LMFDB - Embedded newform 9747.2.a.bk.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9747,2,Mod(1,9747)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9747, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9747.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9747 = 3^{3} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9747.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:	$77.8301868501$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.1272367936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 23x^{4} + 152x^{2} - 301 $ x^6 - 23x^4 + 152x^2 - 301
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.95759$ of defining polynomial
Character	$\chi$	$=$	9747.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.91223 q^{2} +1.65662 q^{4} +1.95759 q^{5} +1.25561 q^{7} -0.656620 q^{8} +3.74336 q^{10} -5.70095 q^{11} -1.28539 q^{13} +2.40101 q^{14} -4.56885 q^{16} +6.98635 q^{17} +3.24299 q^{20} -10.9015 q^{22} +1.95759 q^{23} -1.16784 q^{25} -2.45797 q^{26} +2.08007 q^{28} -4.91223 q^{29} -9.61614 q^{31} -7.42345 q^{32} +13.3595 q^{34} +2.45797 q^{35} -3.74336 q^{37} -1.28539 q^{40} -9.22547 q^{41} -3.40101 q^{43} -9.44432 q^{44} +3.74336 q^{46} +3.07117 q^{47} -5.42345 q^{49} -2.23317 q^{50} -2.12941 q^{52} -6.65662 q^{53} -11.1601 q^{55} -0.824458 q^{56} -9.39331 q^{58} +4.96986 q^{59} +7.70655 q^{61} -18.3883 q^{62} -5.05763 q^{64} -2.51628 q^{65} -4.58738 q^{67} +11.5737 q^{68} +4.70020 q^{70} +8.19533 q^{71} -9.27540 q^{73} -7.15817 q^{74} -7.15817 q^{77} +6.31415 q^{79} -8.94394 q^{80} -17.6412 q^{82} -3.07117 q^{83} +13.6764 q^{85} -6.50351 q^{86} +3.74336 q^{88} +6.01979 q^{89} -1.61395 q^{91} +3.24299 q^{92} +5.87277 q^{94} +13.8009 q^{97} -10.3709 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} + 6 q^{4} - 2 q^{7} + 20 q^{14} - 10 q^{16} + 16 q^{25} - 24 q^{28} - 16 q^{29} - 12 q^{32} - 34 q^{41} - 26 q^{43} - 42 q^{50} - 36 q^{53} + 6 q^{55} + 22 q^{56} - 12 q^{58} + 18 q^{59} - 6 q^{61}+ \cdots - 56 q^{98}+O(q^{100}) $ 6 * q - 2 * q^2 + 6 * q^4 - 2 * q^7 + 20 * q^14 - 10 * q^16 + 16 * q^25 - 24 * q^28 - 16 * q^29 - 12 * q^32 - 34 * q^41 - 26 * q^43 - 42 * q^50 - 36 * q^53 + 6 * q^55 + 22 * q^56 - 12 * q^58 + 18 * q^59 - 6 * q^61 - 32 * q^64 - 24 * q^65 + 16 * q^71 + 14 * q^73 - 2 * q^82 + 18 * q^85 + 30 * q^86 - 24 * q^89 - 56 * q^98

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 $

$2$	1.91223	1.35215	0.676075	−	0.736833i	$-0.263679\pi$
$2$	1.91223	1.35215	0.676075	+	0.736833i	$0.263679\pi$
$3$	0	0
$3$	0	0
$4$	1.65662	0.828310
$5$	1.95759	0.875461	0.437731	−	0.899106i	$-0.355782\pi$
$5$	1.95759	0.875461	0.437731	+	0.899106i	$0.355782\pi$
$6$	0	0
$7$	1.25561	0.474575	0.237288	−	0.971439i	$-0.423742\pi$
$7$	1.25561	0.474575	0.237288	+	0.971439i	$0.423742\pi$
$8$	−0.656620	−0.232150
$9$	0	0
$10$	3.74336	1.18376
$11$	−5.70095	−1.71890	−0.859451	−	0.511218i	$-0.829194\pi$
$11$	−5.70095	−1.71890	−0.859451	+	0.511218i	$0.829194\pi$
$12$	0	0
$13$	−1.28539	−0.356504	−0.178252	−	0.983985i	$-0.557044\pi$
$13$	−1.28539	−0.356504	−0.178252	+	0.983985i	$0.557044\pi$
$14$	2.40101	0.641697
$15$	0	0
$16$	−4.56885	−1.14221
$17$	6.98635	1.69444	0.847219	−	0.531244i	$-0.178275\pi$
$17$	6.98635	1.69444	0.847219	+	0.531244i	$0.178275\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	3.24299	0.725154
$21$	0	0
$22$	−10.9015	−2.32421
$23$	1.95759	0.408186	0.204093	−	0.978952i	$-0.434576\pi$
$23$	1.95759	0.408186	0.204093	+	0.978952i	$0.434576\pi$
$24$	0	0
$25$	−1.16784	−0.233568
$26$	−2.45797	−0.482047
$27$	0	0
$28$	2.08007	0.393096
$29$	−4.91223	−0.912178	−0.456089	−	0.889934i	$-0.650750\pi$
$29$	−4.91223	−0.912178	−0.456089	+	0.889934i	$0.650750\pi$
$30$	0	0
$31$	−9.61614	−1.72711	−0.863554	−	0.504256i	$-0.831767\pi$
$31$	−9.61614	−1.72711	−0.863554	+	0.504256i	$0.831767\pi$
$32$	−7.42345	−1.31229
$33$	0	0
$34$	13.3595	2.29113
$35$	2.45797	0.415472
$36$	0	0
$37$	−3.74336	−0.615405	−0.307702	−	0.951483i	$-0.599560\pi$
$37$	−3.74336	−0.615405	−0.307702	+	0.951483i	$0.599560\pi$
$38$	0	0
$39$	0	0
$40$	−1.28539	−0.203239
$41$	−9.22547	−1.44078	−0.720388	−	0.693571i	$-0.756036\pi$
$41$	−9.22547	−1.44078	−0.720388	+	0.693571i	$0.756036\pi$
$42$	0	0
$43$	−3.40101	−0.518649	−0.259325	−	0.965790i	$-0.583500\pi$
$43$	−3.40101	−0.518649	−0.259325	+	0.965790i	$0.583500\pi$
$44$	−9.44432	−1.42378
$45$	0	0
$46$	3.74336	0.551929
$47$	3.07117	0.447976	0.223988	−	0.974592i	$-0.428092\pi$
$47$	3.07117	0.447976	0.223988	+	0.974592i	$0.428092\pi$
$48$	0	0
$49$	−5.42345	−0.774778
$50$	−2.23317	−0.315818
$51$	0	0
$52$	−2.12941	−0.295296
$53$	−6.65662	−0.914357	−0.457179	−	0.889375i	$-0.651140\pi$
$53$	−6.65662	−0.914357	−0.457179	+	0.889375i	$0.651140\pi$
$54$	0	0
$55$	−11.1601	−1.50483
$56$	−0.824458	−0.110173
$57$	0	0
$58$	−9.39331	−1.23340
$59$	4.96986	0.647021	0.323510	−	0.946225i	$-0.395137\pi$
$59$	4.96986	0.647021	0.323510	+	0.946225i	$0.395137\pi$
$60$	0	0
