LMFDB - Embedded newform 3969.2.a.bi.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, 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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 22 x^{10} + 92 x^{9} + 125 x^{8} - 620 x^{7} - 94 x^{6} + 1280 x^{5} - 234 x^{4} + \cdots - 7 $ x^12 - 4x^11 - 22x^10 + 92x^9 + 125x^8 - 620x^7 - 94x^6 + 1280x^5 - 234x^4 - 736x^3 + 96x^2 + 60*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 441)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.311400$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.10281 q^{2} -0.783802 q^{4} +0.105466 q^{5} -3.07001 q^{8} +O(q^{10})$	$q+1.10281 q^{2} -0.783802 q^{4} +0.105466 q^{5} -3.07001 q^{8} +0.116309 q^{10} -3.33731 q^{11} -2.47995 q^{13} -1.81805 q^{16} +1.61319 q^{17} +7.68266 q^{19} -0.0826646 q^{20} -3.68044 q^{22} +1.89719 q^{23} -4.98888 q^{25} -2.73492 q^{26} +9.29042 q^{29} -9.26162 q^{31} +4.13506 q^{32} +1.77905 q^{34} -1.98254 q^{37} +8.47254 q^{38} -0.323782 q^{40} -7.48537 q^{41} +7.54776 q^{43} +2.61579 q^{44} +2.09224 q^{46} -3.19560 q^{47} -5.50180 q^{50} +1.94379 q^{52} +9.97679 q^{53} -0.351974 q^{55} +10.2456 q^{58} +4.45986 q^{59} +5.67100 q^{61} -10.2138 q^{62} +8.19630 q^{64} -0.261550 q^{65} +9.97141 q^{67} -1.26442 q^{68} -3.29042 q^{71} +4.72378 q^{73} -2.18637 q^{74} -6.02169 q^{76} +7.69409 q^{79} -0.191743 q^{80} -8.25496 q^{82} +1.16886 q^{83} +0.170137 q^{85} +8.32378 q^{86} +10.2456 q^{88} +6.02954 q^{89} -1.48702 q^{92} -3.52415 q^{94} +0.810260 q^{95} -3.80255 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8}+O(q^{10}) $ 12 * q + 4 * q^2 + 12 * q^4 + 12 * q^8	$ 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8} + 20 q^{11} + 12 q^{16} + 32 q^{23} + 12 q^{25} + 16 q^{29} + 48 q^{32} + 12 q^{37} + 56 q^{44} - 24 q^{46} - 4 q^{50} + 32 q^{53} + 48 q^{64} + 60 q^{65} + 12 q^{67} + 56 q^{71} + 68 q^{74} - 12 q^{79} - 12 q^{85} + 76 q^{86} + 16 q^{92} + 64 q^{95}+O(q^{100}) $ 12 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 + 20 * q^11 + 12 * q^16 + 32 * q^23 + 12 * q^25 + 16 * q^29 + 48 * q^32 + 12 * q^37 + 56 * q^44 - 24 * q^46 - 4 * q^50 + 32 * q^53 + 48 * q^64 + 60 * q^65 + 12 * q^67 + 56 * q^71 + 68 * q^74 - 12 * q^79 - 12 * q^85 + 76 * q^86 + 16 * q^92 + 64 * q^95

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.10281	0.779807	0.389903	−	0.920856i	$-0.372508\pi$
$2$	1.10281	0.779807	0.389903	+	0.920856i	$0.372508\pi$
$3$	0	0
$3$	0	0
$4$	−0.783802	−0.391901
$5$	0.105466	0.0471659	0.0235829	−	0.999722i	$-0.492493\pi$
$5$	0.105466	0.0471659	0.0235829	+	0.999722i	$0.492493\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−3.07001	−1.08541
$9$	0	0
$10$	0.116309	0.0367803
$11$	−3.33731	−1.00624	−0.503119	−	0.864217i	$-0.667814\pi$
$11$	−3.33731	−1.00624	−0.503119	+	0.864217i	$0.667814\pi$
$12$	0	0
$13$	−2.47995	−0.687814	−0.343907	−	0.939004i	$-0.611750\pi$
$13$	−2.47995	−0.687814	−0.343907	+	0.939004i	$0.611750\pi$
$14$	0	0
$15$	0	0
$16$	−1.81805	−0.454512
$17$	1.61319	0.391255	0.195628	−	0.980678i	$-0.437326\pi$
$17$	1.61319	0.391255	0.195628	+	0.980678i	$0.437326\pi$
$18$	0	0
$19$	7.68266	1.76252	0.881262	−	0.472629i	$-0.156695\pi$
