249	+	0.773129i	$0.718691\pi$
$354$	0	0
$355$	13.0710	0.693738
$356$	0	0
$357$	3.15467	0.166963
$358$	0	0
$359$	−7.32283	−0.386484	−0.193242	−	0.981151i	$-0.561900\pi$
$359$	−7.32283	−0.386484	−0.193242	+	0.981151i	$0.561900\pi$
$360$	0	0
$361$	4.64350	0.244395
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.894075	0.0467980
$366$	0	0
$367$	33.6772	1.75794	0.878969	−	0.476879i	$-0.158232\pi$
$367$	33.6772	1.75794	0.878969	+	0.476879i	$0.158232\pi$
$368$	0	0
$369$	71.5207	3.72322
$370$	0	0
$371$	0.241910	0.0125593
$372$	0	0
$373$	28.3365	1.46721	0.733605	−	0.679576i	$-0.237836\pi$
$373$	28.3365	1.46721	0.733605	+	0.679576i	$0.237836\pi$
$374$	0	0
$375$	3.31441	0.171155
$376$	0	0
$377$	−0.573754	−0.0295498
$378$	0	0
$379$	14.4046	0.739915	0.369957	−	0.929049i	$-0.379372\pi$
$379$	14.4046	0.739915	0.369957	+	0.929049i	$0.379372\pi$
$380$	0	0
$381$	−3.64143	−0.186556
$382$	0	0
$383$	16.2257	0.829093	0.414547	−	0.910028i	$-0.363940\pi$
$383$	16.2257	0.829093	0.414547	+	0.910028i	$0.363940\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−62.5331	−3.17874
$388$	0	0
$389$	32.3071	1.63803	0.819017	−	0.573770i	$-0.194520\pi$
$389$	32.3071	1.63803	0.819017	+	0.573770i	$0.194520\pi$
$390$	0	0
$391$	31.7541	1.60587
$392$	0	0
$393$	−26.1493	−1.31906
$394$	0	0
$395$	−13.3047	−0.669430
$396$	0	0
$397$	−7.36161	−0.369469	−0.184734	−	0.982788i	$-0.559142\pi$
$397$	−7.36161	−0.369469	−0.184734	+	0.982788i	$0.559142\pi$
$398$	0	0
$399$	2.99693	0.150034
$400$	0	0
$401$	−35.2451	−1.76006	−0.880029	−	0.474920i	$-0.842477\pi$
$401$	−35.2451	−1.76006	−0.880029	+	0.474920i	$0.842477\pi$
$402$	0	0
$403$	24.2917	1.21005
$404$	0	0
$405$	−30.8090	−1.53091
$406$	0	0
$407$	0	0
$408$	0	0
$409$	14.3104	0.707605	0.353803	−	0.935320i	$-0.384888\pi$
$409$	14.3104	0.707605	0.353803	+	0.935320i	$0.384888\pi$
$410$	0	0
$411$	−16.7994	−0.828654
$412$	0	0
$413$	0.110383	0.00543161
$414$	0	0
$415$	16.9499	0.832037
$416$	0	0
$417$	27.6929	1.35613
$418$	0	0
$419$	−8.62183	−0.421204	−0.210602	−	0.977572i	$-0.567542\pi$
$419$	−8.62183	−0.421204	−0.210602	+	0.977572i	$0.567542\pi$
$420$	0	0
$421$	−36.5707	−1.78235	−0.891174	−	0.453661i	$-0.850118\pi$
$421$	−36.5707	−1.78235	−0.891174	+	0.453661i	$0.850118\pi$
$422$	0	0
$423$	42.3280	2.05806
$424$	0	0
$425$	−5.11840	−0.248279
$426$	0	0
$427$	0.211058	0.0102138
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11.4029	−0.549257	−0.274629	−	0.961550i	$-0.588555\pi$
$431$	−11.4029	−0.549257	−0.274629	+	0.961550i	$0.588555\pi$
$432$	0	0
$433$	−3.41134	−0.163938	−0.0819692	−	0.996635i	$-0.526121\pi$
$433$	−3.41134	−0.163938	−0.0819692	+	0.996635i	$0.526121\pi$
$434$	0	0
$435$	0.631040	0.0302561
$436$	0	0
$437$	30.1663	1.44305
$438$	0	0
$439$	2.90901	0.138839	0.0694196	−	0.997588i	$-0.477885\pi$
$439$	2.90901	0.138839	0.0694196	+	0.997588i	$0.477885\pi$