$61$	7.70655	0.986722	0.493361	−	0.869825i	$-0.335768\pi$
$61$	7.70655	0.986722	0.493361	+	0.869825i	$0.335768\pi$
$62$	−18.3883	−2.33531
$63$	0	0
$64$	−5.05763	−0.632204
$65$	−2.51628	−0.312106
$66$	0	0
$67$	−4.58738	−0.560437	−0.280219	−	0.959936i	$-0.590407\pi$
$67$	−4.58738	−0.560437	−0.280219	+	0.959936i	$0.590407\pi$
$68$	11.5737	1.40352
$69$	0	0
$70$	4.70020	0.561781
$71$	8.19533	0.972607	0.486303	−	0.873790i	$-0.338345\pi$
$71$	8.19533	0.972607	0.486303	+	0.873790i	$0.338345\pi$
$72$	0	0
$73$	−9.27540	−1.08560	−0.542802	−	0.839861i	$-0.682636\pi$
$73$	−9.27540	−1.08560	−0.542802	+	0.839861i	$0.682636\pi$
$74$	−7.15817	−0.832120
$75$	0	0
$76$	0	0
$77$	−7.15817	−0.815749
$78$	0	0
$79$	6.31415	0.710397	0.355199	−	0.934791i	$-0.384413\pi$
$79$	6.31415	0.710397	0.355199	+	0.934791i	$0.384413\pi$
$80$	−8.94394	−0.999963
$81$	0	0
$82$	−17.6412	−1.94815
$83$	−3.07117	−0.337104	−0.168552	−	0.985693i	$-0.553909\pi$
$83$	−3.07117	−0.337104	−0.168552	+	0.985693i	$0.553909\pi$
$84$	0	0
$85$	13.6764	1.48341
$86$	−6.50351	−0.701292
$87$	0	0
$88$	3.74336	0.399044
$89$	6.01979	0.638096	0.319048	−	0.947738i	$-0.396637\pi$
$89$	6.01979	0.638096	0.319048	+	0.947738i	$0.396637\pi$
$90$	0	0
$91$	−1.61395	−0.169188
$92$	3.24299	0.338105
$93$	0	0
$94$	5.87277	0.605730
$95$	0	0
$96$	0	0
$97$	13.8009	1.40127	0.700633	−	0.713522i	$-0.252901\pi$
$97$	13.8009	1.40127	0.700633	+	0.713522i	$0.252901\pi$
$98$	−10.3709	−1.04762
$99$	0	0
$100$	−1.93466	−0.193466
$101$	12.6873	1.26243	0.631217	−	0.775606i	$-0.282556\pi$
$101$	12.6873	1.26243	0.631217	+	0.775606i	$0.282556\pi$
$102$	0	0
$103$	2.45797	0.242191	0.121095	−	0.992641i	$-0.461359\pi$
$103$	2.45797	0.242191	0.121095	+	0.992641i	$0.461359\pi$
$104$	0.844016	0.0827626
$105$	0	0
$106$	−12.7290	−1.23635
$107$	−13.4234	−1.29769	−0.648847	−	0.760919i	$-0.724748\pi$
$107$	−13.4234	−1.29769	−0.648847	+	0.760919i	$0.724748\pi$
$108$	0	0
$109$	−13.3595	−1.27961	−0.639804	−	0.768538i	$-0.720984\pi$
$109$	−13.3595	−1.27961	−0.639804	+	0.768538i	$0.720984\pi$
$110$	−21.3407	−2.03476
$111$	0	0
$112$	−5.73669	−0.542066
$113$	−14.7565	−1.38817	−0.694086	−	0.719892i	$-0.744191\pi$
$113$	−14.7565	−1.38817	−0.694086	+	0.719892i	$0.744191\pi$
$114$	0	0
$115$	3.83216	0.357351
$116$	−8.13770	−0.755566
$117$	0	0
$118$	9.50351	0.874869
$119$	8.77212	0.804139
$120$	0	0
$121$	21.5009	1.95462
$122$	14.7367	1.33420
$123$	0	0
$124$	−15.9303	−1.43058
$125$	−12.0741	−1.07994
$126$	0	0
$127$	18.3883	1.63169	0.815847	−	0.578268i	$-0.196271\pi$
$127$	18.3883	1.63169	0.815847	+	0.578268i	$0.196271\pi$
$128$	5.17554	0.457458
$129$	0	0
$130$	−4.81170	−0.422014
$131$	−6.81453	−0.595388	−0.297694	−	0.954661i	$-0.596218\pi$
$131$	−6.81453	−0.595388	−0.297694	+	0.954661i	$0.596218\pi$
$132$	0	0
$133$	0	0
$134$	−8.77212	−0.757796
$135$	0	0
$136$	−4.58738	−0.393364
$137$	−6.54497	−0.559174	−0.279587	−	0.960120i	$-0.590198\pi$
$137$	−6.54497	−0.559174	−0.279587	+	0.960120i	$0.590198\pi$
$138$	0	0
$139$	−14.5886	−1.23739	−0.618696	−	0.785630i	$-0.712339\pi$
$139$	−14.5886	−1.23739	−0.618696	+	0.785630i	$0.712339\pi$
$140$	4.07192	0.344140
$141$	0	0
$142$	15.6714	1.31511
$143$	7.32797	0.612796
$144$	0	0
$145$	−9.61614	−0.798577
$146$	−17.7367	−1.46790
$147$	0	0
$148$	−6.20133	−0.509746
$149$	−14.0317	−1.14952	−0.574761	−	0.818322i	$-0.694905\pi$
$149$	−14.0317	−1.14952	−0.574761	+	0.818322i	$0.694905\pi$
$150$	0	0
$151$	11.2301	0.913892	0.456946	−	0.889495i	$-0.348943\pi$
$151$	11.2301	0.913892	0.456946	+	0.889495i	$0.348943\pi$
$152$	0	0
$153$	0	0
$154$	−13.6881	−1.10301
$155$	−18.8245	−1.51202
$156$	0	0
$157$	15.4509	1.23312	0.616560	−	0.787308i	$-0.288526\pi$
$157$	15.4509	1.23312	0.616560	+	0.787308i	$0.288526\pi$
$158$	12.0741	0.960564
$159$	0	0
$160$	−14.5321	−1.14886
$161$	2.45797	0.193715
$162$	0	0
$163$	−1.53871	−0.120521	−0.0602606	−	0.998183i	$-0.519193\pi$
$163$	−1.53871	−0.120521	−0.0602606	+	0.998183i	$0.519193\pi$
$164$	−15.2831	−1.19341
$165$	0	0
$166$	−5.87277	−0.455816
$167$	−18.5035	−1.43184	−0.715922	−	0.698180i	$-0.753994\pi$
$167$	−18.5035	−1.43184	−0.715922	+	0.698180i	$0.753994\pi$
$168$	0	0
$169$	−11.3478	−0.872905
$170$	26.1524	2.00580
$171$	0	0
$172$	−5.63419	−0.429603
$173$	−5.71690	−0.434648	−0.217324	−	0.976100i	$-0.569733\pi$
$173$	−5.71690	−0.434648	−0.217324	+	0.976100i	$0.569733\pi$
$174$	0	0
$175$	−1.46635	−0.110845
$176$	26.0468	1.96335
$177$	0	0
$178$	11.5112	0.862802
$179$	−14.8667	−1.11119	−0.555594	−	0.831454i	$-0.687509\pi$
$179$	−14.8667	−1.11119	−0.555594	+	0.831454i	$0.687509\pi$
$180$	0	0
$181$	−20.5177	−1.52507	−0.762533	−	0.646949i	$-0.776045\pi$
$181$	−20.5177	−1.52507	−0.762533	+	0.646949i	$0.776045\pi$
$182$	−3.08625	−0.228768
$183$	0	0
$184$	−1.28539	−0.0947605
$185$	−7.32797	−0.538763
$186$	0	0
$187$	−39.8288	−2.91257
$188$	5.08776	0.371063
$189$	0	0
$190$	0	0
$191$	18.4473	1.33480	0.667398	−	0.744701i	$-0.267408\pi$
$191$	18.4473	1.33480	0.667398	+	0.744701i	$0.267408\pi$