$440$	0	0
$441$	−55.6209	−2.64861
$442$	0	0
$443$	−15.0524	−0.715162	−0.357581	−	0.933882i	$-0.616398\pi$
$443$	−15.0524	−0.715162	−0.357581	+	0.933882i	$0.616398\pi$
$444$	0	0
$445$	7.22907	0.342691
$446$	0	0
$447$	67.5742	3.19615
$448$	0	0
$449$	2.60396	0.122889	0.0614443	−	0.998111i	$-0.480429\pi$
$449$	2.60396	0.122889	0.0614443	+	0.998111i	$0.480429\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−43.7309	−2.05466
$454$	0	0
$455$	0.560388	0.0262714
$456$	0	0
$457$	5.13890	0.240388	0.120194	−	0.992750i	$-0.461648\pi$
$457$	5.13890	0.240388	0.120194	+	0.992750i	$0.461648\pi$
$458$	0	0
$459$	84.5728	3.94752
$460$	0	0
$461$	−37.7325	−1.75738	−0.878688	−	0.477396i	$-0.841581\pi$
$461$	−37.7325	−1.75738	−0.878688	+	0.477396i	$0.841581\pi$
$462$	0	0
$463$	14.7816	0.686961	0.343481	−	0.939160i	$-0.388394\pi$
$463$	14.7816	0.686961	0.343481	+	0.939160i	$0.388394\pi$
$464$	0	0
$465$	−26.7171	−1.23897
$466$	0	0
$467$	8.75904	0.405320	0.202660	−	0.979249i	$-0.435041\pi$
$467$	8.75904	0.405320	0.202660	+	0.979249i	$0.435041\pi$
$468$	0	0
$469$	0.322588	0.0148957
$470$	0	0
$471$	60.5504	2.79002
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.86246	−0.223105
$476$	0	0
$477$	10.3880	0.475632
$478$	0	0
$479$	1.14200	0.0521793	0.0260897	−	0.999660i	$-0.491694\pi$
$479$	1.14200	0.0521793	0.0260897	+	0.999660i	$0.491694\pi$
$480$	0	0
$481$	7.71425	0.351739
$482$	0	0
$483$	3.82372	0.173985
$484$	0	0
$485$	−3.39538	−0.154176
$486$	0	0
$487$	−1.40585	−0.0637054	−0.0318527	−	0.999493i	$-0.510141\pi$
$487$	−1.40585	−0.0637054	−0.0318527	+	0.999493i	$0.510141\pi$
$488$	0	0
$489$	44.7905	2.02550
$490$	0	0
$491$	−31.7045	−1.43081	−0.715403	−	0.698712i	$-0.753757\pi$
$491$	−31.7045	−1.43081	−0.715403	+	0.698712i	$0.753757\pi$
$492$	0	0
$493$	−0.974508	−0.0438896
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.43066	−0.109030
$498$	0	0
$499$	9.59073	0.429340	0.214670	−	0.976687i	$-0.431132\pi$
$499$	9.59073	0.429340	0.214670	+	0.976687i	$0.431132\pi$
$500$	0	0
$501$	12.4272	0.555205
$502$	0	0
$503$	−42.3759	−1.88945	−0.944725	−	0.327865i	$-0.893671\pi$
$503$	−42.3759	−1.88945	−0.944725	+	0.327865i	$0.893671\pi$
$504$	0	0
$505$	−0.289237	−0.0128709
$506$	0	0
$507$	12.9881	0.576824
$508$	0	0
$509$	32.1889	1.42675	0.713374	−	0.700784i	$-0.247166\pi$
$509$	32.1889	1.42675	0.713374	+	0.700784i	$0.247166\pi$
$510$	0	0
$511$	−0.166260	−0.00735492
$512$	0	0
$513$	80.3438	3.54726
$514$	0	0
$515$	17.3893	0.766265
$516$	0	0
$517$	0	0
$518$	0	0
$519$	32.0893	1.40857
$520$	0	0
$521$	16.9751	0.743692	0.371846	−	0.928294i	$-0.378725\pi$
$521$	16.9751	0.743692	0.371846	+	0.928294i	$0.378725\pi$
$522$	0	0
$523$	−3.13310	−0.137001	−0.0685004	−	0.997651i	$-0.521821\pi$
$523$	−3.13310	−0.137001	−0.0685004	+	0.997651i	$0.521821\pi$
$524$	0	0