$192$	0	0
$193$	25.4336	1.83075	0.915375	−	0.402602i	$-0.131894\pi$
$193$	25.4336	1.83075	0.915375	+	0.402602i	$0.131894\pi$
$194$	26.3904	1.89472
$195$	0	0
$196$	−8.98459	−0.641757
$197$	−4.52838	−0.322634	−0.161317	−	0.986903i	$-0.551574\pi$
$197$	−4.52838	−0.322634	−0.161317	+	0.986903i	$0.551574\pi$
$198$	0	0
$199$	8.56115	0.606884	0.303442	−	0.952850i	$-0.401864\pi$
$199$	8.56115	0.606884	0.303442	+	0.952850i	$0.401864\pi$
$200$	0.766826	0.0542228
$201$	0	0
$202$	24.2610	1.70700
$203$	−6.16784	−0.432897
$204$	0	0
$205$	−18.0597	−1.26134
$206$	4.70020	0.327478
$207$	0	0
$208$	5.87277	0.407204
$209$	0	0
$210$	0	0
$211$	−14.6449	−1.00820	−0.504098	−	0.863646i	$-0.668175\pi$
$211$	−14.6449	−1.00820	−0.504098	+	0.863646i	$0.668175\pi$
$212$	−11.0275	−0.757371
$213$	0	0
$214$	−25.6687	−1.75468
$215$	−6.65779	−0.454057
$216$	0	0
$217$	−12.0741	−0.819644
$218$	−25.5464	−1.73022
$219$	0	0
$220$	−18.4881	−1.24647
$221$	−8.98021	−0.604074
$222$	0	0
$223$	22.9756	1.53856	0.769281	−	0.638910i	$-0.220615\pi$
$223$	22.9756	1.53856	0.769281	+	0.638910i	$0.220615\pi$
$224$	−9.32094	−0.622782
$225$	0	0
$226$	−28.2178	−1.87702
$227$	8.71690	0.578561	0.289280	−	0.957244i	$-0.406584\pi$
$227$	8.71690	0.578561	0.289280	+	0.957244i	$0.406584\pi$
$228$	0	0
$229$	−18.0724	−1.19426	−0.597128	−	0.802146i	$-0.703691\pi$
$229$	−18.0724	−1.19426	−0.597128	+	0.802146i	$0.703691\pi$
$230$	7.32797	0.483192
$231$	0	0
$232$	3.22547	0.211762
$233$	15.7585	1.03237	0.516186	−	0.856477i	$-0.327351\pi$
$233$	15.7585	1.03237	0.516186	+	0.856477i	$0.327351\pi$
$234$	0	0
$235$	6.01209	0.392185
$236$	8.23317	0.535934
$237$	0	0
$238$	16.7743	1.08732
$239$	26.8908	1.73942	0.869711	−	0.493561i	$-0.164305\pi$
$239$	26.8908	1.73942	0.869711	+	0.493561i	$0.164305\pi$
$240$	0	0
$241$	−26.8318	−1.72839	−0.864195	−	0.503158i	$-0.832171\pi$
$241$	−26.8318	−1.72839	−0.864195	+	0.503158i	$0.832171\pi$
$242$	41.1146	2.64295
$243$	0	0
$244$	12.7668	0.817312
$245$	−10.6169	−0.678288
$246$	0	0
$247$	0	0
$248$	6.31415	0.400949
$249$	0	0
$250$	−23.0885	−1.46024
$251$	15.2581	0.963082	0.481541	−	0.876424i	$-0.340077\pi$
$251$	15.2581	0.963082	0.481541	+	0.876424i	$0.340077\pi$
$252$	0	0
$253$	−11.1601	−0.701632
$254$	35.1626	2.20629
$255$	0	0
$256$	20.0121	1.25076
$257$	−23.7565	−1.48189	−0.740944	−	0.671567i	$-0.765622\pi$
$257$	−23.7565	−1.48189	−0.740944	+	0.671567i	$0.765622\pi$
$258$	0	0
$259$	−4.70020	−0.292056
$260$	−4.16851	−0.258520
$261$	0	0
$262$	−13.0309	−0.805054
$263$	7.21717	0.445030	0.222515	−	0.974929i	$-0.428573\pi$
$263$	7.21717	0.445030	0.222515	+	0.974929i	$0.428573\pi$
$264$	0	0
$265$	−13.0309	−0.800484
$266$	0	0
$267$	0	0
$268$	−7.59954	−0.464216
$269$	2.53101	0.154318	0.0771591	−	0.997019i	$-0.475415\pi$
$269$	2.53101	0.154318	0.0771591	+	0.997019i	$0.475415\pi$
$270$	0	0
$271$	3.02243	0.183600	0.0917999	−	0.995777i	$-0.470738\pi$
$271$	3.02243	0.183600	0.0917999	+	0.995777i	$0.470738\pi$
$272$	−31.9196	−1.93541
$273$	0	0
$274$	−12.5155	−0.756088
$275$	6.65779	0.401480
$276$	0	0
$277$	−5.08513	−0.305536	−0.152768	−	0.988262i	$-0.548819\pi$
$277$	−5.08513	−0.305536	−0.152768	+	0.988262i	$0.548819\pi$
$278$	−27.8968	−1.67314
$279$	0	0
$280$	−1.61395	−0.0964521
$281$	8.25296	0.492331	0.246165	−	0.969228i	$-0.420829\pi$
$281$	8.25296	0.492331	0.246165	+	0.969228i	$0.420829\pi$
$282$	0	0
$283$	−23.7488	−1.41172	−0.705859	−	0.708352i	$-0.749439\pi$
$283$	−23.7488	−1.41172	−0.705859	+	0.708352i	$0.749439\pi$
$284$	13.5766	0.805620
$285$	0	0
$286$	14.0128	0.828592
$287$	−11.5836	−0.683757
$288$	0	0
$289$	31.8091	1.87112
$290$	−18.3883	−1.07980
$291$	0	0
$292$	−15.3658	−0.899216
$293$	25.2901	1.47747	0.738733	−	0.673999i	$-0.235425\pi$
$293$	25.2901	1.47747	0.738733	+	0.673999i	$0.235425\pi$
$294$	0	0
$295$	9.72896	0.566442
$296$	2.45797	0.142867
$297$	0	0
$298$	−26.8318	−1.55433
$299$	−2.51628	−0.145520
$300$	0	0
$301$	−4.27034	−0.246138
$302$	21.4745	1.23572
$303$	0	0
$304$	0	0
$305$	15.0863	0.863837
$306$	0	0
$307$	−9.21350	−0.525842	−0.262921	−	0.964817i	$-0.584686\pi$
$307$	−9.21350	−0.525842	−0.262921	+	0.964817i	$0.584686\pi$
$308$	−11.8584	−0.675693
$309$	0	0
$310$	−35.9967	−2.04447
$311$	−7.71754	−0.437622	−0.218811	−	0.975767i	$-0.570218\pi$
$311$	−7.71754	−0.437622	−0.218811	+	0.975767i	$0.570218\pi$
$312$	0	0
$313$	−7.05499	−0.398771	−0.199386	−	0.979921i	$-0.563895\pi$
$313$	−7.05499	−0.398771	−0.199386	+	0.979921i	$0.563895\pi$
$314$	29.5457	1.66736
$315$	0	0
$316$	10.4602	0.588429
$317$	−16.4285	−0.922717	−0.461358	−	0.887214i	$-0.652638\pi$
$317$	−16.4285	−0.922717	−0.461358	+	0.887214i	$0.652638\pi$
$318$	0	0
$319$	28.0044	1.56794
$320$	−9.90077	−0.553470
$321$	0	0
$322$	4.70020	0.261932
$323$	0	0
$324$	0	0
$325$	1.50113	0.0832678
$326$	−2.94237	−0.162963
$327$	0	0
$328$	6.05763	0.334477
$329$	3.85618	0.212598
$330$	0	0
$331$	19.1195	1.05090	0.525450	−	0.850824i	$-0.323897\pi$