$525$	−0.616340	−0.0268993
$526$	0	0
$527$	41.2588	1.79726
$528$	0	0
$529$	15.4885	0.673415
$530$	0	0
$531$	4.74002	0.205699
$532$	0	0
$533$	−26.9907	−1.16910
$534$	0	0
$535$	12.1746	0.526355
$536$	0	0
$537$	36.1722	1.56094
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−20.9558	−0.900961	−0.450480	−	0.892786i	$-0.648747\pi$
$541$	−20.9558	−0.900961	−0.450480	+	0.892786i	$0.648747\pi$
$542$	0	0
$543$	26.7240	1.14684
$544$	0	0
$545$	14.0940	0.603719
$546$	0	0
$547$	−16.9309	−0.723914	−0.361957	−	0.932195i	$-0.617891\pi$
$547$	−16.9309	−0.723914	−0.361957	+	0.932195i	$0.617891\pi$
$548$	0	0
$549$	9.06313	0.386805
$550$	0	0
$551$	−0.925779	−0.0394395
$552$	0	0
$553$	2.47411	0.105210
$554$	0	0
$555$	−8.48448	−0.360146
$556$	0	0
$557$	−0.152225	−0.00644999	−0.00322499	−	0.999995i	$-0.501027\pi$
$557$	−0.152225	−0.00644999	−0.00322499	+	0.999995i	$0.501027\pi$
$558$	0	0
$559$	23.5990	0.998131
$560$	0	0
$561$	0	0
$562$	0	0
$563$	35.3689	1.49062	0.745311	−	0.666717i	$-0.232301\pi$
$563$	35.3689	1.49062	0.745311	+	0.666717i	$0.232301\pi$
$564$	0	0
$565$	−10.4723	−0.440572
$566$	0	0
$567$	5.72918	0.240603
$568$	0	0
$569$	−27.2403	−1.14197	−0.570986	−	0.820960i	$-0.693439\pi$
$569$	−27.2403	−1.14197	−0.570986	+	0.820960i	$0.693439\pi$
$570$	0	0
$571$	17.9295	0.750325	0.375162	−	0.926959i	$-0.377587\pi$
$571$	17.9295	0.750325	0.375162	+	0.926959i	$0.377587\pi$
$572$	0	0
$573$	−42.3668	−1.76990
$574$	0	0
$575$	−6.20391	−0.258721
$576$	0	0
$577$	−8.63847	−0.359624	−0.179812	−	0.983701i	$-0.557549\pi$
$577$	−8.63847	−0.359624	−0.179812	+	0.983701i	$0.557549\pi$
$578$	0	0
$579$	34.6006	1.43795
$580$	0	0
$581$	−3.15196	−0.130766
$582$	0	0
$583$	0	0
$584$	0	0
$585$	24.0638	0.994917
$586$	0	0
$587$	−16.4297	−0.678128	−0.339064	−	0.940763i	$-0.610110\pi$
$587$	−16.4297	−0.678128	−0.339064	+	0.940763i	$0.610110\pi$
$588$	0	0
$589$	39.1957	1.61503
$590$	0	0
$591$	−64.5154	−2.65381
$592$	0	0
$593$	9.98760	0.410141	0.205071	−	0.978747i	$-0.434258\pi$
$593$	9.98760	0.410141	0.205071	+	0.978747i	$0.434258\pi$
$594$	0	0
$595$	0.951806	0.0390203
$596$	0	0
$597$	−53.1032	−2.17337
$598$	0	0
$599$	−20.4634	−0.836110	−0.418055	−	0.908422i	$-0.637288\pi$
$599$	−20.4634	−0.836110	−0.418055	+	0.908422i	$0.637288\pi$
$600$	0	0
$601$	23.0934	0.941998	0.470999	−	0.882134i	$-0.343894\pi$
$601$	23.0934	0.941998	0.470999	+	0.882134i	$0.343894\pi$
$602$	0	0
$603$	13.8524	0.564113
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.88119	0.0763551	0.0381776	−	0.999271i	$-0.487845\pi$
$607$	1.88119	0.0763551	0.0381776	+	0.999271i	$0.487845\pi$
$608$	0	0
$609$	−0.117347	−0.00475514
$610$	0	0
$611$	−15.9739	−0.646235
$612$	0	0
$613$	17.2042	0.694871	0.347435	−	0.937704i	$-0.387053\pi$
$613$	17.2042	0.694871	0.347435	+	0.937704i	$0.387053\pi$
$614$	0	0
$615$	29.6856	1.19704