$331$	19.1195	1.05090	0.525450	+	0.850824i	$0.323897\pi$
$332$	−5.08776	−0.279227
$333$	0	0
$334$	−35.3830	−1.93607
$335$	−8.98021	−0.490641
$336$	0	0
$337$	−6.64271	−0.361851	−0.180926	−	0.983497i	$-0.557909\pi$
$337$	−6.64271	−0.361851	−0.180926	+	0.983497i	$0.557909\pi$
$338$	−21.6995	−1.18030
$339$	0	0
$340$	22.6566	1.22873
$341$	54.8211	2.96873
$342$	0	0
$343$	−15.5990	−0.842266
$344$	2.23317	0.120405
$345$	0	0
$346$	−10.9320	−0.587709
$347$	−33.8182	−1.81545	−0.907727	−	0.419561i	$-0.862184\pi$
$347$	−33.8182	−1.81545	−0.907727	+	0.419561i	$0.862184\pi$
$348$	0	0
$349$	14.7015	0.786953	0.393476	−	0.919335i	$-0.371272\pi$
$349$	14.7015	0.786953	0.393476	+	0.919335i	$0.371272\pi$
$350$	−2.80399	−0.149880
$351$	0	0
$352$	42.3207	2.25570
$353$	7.09917	0.377851	0.188925	−	0.981991i	$-0.439500\pi$
$353$	7.09917	0.377851	0.188925	+	0.981991i	$0.439500\pi$
$354$	0	0
$355$	16.0431	0.851480
$356$	9.97251	0.528542
$357$	0	0
$358$	−28.4285	−1.50249
$359$	−24.6047	−1.29858	−0.649292	−	0.760539i	$-0.724935\pi$
$359$	−24.6047	−1.29858	−0.649292	+	0.760539i	$0.724935\pi$
$360$	0	0
$361$	0	0
$362$	−39.2345	−2.06212
$363$	0	0
$364$	−2.67371	−0.140140
$365$	−18.1574	−0.950404
$366$	0	0
$367$	12.3709	0.645754	0.322877	−	0.946441i	$-0.395350\pi$
$367$	12.3709	0.645754	0.322877	+	0.946441i	$0.395350\pi$
$368$	−8.94394	−0.466235
$369$	0	0
$370$	−14.0128	−0.728489
$371$	−8.35811	−0.433931
$372$	0	0
$373$	7.15817	0.370636	0.185318	−	0.982679i	$-0.440669\pi$
$373$	7.15817	0.370636	0.185318	+	0.982679i	$0.440669\pi$
$374$	−76.1619	−3.93824
$375$	0	0
$376$	−2.01659	−0.103998
$377$	6.31415	0.325195
$378$	0	0
$379$	−14.2035	−0.729585	−0.364793	−	0.931089i	$-0.618860\pi$
$379$	−14.2035	−0.729585	−0.364793	+	0.931089i	$0.618860\pi$
$380$	0	0
$381$	0	0
$382$	35.2754	1.80485
$383$	−19.4184	−0.992233	−0.496117	−	0.868256i	$-0.665241\pi$
$383$	−19.4184	−0.992233	−0.496117	+	0.868256i	$0.665241\pi$
$384$	0	0
$385$	−14.0128	−0.714156
$386$	48.6349	2.47545
$387$	0	0
$388$	22.8628	1.16068
$389$	7.31491	0.370880	0.185440	−	0.982656i	$-0.440629\pi$
$389$	7.31491	0.370880	0.185440	+	0.982656i	$0.440629\pi$
$390$	0	0
$391$	13.6764	0.691646
$392$	3.56115	0.179865
$393$	0	0
$394$	−8.65930	−0.436249
$395$	12.3605	0.621925
$396$	0	0
$397$	−17.6489	−0.885774	−0.442887	−	0.896578i	$-0.646046\pi$
$397$	−17.6489	−0.885774	−0.442887	+	0.896578i	$0.646046\pi$
$398$	16.3709	0.820598
$399$	0	0
$400$	5.33568	0.266784
$401$	−20.0801	−1.00275	−0.501375	−	0.865230i	$-0.667172\pi$
$401$	−20.0801	−1.00275	−0.501375	+	0.865230i	$0.667172\pi$
$402$	0	0
$403$	12.3605	0.615722
$404$	21.0180	1.04569
$405$	0	0
$406$	−11.7943	−0.585342
$407$	21.3407	1.05782
$408$	0	0
$409$	−10.1703	−0.502891	−0.251445	−	0.967872i	$-0.580906\pi$
$409$	−10.1703	−0.502891	−0.251445	+	0.967872i	$0.580906\pi$
$410$	−34.5343	−1.70553
$411$	0	0
$412$	4.07192	0.200609
$413$	6.24020	0.307060
$414$	0	0
$415$	−6.01209	−0.295122
$416$	9.54206	0.467838
$417$	0	0
$418$	0	0
$419$	−16.4897	−0.805573	−0.402786	−	0.915294i	$-0.631958\pi$
$419$	−16.4897	−0.805573	−0.402786	+	0.915294i	$0.631958\pi$
$420$	0	0
$421$	22.2057	1.08224	0.541120	−	0.840946i	$-0.318001\pi$
$421$	22.2057	1.08224	0.541120	+	0.840946i	$0.318001\pi$
$422$	−28.0044	−1.36323
$423$	0	0
$424$	4.37087	0.212268
$425$	−8.15892	−0.395766
$426$	0	0
$427$	9.67641	0.468274
$428$	−22.2376	−1.07489
$429$	0	0
$430$	−12.7312	−0.613954
$431$	31.1421	1.50006	0.750031	−	0.661403i	$-0.230039\pi$
$431$	31.1421	1.50006	0.750031	+	0.661403i	$0.230039\pi$
$432$	0	0
$433$	7.59954	0.365211	0.182605	−	0.983186i	$-0.441547\pi$
$433$	7.59954	0.365211	0.182605	+	0.983186i	$0.441547\pi$
$434$	−23.0885	−1.10828
$435$	0	0
$436$	−22.1316	−1.05991
$437$	0	0
$438$	0	0
$439$	−25.8750	−1.23495	−0.617473	−	0.786592i	$-0.711843\pi$
$439$	−25.8750	−1.23495	−0.617473	+	0.786592i	$0.711843\pi$
$440$	7.32797	0.349347
$441$	0	0
$442$	−17.1722	−0.816799
$443$	9.84695	0.467843	0.233921	−	0.972256i	$-0.424844\pi$
$443$	9.84695	0.467843	0.233921	+	0.972256i	$0.424844\pi$
$444$	0	0
$445$	11.7843	0.558629
$446$	43.9347	2.08037
$447$	0	0
$448$	−6.35041	−0.300029
$449$	−40.0647	−1.89077	−0.945384	−	0.325958i	$-0.894313\pi$
$449$	−40.0647	−1.89077	−0.945384	+	0.325958i	$0.894313\pi$
$450$	0	0
$451$	52.5940	2.47655
$452$	−24.4459	−1.14984
$453$	0	0
$454$	16.6687	0.782301
$455$	−3.15946	−0.148118
$456$	0	0
$457$	36.0372	1.68575	0.842874	−	0.538111i	$-0.180862\pi$
$457$	36.0372	1.68575	0.842874	+	0.538111i	$0.180862\pi$
$458$	−34.5585	−1.61481
$459$	0	0
$460$	6.34844	0.295997
$461$	−18.3293	−0.853679	−0.426839	−	0.904328i	$-0.640373\pi$
$461$	−18.3293	−0.853679	−0.426839	+	0.904328i	$0.640373\pi$
$462$	0	0
$463$	25.9769	1.20725	0.603624	−	0.797269i	$-0.293723\pi$
$463$	25.9769	1.20725	0.603624	+	0.797269i	$0.293723\pi$
$464$	22.4432	1.04190
$465$	0	0
$466$	30.1338	1.39592
$467$	−12.8001	−0.592319	−0.296159	−	0.955139i	$-0.595706\pi$
$467$	−12.8001	−0.592319	−0.296159	+	0.955139i	$0.595706\pi$
$468$	0	0
$469$	−5.75995	−0.265970
$470$	11.4965	0.530293
$471$	0	0
$472$	−3.26331	−0.150206
$473$	19.3890	0.891508
$474$	0	0
$475$	0	0
$476$	14.5321	0.666076
$477$	0	0
$478$	51.4214	2.35196
$479$	−3.91518	−0.178889	−0.0894446	−	0.995992i	$-0.528509\pi$
$479$	−3.91518	−0.178889	−0.0894446	+	0.995992i	$0.528509\pi$
$480$	0	0
$481$	4.81170	0.219394
$482$	−51.3086	−2.33704
$483$	0	0
$484$	35.6188	1.61904
$485$	27.0165	1.22675
$486$	0	0
$487$	11.3429	0.513996	0.256998	−	0.966412i	$-0.417267\pi$
$487$	11.3429	0.513996	0.256998	+	0.966412i	$0.417267\pi$
$488$	−5.06028	−0.229068
$489$	0	0
$490$	−20.3019	−0.917148
$491$	22.8038	1.02912	0.514561	−	0.857454i	$-0.327955\pi$
$491$	22.8038	1.02912	0.514561	+	0.857454i	$0.327955\pi$
$492$	0	0
$493$	−34.3185	−1.54563
$494$	0	0
$495$	0	0
$496$	43.9347	1.97273
$497$	10.2901	0.461575
$498$	0	0
$499$	30.7538	1.37673	0.688365	−	0.725364i	$-0.258329\pi$
$499$	30.7538	1.37673	0.688365	+	0.725364i	$0.258329\pi$
$500$	−20.0022	−0.894526
$501$	0	0
$502$	29.1770	1.30223
$503$	8.94394	0.398791	0.199395	−	0.979919i	$-0.436102\pi$
$503$	8.94394	0.398791	0.199395	+	0.979919i	$0.436102\pi$
$504$	0	0
$505$	24.8365	1.10521
$506$	−21.3407	−0.948711
$507$	0	0
$508$	30.4624	1.35155
$509$	−2.41574	−0.107076	−0.0535380	−	0.998566i	$-0.517050\pi$
$509$	−2.41574	−0.107076	−0.0535380	+	0.998566i	$0.517050\pi$
$510$	0	0
$511$	−11.6463	−0.515201
$512$	27.9166	1.23375
$513$	0	0
$514$	−45.4278	−2.00374
$515$	4.81170	0.212029
$516$	0	0
$517$	−17.5086	−0.770026
$518$	−8.98786	−0.394904
$519$	0	0
$520$	1.65224	0.0724554
$521$	−2.43553	−0.106703	−0.0533513	−	0.998576i	$-0.516990\pi$
$521$	−2.43553	−0.106703	−0.0533513	+	0.998576i	$0.516990\pi$
$522$	0	0
$523$	7.88936	0.344978	0.172489	−	0.985011i	$-0.444819\pi$
$523$	7.88936	0.344978	0.172489	+	0.985011i	$0.444819\pi$
$524$	−11.2891	−0.493166
$525$	0	0
$526$	13.8009	0.601747
$527$	−67.1817	−2.92648
$528$	0	0
$529$	−19.1678	−0.833384
$530$	−24.9181	−1.08237
$531$	0	0
$532$	0	0
$533$	11.8584	0.513643
$534$	0	0
$535$	−26.2776	−1.13608
$536$	3.01217	0.130106
$537$	0	0
$538$	4.83987	0.208661
$539$	30.9188	1.33177
$540$	0	0
$541$	−24.4683	−1.05198	−0.525988	−	0.850492i	$-0.676304\pi$
$541$	−24.4683	−1.05198	−0.525988	+	0.850492i	$0.676304\pi$
$542$	5.77959	0.248255
$543$	0	0
$544$	−51.8628	−2.22360
$545$	−26.1524	−1.12025
$546$	0	0
$547$	18.9037	0.808264	0.404132	−	0.914701i	$-0.367574\pi$
$547$	18.9037	0.808264	0.404132	+	0.914701i	$0.367574\pi$
$548$	−10.8425	−0.463170
$549$	0	0
$550$	12.7312	0.542861
$551$	0	0
$552$	0	0
$553$	7.92810	0.337137
$554$	−9.72393	−0.413130
$555$	0	0
$556$	−24.1678	−1.02494
$557$	−28.6766	−1.21507	−0.607533	−	0.794294i	$-0.707841\pi$
$557$	−28.6766	−1.21507	−0.607533	+	0.794294i	$0.707841\pi$
$558$	0	0
$559$	4.37164	0.184901
$560$	−11.2301	−0.474558
$561$	0	0
$562$	15.7816	0.665705
$563$	15.6386	0.659087	0.329544	−	0.944140i	$-0.393105\pi$
$563$	15.6386	0.659087	0.329544	+	0.944140i	$0.393105\pi$
$564$	0	0
$565$	−28.8871	−1.21529
$566$	−45.4131	−1.90885
$567$	0	0
$568$	−5.38122	−0.225791
$569$	9.76683	0.409447	0.204723	−	0.978820i	$-0.434371\pi$
$569$	9.76683	0.409447	0.204723	+	0.978820i	$0.434371\pi$
$570$	0	0
$571$	0.453585	0.0189820	0.00949098	−	0.999955i	$-0.496979\pi$
$571$	0.453585	0.0189820	0.00949098	+	0.999955i	$0.496979\pi$
$572$	12.1397	0.507585
$573$	0	0
$574$	−22.1505	−0.924542
$575$	−2.28615	−0.0953390
$576$	0	0
$577$	−9.78994	−0.407560	−0.203780	−	0.979017i	$-0.565323\pi$
$577$	−9.78994	−0.407560	−0.203780	+	0.979017i	$0.565323\pi$
$578$	60.8262	2.53004
$579$	0	0
$580$	−15.9303	−0.661469
$581$	−3.85618	−0.159981
$582$	0	0
$583$	37.9491	1.57169
$584$	6.09042	0.252023
$585$	0	0
$586$	48.3605	1.99775
$587$	33.1511	1.36829	0.684147	−	0.729344i	$-0.260175\pi$
$587$	33.1511	1.36829	0.684147	+	0.729344i	$0.260175\pi$
$588$	0	0
$589$	0	0
$590$	18.6040	0.765914
$591$	0	0
$592$	17.1029	0.702923
$593$	36.3941	1.49453	0.747264	−	0.664527i	$-0.231367\pi$
$593$	36.3941	1.49453	0.747264	+	0.664527i	$0.231367\pi$
$594$	0	0
$595$	17.1722	0.703992
$596$	−23.2452	−0.952160
$597$	0	0
$598$	−4.81170	−0.196765
$599$	−26.4734	−1.08167	−0.540836	−	0.841128i	$-0.681892\pi$
$599$	−26.4734	−1.08167	−0.540836	+	0.841128i	$0.681892\pi$
$600$	0	0
$601$	13.0309	0.531543	0.265772	−	0.964036i	$-0.414373\pi$
$601$	13.0309	0.531543	0.265772	+	0.964036i	$0.414373\pi$
$602$	−8.16587	−0.332816
$603$	0	0
$604$	18.6040	0.756986
$605$	42.0899	1.71120
$606$	0	0
$607$	43.5320	1.76691	0.883455	−	0.468515i	$-0.155211\pi$
$607$	43.5320	1.76691	0.883455	+	0.468515i	$0.155211\pi$
$608$	0	0
$609$	0	0
$610$	28.8484	1.16804
$611$	−3.94766	−0.159705
$612$	0	0
$613$	46.6111	1.88260	0.941302	−	0.337566i	$-0.109604\pi$
$613$	46.6111	1.88260	0.941302	+	0.337566i	$0.109604\pi$
$614$	−17.6183	−0.711017
$615$	0	0
$616$	4.70020	0.189376
$617$	−28.0582	−1.12958	−0.564791	−	0.825234i	$-0.691043\pi$
$617$	−28.0582	−1.12958	−0.564791	+	0.825234i	$0.691043\pi$