$616$	0	0
$617$	−17.5502	−0.706546	−0.353273	−	0.935520i	$-0.614931\pi$
$617$	−17.5502	−0.706546	−0.353273	+	0.935520i	$0.614931\pi$
$618$	0	0
$619$	33.8060	1.35878	0.679389	−	0.733778i	$-0.262245\pi$
$619$	33.8060	1.35878	0.679389	+	0.733778i	$0.262245\pi$
$620$	0	0
$621$	102.509	4.11355
$622$	0	0
$623$	−1.34430	−0.0538584
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13.1025	0.522430
$630$	0	0
$631$	30.3134	1.20676	0.603380	−	0.797454i	$-0.293820\pi$
$631$	30.3134	1.20676	0.603380	+	0.797454i	$0.293820\pi$
$632$	0	0
$633$	−29.1043	−1.15679
$634$	0	0
$635$	−1.09867	−0.0435992
$636$	0	0
$637$	20.9904	0.831671
$638$	0	0
$639$	−104.376	−4.12905
$640$	0	0
$641$	−3.68875	−0.145697	−0.0728484	−	0.997343i	$-0.523209\pi$
$641$	−3.68875	−0.145697	−0.0728484	+	0.997343i	$0.523209\pi$
$642$	0	0
$643$	4.76467	0.187900	0.0939501	−	0.995577i	$-0.470051\pi$
$643$	4.76467	0.187900	0.0939501	+	0.995577i	$0.470051\pi$
$644$	0	0
$645$	−25.9552	−1.02199
$646$	0	0
$647$	−34.6861	−1.36365	−0.681826	−	0.731514i	$-0.738814\pi$
$647$	−34.6861	−1.36365	−0.681826	+	0.731514i	$0.738814\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	4.96825	0.194721
$652$	0	0
$653$	−6.35710	−0.248773	−0.124386	−	0.992234i	$-0.539696\pi$
$653$	−6.35710	−0.248773	−0.124386	+	0.992234i	$0.539696\pi$
$654$	0	0
$655$	−7.88959	−0.308272
$656$	0	0
$657$	−7.13945	−0.278536
$658$	0	0
$659$	8.61834	0.335723	0.167861	−	0.985811i	$-0.446314\pi$
$659$	8.61834	0.335723	0.167861	+	0.985811i	$0.446314\pi$
$660$	0	0
$661$	−17.5983	−0.684494	−0.342247	−	0.939610i	$-0.611188\pi$
$661$	−17.5983	−0.684494	−0.342247	+	0.939610i	$0.611188\pi$
$662$	0	0
$663$	−51.1227	−1.98544
$664$	0	0
$665$	0.904212	0.0350638
$666$	0	0
$667$	−1.18118	−0.0457356
$668$	0	0
$669$	9.37497	0.362457
$670$	0	0
$671$	0	0
$672$	0	0
$673$	51.0036	1.96604	0.983022	−	0.183486i	$-0.0587383\pi$
$673$	51.0036	1.96604	0.983022	+	0.183486i	$0.0587383\pi$
$674$	0	0
$675$	−16.5233	−0.635982
$676$	0	0
$677$	−4.86576	−0.187007	−0.0935033	−	0.995619i	$-0.529807\pi$
$677$	−4.86576	−0.187007	−0.0935033	+	0.995619i	$0.529807\pi$
$678$	0	0
$679$	0.631397	0.0242308
$680$	0	0
$681$	57.1245	2.18902
$682$	0	0
$683$	9.50423	0.363669	0.181835	−	0.983329i	$-0.441796\pi$
$683$	9.50423	0.363669	0.181835	+	0.983329i	$0.441796\pi$
$684$	0	0
$685$	−5.06861	−0.193661
$686$	0	0
$687$	36.5909	1.39603
$688$	0	0
$689$	−3.92025	−0.149350
$690$	0	0
$691$	26.8072	1.01979	0.509897	−	0.860235i	$-0.329684\pi$
$691$	26.8072	1.01979	0.509897	+	0.860235i	$0.329684\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.35531	0.316935
$696$	0	0
$697$	−45.8432	−1.73643
$698$	0	0
$699$	78.7888	2.98007
$700$	0	0
$701$	26.0457	0.983732	0.491866	−	0.870671i	$-0.336315\pi$
$701$	26.0457	0.983732	0.491866	+	0.870671i	$0.336315\pi$
$702$	0	0
$703$	12.4473	0.469459
$704$	0	0