$618$	0	0
$619$	15.0922	0.606605	0.303302	−	0.952894i	$-0.401911\pi$
$619$	15.0922	0.606605	0.303302	+	0.952894i	$0.401911\pi$
$620$	−31.1850	−1.25242
$621$	0	0
$622$	−14.7577	−0.591730
$623$	7.55850	0.302825
$624$	0	0
$625$	−17.7970	−0.711879
$626$	−13.4908	−0.539199
$627$	0	0
$628$	25.5963	1.02141
$629$	−26.1524	−1.04277
$630$	0	0
$631$	10.4663	0.416659	0.208329	−	0.978059i	$-0.433197\pi$
$631$	10.4663	0.416659	0.208329	+	0.978059i	$0.433197\pi$
$632$	−4.14600	−0.164919
$633$	0	0
$634$	−31.4151	−1.24765
$635$	35.9967	1.42848
$636$	0	0
$637$	6.97127	0.276212
$638$	53.5508	2.12010
$639$	0	0
$640$	10.1316	0.400486
$641$	−11.5638	−0.456742	−0.228371	−	0.973574i	$-0.573340\pi$
$641$	−11.5638	−0.456742	−0.228371	+	0.973574i	$0.573340\pi$
$642$	0	0
$643$	48.7789	1.92365	0.961826	−	0.273661i	$-0.0882346\pi$
$643$	48.7789	1.92365	0.961826	+	0.273661i	$0.0882346\pi$
$644$	4.07192	0.160456
$645$	0	0
$646$	0	0
$647$	−6.14233	−0.241480	−0.120740	−	0.992684i	$-0.538527\pi$
$647$	−6.14233	−0.241480	−0.120740	+	0.992684i	$0.538527\pi$
$648$	0	0
$649$	−28.3329	−1.11217
$650$	2.87051	0.112591
$651$	0	0
$652$	−2.54906	−0.0998289
$653$	22.2496	0.870695	0.435347	−	0.900263i	$-0.356626\pi$
$653$	22.2496	0.870695	0.435347	+	0.900263i	$0.356626\pi$
$654$	0	0
$655$	−13.3401	−0.521239
$656$	42.1498	1.64567
$657$	0	0
$658$	7.37390	0.287465
$659$	−43.4003	−1.69064	−0.845319	−	0.534263i	$-0.820589\pi$
$659$	−43.4003	−1.69064	−0.845319	+	0.534263i	$0.820589\pi$
$660$	0	0
$661$	−41.0353	−1.59609	−0.798045	−	0.602598i	$-0.794132\pi$
$661$	−41.0353	−1.59609	−0.798045	+	0.602598i	$0.794132\pi$
$662$	36.5608	1.42097
$663$	0	0
$664$	2.01659	0.0782589
$665$	0	0
$666$	0	0
$667$	−9.61614	−0.372338
$668$	−30.6533	−1.18601
$669$	0	0
$670$	−17.1722	−0.663421
$671$	−43.9347	−1.69608
$672$	0	0
$673$	−34.7212	−1.33840	−0.669202	−	0.743081i	$-0.733364\pi$
$673$	−34.7212	−1.33840	−0.669202	+	0.743081i	$0.733364\pi$
$674$	−12.7024	−0.489277
$675$	0	0
$676$	−18.7989	−0.723036
$677$	−5.52927	−0.212507	−0.106254	−	0.994339i	$-0.533886\pi$
$677$	−5.52927	−0.212507	−0.106254	+	0.994339i	$0.533886\pi$
$678$	0	0
$679$	17.3285	0.665007
$680$	−8.98021	−0.344375
$681$	0	0
$682$	104.831	4.01417
$683$	47.5130	1.81803	0.909016	−	0.416760i	$-0.136835\pi$
$683$	47.5130	1.81803	0.909016	+	0.416760i	$0.136835\pi$
$684$	0	0
$685$	−12.8124	−0.489536
$686$	−29.8288	−1.13887
$687$	0	0
$688$	15.5387	0.592408
$689$	8.55638	0.325972
$690$	0	0
$691$	19.8623	0.755598	0.377799	−	0.925888i	$-0.376681\pi$
$691$	19.8623	0.755598	0.377799	+	0.925888i	$0.376681\pi$
$692$	−9.47073	−0.360023
$693$	0	0
$694$	−64.6681	−2.45477
$695$	−28.5586	−1.08329
$696$	0	0
$697$	−64.4523	−2.44131
$698$	28.1126	1.06408
$699$	0	0
$700$	−2.42918	−0.0918144
$701$	21.3027	0.804591	0.402296	−	0.915510i	$-0.368212\pi$
$701$	21.3027	0.804591	0.402296	+	0.915510i	$0.368212\pi$
$702$	0	0
$703$	0	0
$704$	28.8333	1.08670
$705$	0	0
$706$	13.5752	0.510911
$707$	15.9303	0.599120
$708$	0	0
$709$	21.5457	0.809167	0.404584	−	0.914501i	$-0.367416\pi$
$709$	21.5457	0.809167	0.404584	+	0.914501i	$0.367416\pi$
$710$	30.6781	1.15133
$711$	0	0
$712$	−3.95272	−0.148134
$713$	−18.8245	−0.704982
$714$	0	0
$715$	14.3452	0.536479
$716$	−24.6285	−0.920408
$717$	0	0
$718$	−47.0498	−1.75588
$719$	−47.4085	−1.76804	−0.884019	−	0.467451i	$-0.845172\pi$
$719$	−47.4085	−1.76804	−0.884019	+	0.467451i	$0.845172\pi$
$720$	0	0
$721$	3.08625	0.114938
$722$	0	0
$723$	0	0
$724$	−33.9900	−1.26323
$725$	5.73669	0.213055
$726$	0	0
$727$	16.5860	0.615140	0.307570	−	0.951525i	$-0.400484\pi$
$727$	16.5860	0.615140	0.307570	+	0.951525i	$0.400484\pi$
$728$	1.05975	0.0392771
$729$	0	0
$730$	−34.7212	−1.28509
$731$	−23.7606	−0.878819
$732$	0	0
$733$	7.00438	0.258713	0.129356	−	0.991598i	$-0.458709\pi$
$733$	7.00438	0.258713	0.129356	+	0.991598i	$0.458709\pi$
$734$	23.6559	0.873157
$735$	0	0
$736$	−14.5321	−0.535659
$737$	26.1524	0.963337
$738$	0	0
$739$	−7.57149	−0.278522	−0.139261	−	0.990256i	$-0.544473\pi$
$739$	−7.57149	−0.278522	−0.139261	+	0.990256i	$0.544473\pi$
$740$	−12.1397	−0.446263
$741$	0	0
$742$	−15.9826	−0.586741
$743$	−0.0576321	−0.00211432	−0.00105716	−	0.999999i	$-0.500337\pi$
$743$	−0.0576321	−0.00211432	−0.00105716	+	0.999999i	$0.500337\pi$
$744$	0	0
$745$	−27.4683	−1.00636
$746$	13.6881	0.501155
$747$	0	0
$748$	−65.9813	−2.41251
$749$	−16.8546	−0.615854
$750$	0	0
$751$	28.1172	1.02601	0.513006	−	0.858385i	$-0.328532\pi$
$751$	28.1172	1.02601	0.513006	+	0.858385i	$0.328532\pi$
$752$	−14.0317	−0.511683
$753$	0	0
$754$	12.0741	0.439713
$755$	21.9839	0.800077
$756$	0	0
$757$	−0.934664	−0.0339709	−0.0169855	−	0.999856i	$-0.505407\pi$
$757$	−0.934664	−0.0339709	−0.0169855	+	0.999856i	$0.505407\pi$
$758$	−27.1604	−0.986509
$759$	0	0
$760$	0	0
$761$	−42.7470	−1.54958	−0.774789	−	0.632220i	$-0.782144\pi$
$761$	−42.7470	−1.54958	−0.774789	+	0.632220i	$0.782144\pi$
$762$	0	0
$763$	−16.7743	−0.607271
$764$	30.5601	1.10563
$765$	0	0
$766$	−37.1324	−1.34165
$767$	−6.38823	−0.230666
$768$	0	0
$769$	−39.9742	−1.44151	−0.720754	−	0.693191i	$-0.756204\pi$
$769$	−39.9742	−1.44151	−0.720754	+	0.693191i	$0.756204\pi$
$770$	−26.7956	−0.965647
$771$	0	0
$772$	42.1338	1.51643
$773$	23.7763	0.855173	0.427586	−	0.903974i	$-0.359364\pi$
$773$	23.7763	0.855173	0.427586	+	0.903974i	$0.359364\pi$
$774$	0	0
$775$	11.2301	0.403397
$776$	−9.06194	−0.325305
$777$	0	0
$778$	13.9878	0.501486
$779$	0	0
$780$	0	0
$781$	−46.7212	−1.67182
$782$	26.1524	0.935209
$783$	0	0
$784$	24.7789	0.884961
$785$	30.2466	1.07955
$786$	0	0
$787$	16.2201	0.578184	0.289092	−	0.957301i	$-0.406647\pi$
$787$	16.2201	0.578184	0.289092	+	0.957301i	$0.406647\pi$
$788$	−7.50181	−0.267241
$789$	0	0
$790$	23.6362	0.840937
$791$	−18.5284	−0.658793
$792$	0	0
$793$	−9.90595	−0.351771
$794$	−33.7488	−1.19770
$795$	0	0
$796$	14.1826	0.502688
$797$	−31.8038	−1.12655	−0.563273	−	0.826271i	$-0.690458\pi$
$797$	−31.8038	−1.12655	−0.563273	+	0.826271i	$0.690458\pi$
$798$	0	0
$799$	21.4562	0.759067
$800$	8.66938	0.306509
$801$	0	0
$802$	−38.3977	−1.35587
$803$	52.8786	1.86605
$804$	0	0
$805$	4.81170	0.169590
$806$	23.6362	0.832548
$807$	0	0
$808$	−8.33074	−0.293074
$809$	18.4860	0.649933	0.324966	−	0.945726i	$-0.394647\pi$
$809$	18.4860	0.649933	0.324966	+	0.945726i	$0.394647\pi$
$810$	0	0
$811$	−29.4026	−1.03247	−0.516233	−	0.856448i	$-0.672666\pi$
$811$	−29.4026	−1.03247	−0.516233	+	0.856448i	$0.672666\pi$
$812$	−10.2178	−0.358573
$813$	0	0
$814$	40.8084	1.43033
$815$	−3.01217	−0.105512
$816$	0	0
$817$	0	0
$818$	−19.4480	−0.679984
$819$	0	0
$820$	−29.9181	−1.04478
$821$	−8.61538	−0.300679	−0.150339	−	0.988634i	$-0.548037\pi$
$821$	−8.61538	−0.300679	−0.150339	+	0.988634i	$0.548037\pi$
$822$	0	0
$823$	50.9786	1.77700	0.888502	−	0.458874i	$-0.151747\pi$
$823$	50.9786	1.77700	0.888502	+	0.458874i	$0.151747\pi$
$824$	−1.61395	−0.0562247
$825$	0	0
$826$	11.9327	0.415192
$827$	−18.8047	−0.653902	−0.326951	−	0.945041i	$-0.606021\pi$
$827$	−18.8047	−0.653902	−0.326951	+	0.945041i	$0.606021\pi$
$828$	0	0
$829$	−27.0476	−0.939400	−0.469700	−	0.882826i	$-0.655638\pi$
$829$	−27.0476	−0.939400	−0.469700	+	0.882826i	$0.655638\pi$
$830$	−11.4965	−0.399049
$831$	0	0
$832$	6.50105	0.225383
$833$	−37.8901	−1.31281
$834$	0	0
$835$	−36.2223	−1.25352
$836$	0	0
$837$	0	0
$838$	−31.5320	−1.08926
$839$	−34.7591	−1.20002	−0.600009	−	0.799993i	$-0.704836\pi$
$839$	−34.7591	−1.20002	−0.600009	+	0.799993i	$0.704836\pi$
$840$	0	0
$841$	−4.87000	−0.167931
$842$	42.4624	1.46335
$843$	0	0
$844$	−24.2610	−0.835099
$845$	−22.2143	−0.764194
$846$	0	0
$847$	26.9967	0.927617
$848$	30.4131	1.04439
$849$	0	0
$850$	−15.6017	−0.535135
$851$	−7.32797	−0.251200
$852$	0	0
$853$	23.6291	0.809046	0.404523	−	0.914528i	$-0.367438\pi$
$853$	23.6291	0.809046	0.404523	+	0.914528i	$0.367438\pi$
$854$	18.5035	0.633177
$855$	0	0
$856$	8.81411	0.301260
$857$	18.0871	0.617843	0.308922	−	0.951087i	$-0.400032\pi$
$857$	18.0871	0.617843	0.308922	+	0.951087i	$0.400032\pi$
$858$	0	0
$859$	5.56379	0.189834	0.0949171	−	0.995485i	$-0.469741\pi$
$859$	5.56379	0.189834	0.0949171	+	0.995485i	$0.469741\pi$
$860$	−11.0294	−0.376100
$861$	0	0
$862$	59.5508	2.02831
$863$	−35.8865	−1.22159	−0.610795	−	0.791789i	$-0.709150\pi$
$863$	−35.8865	−1.22159	−0.610795	+	0.791789i	$0.709150\pi$
$864$	0	0
$865$	−11.1913	−0.380517
$866$	14.5321	0.493820
$867$	0	0
$868$	−20.0022	−0.678919
$869$	−35.9967	−1.22110
$870$	0	0
$871$	5.89659	0.199798
$872$	8.77212	0.297061
$873$	0	0
$874$	0	0
$875$	−15.1603	−0.512513
$876$	0	0
$877$	8.33074	0.281309	0.140655	−	0.990059i	$-0.455079\pi$
$877$	8.33074	0.281309	0.140655	+	0.990059i	$0.455079\pi$
$878$	−49.4789	−1.66983
$879$	0	0
$880$	50.9890	1.71884
$881$	−0.117999	−0.00397547	−0.00198774	−	0.999998i	$-0.500633\pi$
$881$	−0.117999	−0.00397547	−0.00198774	+	0.999998i	$0.500633\pi$
$882$	0	0
$883$	−34.9140	−1.17495	−0.587474	−	0.809243i	$-0.699878\pi$
$883$	−34.9140	−1.17495	−0.587474	+	0.809243i	$0.699878\pi$
$884$	−14.8768	−0.500361
$885$	0	0
$886$	18.8296	0.632594
$887$	8.54400	0.286879	0.143440	−	0.989659i	$-0.454184\pi$
$887$	8.54400	0.286879	0.143440	+	0.989659i	$0.454184\pi$
$888$	0	0
$889$	23.0885	0.774362
$890$	22.5343	0.755350
$891$	0	0
$892$	38.0619	1.27441
$893$	0	0
$894$	0	0
$895$	−29.1029	−0.972802
$896$	6.49846	0.217098
$897$	0	0
$898$	−76.6128	−2.55660
$899$	47.2367	1.57543
$900$	0	0
$901$	−46.5055	−1.54932
$902$	100.572	3.34867
$903$	0	0
$904$	9.68940	0.322265
$905$	−40.1652	−1.33514
$906$	0	0
$907$	−21.6902	−0.720213	−0.360106	−	0.932911i	$-0.617260\pi$
$907$	−21.6902	−0.720213	−0.360106	+	0.932911i	$0.617260\pi$
$908$	14.4406	0.479228
$909$	0	0
$910$	−6.04161	−0.200277
$911$	26.5361	0.879179	0.439590	−	0.898199i	$-0.355124\pi$
$911$	26.5361	0.879179	0.439590	+	0.898199i	$0.355124\pi$
$912$	0	0
$913$	17.5086	0.579449
$914$	68.9113	2.27938
$915$	0	0
$916$	−29.9390	−0.989214