$705$	17.5688	0.661680
$706$	0	0
$707$	0.0537859	0.00202283
$708$	0	0
$709$	18.7635	0.704679	0.352340	−	0.935872i	$-0.385386\pi$
$709$	18.7635	0.704679	0.352340	+	0.935872i	$0.385386\pi$
$710$	0	0
$711$	106.242	3.98437
$712$	0	0
$713$	50.0091	1.87285
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−70.0471	−2.61596
$718$	0	0
$719$	−30.7426	−1.14651	−0.573253	−	0.819378i	$-0.694319\pi$
$719$	−30.7426	−1.14651	−0.573253	+	0.819378i	$0.694319\pi$
$720$	0	0
$721$	−3.23368	−0.120429
$722$	0	0
$723$	7.66011	0.284883
$724$	0	0
$725$	0.190393	0.00707102
$726$	0	0
$727$	15.2620	0.566037	0.283019	−	0.959114i	$-0.408664\pi$
$727$	15.2620	0.566037	0.283019	+	0.959114i	$0.408664\pi$
$728$	0	0
$729$	81.7243	3.02683
$730$	0	0
$731$	40.0823	1.48250
$732$	0	0
$733$	7.80089	0.288133	0.144066	−	0.989568i	$-0.453982\pi$
$733$	7.80089	0.288133	0.144066	+	0.989568i	$0.453982\pi$
$734$	0	0
$735$	−23.0862	−0.851549
$736$	0	0
$737$	0	0
$738$	0	0
$739$	20.0951	0.739209	0.369604	−	0.929189i	$-0.379493\pi$
$739$	20.0951	0.739209	0.369604	+	0.929189i	$0.379493\pi$
$740$	0	0
$741$	−48.5664	−1.78413
$742$	0	0
$743$	27.9809	1.02652	0.513259	−	0.858234i	$-0.328438\pi$
$743$	27.9809	1.02652	0.513259	+	0.858234i	$0.328438\pi$
$744$	0	0
$745$	20.3880	0.746960
$746$	0	0
$747$	−135.350	−4.95219
$748$	0	0
$749$	−2.26397	−0.0827236
$750$	0	0
$751$	−29.8830	−1.09045	−0.545223	−	0.838291i	$-0.683555\pi$
$751$	−29.8830	−1.09045	−0.545223	+	0.838291i	$0.683555\pi$
$752$	0	0
$753$	20.2186	0.736808
$754$	0	0
$755$	−13.1942	−0.480185
$756$	0	0
$757$	6.89865	0.250736	0.125368	−	0.992110i	$-0.459989\pi$
$757$	6.89865	0.250736	0.125368	+	0.992110i	$0.459989\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.6369	−0.675588	−0.337794	−	0.941220i	$-0.609681\pi$
$761$	−18.6369	−0.675588	−0.337794	+	0.941220i	$0.609681\pi$
$762$	0	0
$763$	−2.62088	−0.0948822
$764$	0	0
$765$	40.8719	1.47773
$766$	0	0
$767$	−1.78881	−0.0645901
$768$	0	0
$769$	23.0644	0.831723	0.415861	−	0.909428i	$-0.363480\pi$
$769$	23.0644	0.831723	0.415861	+	0.909428i	$0.363480\pi$
$770$	0	0
$771$	−27.8289	−1.00223
$772$	0	0
$773$	−1.58562	−0.0570309	−0.0285155	−	0.999593i	$-0.509078\pi$
$773$	−1.58562	−0.0570309	−0.0285155	+	0.999593i	$0.509078\pi$
$774$	0	0
$775$	−8.06089	−0.289556
$776$	0	0
$777$	1.57775	0.0566016
$778$	0	0
$779$	−43.5508	−1.56037
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−3.14592	−0.112426
$784$	0	0
$785$	18.2689	0.652044
$786$	0	0
$787$	20.8524	0.743309	0.371655	−	0.928371i	$-0.378791\pi$
$787$	20.8524	0.743309	0.371655	+	0.928371i	$0.378791\pi$
$788$	0	0
$789$	−6.78655	−0.241608
$790$	0	0
$791$	1.94740	0.0692416
$792$	0	0
$793$	−3.42028	−0.121458
$794$	0	0
$795$	4.31167	0.152919
$796$	0	0
$797$	12.0560	0.427045	0.213522	−	0.976938i	$-0.431506\pi$
$797$	12.0560	0.427045	0.213522	+	0.976938i	$0.431506\pi$
$798$	0	0
$799$	−27.1313	−0.959837
$800$	0	0
$801$	−57.7263	−2.03966
$802$	0	0
$803$	0	0
$804$	0	0
$805$	1.15367	0.0406614
$806$	0	0
$807$	63.8541	2.24777
$808$	0	0
$809$	52.6231	1.85013	0.925064	−	0.379810i	$-0.124011\pi$
$809$	52.6231	1.85013	0.925064	+	0.379810i	$0.124011\pi$
$810$	0	0
$811$	48.1470	1.69067	0.845334	−	0.534238i	$-0.179401\pi$
$811$	48.1470	1.69067	0.845334	+	0.534238i	$0.179401\pi$
$812$	0	0
$813$	64.0626	2.24677
$814$	0	0
$815$	13.5139	0.473371
$816$	0	0
$817$	38.0781	1.33218
$818$	0	0
$819$	−4.47486	−0.156364
$820$	0	0
$821$	16.2528	0.567226	0.283613	−	0.958939i	$-0.408467\pi$
$821$	16.2528	0.567226	0.283613	+	0.958939i	$0.408467\pi$
$822$	0	0
$823$	−23.1040	−0.805356	−0.402678	−	0.915342i	$-0.631921\pi$
$823$	−23.1040	−0.805356	−0.402678	+	0.915342i	$0.631921\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	47.2149	1.64182	0.820911	−	0.571057i	$-0.193466\pi$
$827$	47.2149	1.64182	0.820911	+	0.571057i	$0.193466\pi$
$828$	0	0
$829$	−40.7983	−1.41698	−0.708491	−	0.705720i	$-0.750624\pi$
$829$	−40.7983	−1.41698	−0.708491	+	0.705720i	$0.750624\pi$
$830$	0	0
$831$	−86.9884	−3.01759
$832$	0	0
$833$	35.6518	1.23526
$834$	0	0
$835$	3.74944	0.129755
$836$	0	0
$837$	133.192	4.60380
$838$	0	0
$839$	−43.7775	−1.51137	−0.755683	−	0.654938i	$-0.772695\pi$
$839$	−43.7775	−1.51137	−0.755683	+	0.654938i	$0.772695\pi$
$840$	0	0
$841$	−28.9638	−0.998750
$842$	0	0
$843$	13.3178	0.458690
$844$	0	0
$845$	3.91869	0.134807
$846$	0	0
$847$	0	0
$848$	0	0
$849$	22.1574	0.760441
$850$	0	0
$851$	15.8813	0.544403
$852$	0	0
$853$	−53.4183	−1.82901	−0.914503	−	0.404579i	$-0.867418\pi$
$853$	−53.4183	−1.82901	−0.914503	+	0.404579i	$0.867418\pi$
$854$	0	0
$855$	38.8281	1.32789
$856$	0	0
$857$	21.6966	0.741143	0.370572	−	0.928804i	$-0.379162\pi$
$857$	21.6966	0.741143	0.370572	+	0.928804i	$0.379162\pi$
$858$	0	0
$859$	−18.9254	−0.645725	−0.322863	−	0.946446i	$-0.604645\pi$
$859$	−18.9254	−0.645725	−0.322863	+	0.946446i	$0.604645\pi$
$860$	0	0
$861$	−5.52028	−0.188131
$862$	0	0
$863$	−25.0747	−0.853553	−0.426776	−	0.904357i	$-0.640351\pi$
$863$	−25.0747	−0.853553	−0.426776	+	0.904357i	$0.640351\pi$
$864$	0	0
$865$	9.68177	0.329190
$866$	0	0
$867$	−30.4859	−1.03536
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−5.22767	−0.177133
$872$	0	0
$873$	27.1131	0.917639
$874$	0	0
$875$	−0.185958	−0.00628652
$876$	0	0
$877$	3.41891	0.115448	0.0577242	−	0.998333i	$-0.481616\pi$
$877$	3.41891	0.115448	0.0577242	+	0.998333i	$0.481616\pi$
$878$	0	0
$879$	−27.7847	−0.937155
$880$	0	0
$881$	1.37318	0.0462636	0.0231318	−	0.999732i	$-0.492636\pi$
$881$	1.37318	0.0462636	0.0231318	+	0.999732i	$0.492636\pi$
$882$	0	0
$883$	−46.2204	−1.55544	−0.777720	−	0.628610i	$-0.783624\pi$
$883$	−46.2204	−1.55544	−0.777720	+	0.628610i	$0.783624\pi$