$917$	−8.55638	−0.282557
$918$	0	0
$919$	−31.4811	−1.03846	−0.519232	−	0.854633i	$-0.673782\pi$
$919$	−31.4811	−1.03846	−0.519232	+	0.854633i	$0.673782\pi$
$920$	−2.51628	−0.0829592
$921$	0	0
$922$	−35.0497	−1.15430
$923$	−10.5342	−0.346738
$924$	0	0
$925$	4.37164	0.143739
$926$	49.6738	1.63238
$927$	0	0
$928$	36.4657	1.19704
$929$	−44.7636	−1.46865	−0.734323	−	0.678800i	$-0.762500\pi$
$929$	−44.7636	−1.46865	−0.734323	+	0.678800i	$0.762500\pi$
$930$	0	0
$931$	0	0
$932$	26.1058	0.855124
$933$	0	0
$934$	−24.4768	−0.800904
$935$	−77.9686	−2.54985
$936$	0	0
$937$	53.9287	1.76177	0.880887	−	0.473327i	$-0.156947\pi$
$937$	53.9287	1.76177	0.880887	+	0.473327i	$0.156947\pi$
$938$	−11.0143	−0.359631
$939$	0	0
$940$	9.95974	0.324851
$941$	−21.0000	−0.684580	−0.342290	−	0.939594i	$-0.611203\pi$
$941$	−21.0000	−0.684580	−0.342290	+	0.939594i	$0.611203\pi$
$942$	0	0
$943$	−18.0597	−0.588105
$944$	−22.7065	−0.739035
$945$	0	0
$946$	37.0762	1.20545
$947$	−45.3178	−1.47263	−0.736315	−	0.676638i	$-0.763436\pi$
$947$	−45.3178	−1.47263	−0.736315	+	0.676638i	$0.763436\pi$
$948$	0	0
$949$	11.9225	0.387022
$950$	0	0
$951$	0	0
$952$	−5.75995	−0.186681
$953$	−17.5163	−0.567408	−0.283704	−	0.958912i	$-0.591563\pi$
$953$	−17.5163	−0.567408	−0.283704	+	0.958912i	$0.591563\pi$
$954$	0	0
$955$	36.1122	1.16856
$956$	44.5479	1.44078
$957$	0	0
$958$	−7.48672	−0.241885
$959$	−8.21792	−0.265371
$960$	0	0
$961$	61.4701	1.98290
$962$	9.20107	0.296654
$963$	0	0
$964$	−44.4501	−1.43164
$965$	49.7886	1.60275
$966$	0	0
$967$	34.5930	1.11244	0.556218	−	0.831036i	$-0.312252\pi$
$967$	34.5930	1.11244	0.556218	+	0.831036i	$0.312252\pi$
$968$	−14.1179	−0.453767
$969$	0	0
$970$	51.6617	1.65876
$971$	10.2127	0.327741	0.163871	−	0.986482i	$-0.447602\pi$
$971$	10.2127	0.327741	0.163871	+	0.986482i	$0.447602\pi$
$972$	0	0
$973$	−18.3176	−0.587236
$974$	21.6902	0.695000
$975$	0	0
$976$	−35.2101	−1.12705
$977$	37.6834	1.20560	0.602800	−	0.797892i	$-0.294052\pi$
$977$	37.6834	1.20560	0.602800	+	0.797892i	$0.294052\pi$
$978$	0	0
$979$	−34.3185	−1.09683
$980$	−17.5882	−0.561833
$981$	0	0
$982$	43.6061	1.39153
$983$	27.8409	0.887988	0.443994	−	0.896030i	$-0.353561\pi$
$983$	27.8409	0.887988	0.443994	+	0.896030i	$0.353561\pi$
$984$	0	0
$985$	−8.86471	−0.282453
$986$	−65.6249	−2.08992
$987$	0	0
$988$	0	0
$989$	−6.65779	−0.211705
$990$	0	0
$991$	10.4989	0.333508	0.166754	−	0.985999i	$-0.446671\pi$
$991$	10.4989	0.333508	0.166754	+	0.985999i	$0.446671\pi$
$992$	71.3849	2.26647
$993$	0	0
$994$	19.6771	0.624119
$995$	16.7592	0.531303
$996$	0	0
$997$	13.3460	0.422673	0.211336	−	0.977413i	$-0.432218\pi$
$997$	13.3460	0.422673	0.211336	+	0.977413i	$0.432218\pi$
$998$	58.8084	1.86155
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9747.2.a.bk.1.6	yes	6
3.2	odd	2		9747.2.a.bt.1.1	yes	6
19.18	odd	2		9747.2.a.bt.1.2	yes	6
57.56	even	2	inner	9747.2.a.bk.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9747.2.a.bk.1.5	✓	6	57.56	even	2	inner
9747.2.a.bk.1.6	yes	6	1.1	even	1	trivial
9747.2.a.bt.1.1	yes	6	3.2	odd	2
9747.2.a.bt.1.2	yes	6	19.18	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:	\( N \)	\(=\)	\( 9747 = 3^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9747.a (trivial)

\(f(q)\)	\(=\)	\(q+1.91223 q^{2} +1.65662 q^{4} +1.95759 q^{5} +1.25561 q^{7} -0.656620 q^{8} +3.74336 q^{10} -5.70095 q^{11} -1.28539 q^{13} +2.40101 q^{14} -4.56885 q^{16} +6.98635 q^{17} +3.24299 q^{20} -10.9015 q^{22} +1.95759 q^{23} -1.16784 q^{25} -2.45797 q^{26} +2.08007 q^{28} -4.91223 q^{29} -9.61614 q^{31} -7.42345 q^{32} +13.3595 q^{34} +2.45797 q^{35} -3.74336 q^{37} -1.28539 q^{40} -9.22547 q^{41} -3.40101 q^{43} -9.44432 q^{44} +3.74336 q^{46} +3.07117 q^{47} -5.42345 q^{49} -2.23317 q^{50} -2.12941 q^{52} -6.65662 q^{53} -11.1601 q^{55} -0.824458 q^{56} -9.39331 q^{58} +4.96986 q^{59} +7.70655 q^{61} -18.3883 q^{62} -5.05763 q^{64} -2.51628 q^{65} -4.58738 q^{67} +11.5737 q^{68} +4.70020 q^{70} +8.19533 q^{71} -9.27540 q^{73} -7.15817 q^{74} -7.15817 q^{77} +6.31415 q^{79} -8.94394 q^{80} -17.6412 q^{82} -3.07117 q^{83} +13.6764 q^{85} -6.50351 q^{86} +3.74336 q^{88} +6.01979 q^{89} -1.61395 q^{91} +3.24299 q^{92} +5.87277 q^{94} +13.8009 q^{97} -10.3709 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + 6 q^{4} - 2 q^{7} + 20 q^{14} - 10 q^{16} + 16 q^{25} - 24 q^{28} - 16 q^{29} - 12 q^{32} - 34 q^{41} - 26 q^{43} - 42 q^{50} - 36 q^{53} + 6 q^{55} + 22 q^{56} - 12 q^{58} + 18 q^{59} - 6 q^{61}+ \cdots - 56 q^{98}+O(q^{100}) \) 6 * q - 2 * q^2 + 6 * q^4 - 2 * q^7 + 20 * q^14 - 10 * q^16 + 16 * q^25 - 24 * q^28 - 16 * q^29 - 12 * q^32 - 34 * q^41 - 26 * q^43 - 42 * q^50 - 36 * q^53 + 6 * q^55 + 22 * q^56 - 12 * q^58 + 18 * q^59 - 6 * q^61 - 32 * q^64 - 24 * q^65 + 16 * q^71 + 14 * q^73 - 2 * q^82 + 18 * q^85 + 30 * q^86 - 24 * q^89 - 56 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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