$884$	0	0
$885$	1.96741	0.0661338
$886$	0	0
$887$	−32.4716	−1.09029	−0.545145	−	0.838342i	$-0.683525\pi$
$887$	−32.4716	−1.09029	−0.545145	+	0.838342i	$0.683525\pi$
$888$	0	0
$889$	0.204306	0.00685219
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−25.7747	−0.862516
$894$	0	0
$895$	10.9136	0.364802
$896$	0	0
$897$	−61.9649	−2.06895
$898$	0	0
$899$	−1.53474	−0.0511864
$900$	0	0
$901$	−6.65845	−0.221825
$902$	0	0
$903$	4.82658	0.160618
$904$	0	0
$905$	8.06298	0.268023
$906$	0	0
$907$	−35.6120	−1.18248	−0.591238	−	0.806497i	$-0.701361\pi$
$907$	−35.6120	−1.18248	−0.591238	+	0.806497i	$0.701361\pi$
$908$	0	0
$909$	2.30964	0.0766060
$910$	0	0
$911$	5.40562	0.179096	0.0895481	−	0.995982i	$-0.471458\pi$
$911$	5.40562	0.179096	0.0895481	+	0.995982i	$0.471458\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3.76178	0.124361
$916$	0	0
$917$	1.46713	0.0484490
$918$	0	0
$919$	46.5266	1.53477	0.767386	−	0.641185i	$-0.221557\pi$
$919$	46.5266	1.53477	0.767386	+	0.641185i	$0.221557\pi$
$920$	0	0
$921$	−19.0910	−0.629069
$922$	0	0
$923$	39.3898	1.29653
$924$	0	0
$925$	−2.55988	−0.0841683
$926$	0	0
$927$	−138.859	−4.56072
$928$	0	0
$929$	−31.5053	−1.03365	−0.516827	−	0.856090i	$-0.672887\pi$
$929$	−31.5053	−1.03365	−0.516827	+	0.856090i	$0.672887\pi$
$930$	0	0
$931$	33.8691	1.11001
$932$	0	0
$933$	−3.41329	−0.111746
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−43.5043	−1.42122	−0.710612	−	0.703584i	$-0.751582\pi$
$937$	−43.5043	−1.42122	−0.710612	+	0.703584i	$0.751582\pi$
$938$	0	0
$939$	26.2966	0.858156
$940$	0	0
$941$	−24.2949	−0.791991	−0.395996	−	0.918252i	$-0.629600\pi$
$941$	−24.2949	−0.791991	−0.395996	+	0.918252i	$0.629600\pi$
$942$	0	0
$943$	−55.5657	−1.80947
$944$	0	0
$945$	3.07263	0.0999528
$946$	0	0
$947$	37.0312	1.20335	0.601676	−	0.798740i	$-0.294500\pi$
$947$	37.0312	1.20335	0.601676	+	0.798740i	$0.294500\pi$
$948$	0	0
$949$	2.69431	0.0874611
$950$	0	0
$951$	70.9164	2.29962
$952$	0	0
$953$	−12.5995	−0.408139	−0.204070	−	0.978956i	$-0.565417\pi$
$953$	−12.5995	−0.408139	−0.204070	+	0.978956i	$0.565417\pi$
$954$	0	0
$955$	−12.7826	−0.413636
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.942547	0.0304364
$960$	0	0
$961$	33.9779	1.09606
$962$	0	0
$963$	−97.2180	−3.13281
$964$	0	0
$965$	10.4395	0.336058
$966$	0	0
$967$	18.4170	0.592251	0.296126	−	0.955149i	$-0.404305\pi$
$967$	18.4170	0.592251	0.296126	+	0.955149i	$0.404305\pi$
$968$	0	0
$969$	−82.4890	−2.64993
$970$	0	0
$971$	1.95940	0.0628800	0.0314400	−	0.999506i	$-0.489991\pi$
$971$	1.95940	0.0628800	0.0314400	+	0.999506i	$0.489991\pi$
$972$	0	0
$973$	−1.55374	−0.0498105
$974$	0	0
$975$	9.98803	0.319873
$976$	0	0
$977$	−8.09382	−0.258944	−0.129472	−	0.991583i	$-0.541328\pi$
$977$	−8.09382	−0.258944	−0.129472	+	0.991583i	$0.541328\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−112.544	−3.59326
$982$	0	0
$983$	9.43861	0.301045	0.150522	−	0.988607i	$-0.451904\pi$
$983$	9.43861	0.301045	0.150522	+	0.988607i	$0.451904\pi$
$984$	0	0
$985$	−19.4651	−0.620211
$986$	0	0
$987$	−3.26706	−0.103992
$988$	0	0
$989$	48.5831	1.54485
$990$	0	0
$991$	44.9960	1.42934	0.714672	−	0.699459i	$-0.246576\pi$
$991$	44.9960	1.42934	0.714672	+	0.699459i	$0.246576\pi$
$992$	0	0
$993$	−59.8811	−1.90027
$994$	0	0
$995$	−16.0219	−0.507929
$996$	0	0
$997$	−23.5984	−0.747369	−0.373684	−	0.927556i	$-0.621906\pi$
$997$	−23.5984	−0.747369	−0.373684	+	0.927556i	$0.621906\pi$
$998$	0	0
$999$	42.2976	1.33824

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.cy.1.1		6
4.3	odd	2		4840.2.a.be.1.6		6
11.7	odd	10		880.2.bo.j.401.3		12
11.8	odd	10		880.2.bo.j.801.3		12
11.10	odd	2		9680.2.a.cx.1.1		6
44.7	even	10		440.2.y.b.401.1	yes	12
44.19	even	10		440.2.y.b.361.1	✓	12
44.43	even	2		4840.2.a.bf.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.b.361.1	✓	12	44.19	even	10
440.2.y.b.401.1	yes	12	44.7	even	10
880.2.bo.j.401.3		12	11.7	odd	10
880.2.bo.j.801.3		12	11.8	odd	10
4840.2.a.be.1.6		6	4.3	odd	2
4840.2.a.bf.1.6		6	44.43	even	2
9680.2.a.cx.1.1		6	11.10	odd	2
9680.2.a.cy.1.1		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q-3.31441 q^{3} -1.00000 q^{5} +0.185958 q^{7} +7.98529 q^{9} -3.01352 q^{13} +3.31441 q^{15} -5.11840 q^{17} -4.86246 q^{19} -0.616340 q^{21} -6.20391 q^{23} +1.00000 q^{25} -16.5233 q^{27} +0.190393 q^{29} -8.06089 q^{31} -0.185958 q^{35} -2.55988 q^{37} +9.98803 q^{39} +8.95655 q^{41} -7.83103 q^{43} -7.98529 q^{45} +5.30075 q^{47} -6.96542 q^{49} +16.9645 q^{51} +1.30089 q^{53} +16.1162 q^{57} +0.593594 q^{59} +1.13498 q^{61} +1.48493 q^{63} +3.01352 q^{65} +1.73474 q^{67} +20.5623 q^{69} -13.0710 q^{71} -0.894075 q^{73} -3.31441 q^{75} +13.3047 q^{79} +30.8090 q^{81} -16.9499 q^{83} +5.11840 q^{85} -0.631040 q^{87} -7.22907 q^{89} -0.560388 q^{91} +26.7171 q^{93} +4.86246 q^{95} +3.39538 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} + 6 q^{13} + 2 q^{15} - 11 q^{17} - 11 q^{19} - 2 q^{21} - 18 q^{23} + 6 q^{25} + q^{27} + 6 q^{29} - q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} + 4 q^{41} + 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) \) 6 * q - 2 * q^3 - 6 * q^5 + 6 * q^7 + 10 * q^9 + 6 * q^13 + 2 * q^15 - 11 * q^17 - 11 * q^19 - 2 * q^21 - 18 * q^23 + 6 * q^25 + q^27 + 6 * q^29 - q^31 - 6 * q^35 + 4 * q^37 + 27 * q^39 + 4 * q^41 + 3 * q^43 - 10 * q^45 - 14 * q^47 + 8 * q^49 + 31 * q^51 + 14 * q^53 + 5 * q^57 - 2 * q^59 + 4 * q^61 - 16 * q^63 - 6 * q^65 - 11 * q^67 + 8 * q^69 - 7 * q^71 + 9 * q^73 - 2 * q^75 + 36 * q^79 + 30 * q^81 - 45 * q^83 + 11 * q^85 + 25 * q^87 + q^89 + 8 * q^91 + 55 * q^93 + 11 * q^95 + 20 * q^97

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