$19$	7.68266	1.76252	0.881262	+	0.472629i	$0.156695\pi$
$20$	−0.0826646	−0.0184844
$21$	0	0
$22$	−3.68044	−0.784672
$23$	1.89719	0.395591	0.197795	−	0.980243i	$-0.436622\pi$
$23$	1.89719	0.395591	0.197795	+	0.980243i	$0.436622\pi$
$24$	0	0
$25$	−4.98888	−0.997775
$26$	−2.73492	−0.536362
$27$	0	0
$28$	0	0
$29$	9.29042	1.72519	0.862594	−	0.505896i	$-0.168838\pi$
$29$	9.29042	1.72519	0.862594	+	0.505896i	$0.168838\pi$
$30$	0	0
$31$	−9.26162	−1.66344	−0.831718	−	0.555199i	$-0.812642\pi$
$31$	−9.26162	−1.66344	−0.831718	+	0.555199i	$0.812642\pi$
$32$	4.13506	0.730982
$33$	0	0
$34$	1.77905	0.305104
$35$	0	0
$36$	0	0
$37$	−1.98254	−0.325927	−0.162963	−	0.986632i	$-0.552105\pi$
$37$	−1.98254	−0.325927	−0.162963	+	0.986632i	$0.552105\pi$
$38$	8.47254	1.37443
$39$	0	0
$40$	−0.323782	−0.0511945
$41$	−7.48537	−1.16902	−0.584509	−	0.811387i	$-0.698713\pi$
$41$	−7.48537	−1.16902	−0.584509	+	0.811387i	$0.698713\pi$
$42$	0	0
$43$	7.54776	1.15102	0.575512	−	0.817794i	$-0.304803\pi$
$43$	7.54776	1.15102	0.575512	+	0.817794i	$0.304803\pi$
$44$	2.61579	0.394346
$45$	0	0
$46$	2.09224	0.308484
$47$	−3.19560	−0.466127	−0.233063	−	0.972462i	$-0.574875\pi$
$47$	−3.19560	−0.466127	−0.233063	+	0.972462i	$0.574875\pi$
$48$	0	0
$49$	0	0
$50$	−5.50180	−0.778072
$51$	0	0
$52$	1.94379	0.269555
$53$	9.97679	1.37042	0.685209	−	0.728347i	$-0.259711\pi$
$53$	9.97679	1.37042	0.685209	+	0.728347i	$0.259711\pi$
$54$	0	0
$55$	−0.351974	−0.0474601
$56$	0	0
$57$	0	0
$58$	10.2456	1.34531
$59$	4.45986	0.580625	0.290312	−	0.956932i	$-0.406241\pi$
$59$	4.45986	0.580625	0.290312	+	0.956932i	$0.406241\pi$
$60$	0	0
$61$	5.67100	0.726097	0.363048	−	0.931770i	$-0.381736\pi$
$61$	5.67100	0.726097	0.363048	+	0.931770i	$0.381736\pi$
$62$	−10.2138	−1.29716
$63$	0	0
$64$	8.19630	1.02454
$65$	−0.261550	−0.0324413
$66$	0	0
$67$	9.97141	1.21820	0.609101	−	0.793093i	$-0.291530\pi$
$67$	9.97141	1.21820	0.609101	+	0.793093i	$0.291530\pi$
$68$	−1.26442	−0.153333
$69$	0	0
$70$	0	0
$71$	−3.29042	−0.390502	−0.195251	−	0.980753i	$-0.562552\pi$
$71$	−3.29042	−0.390502	−0.195251	+	0.980753i	$0.562552\pi$
$72$	0	0
$73$	4.72378	0.552877	0.276438	−	0.961032i	$-0.410846\pi$
$73$	4.72378	0.552877	0.276438	+	0.961032i	$0.410846\pi$
$74$	−2.18637	−0.254160
$75$	0	0
$76$	−6.02169	−0.690735
$77$	0	0
$78$	0	0
$79$	7.69409	0.865653	0.432827	−	0.901477i	$-0.357516\pi$
$79$	7.69409	0.865653	0.432827	+	0.901477i	$0.357516\pi$
$80$	−0.191743	−0.0214375
$81$	0	0
$82$	−8.25496	−0.911608
$83$	1.16886	0.128299	0.0641493	−	0.997940i	$-0.479567\pi$
$83$	1.16886	0.128299	0.0641493	+	0.997940i	$0.479567\pi$
$84$	0	0
$85$	0.170137	0.0184539
$86$	8.32378	0.897576
$87$	0	0
$88$	10.2456	1.09219
$89$	6.02954	0.639130	0.319565	−	0.947564i	$-0.396463\pi$
$89$	6.02954	0.639130	0.319565	+	0.947564i	$0.396463\pi$
$90$	0	0
$91$	0	0
$92$	−1.48702	−0.155032
$93$	0	0
$94$	−3.52415	−0.363489
$95$	0.810260	0.0831309
$96$	0	0
$97$	−3.80255	−0.386090	−0.193045	−	0.981190i	$-0.561836\pi$
$97$	−3.80255	−0.386090	−0.193045	+	0.981190i	$0.561836\pi$
$98$	0	0
$99$	0	0
$100$	3.91029	0.391029
$101$	17.4702	1.73835	0.869177	−	0.494501i	$-0.164649\pi$
$101$	17.4702	1.73835	0.869177	+	0.494501i	$0.164649\pi$
$102$	0	0
$103$	8.73204	0.860394	0.430197	−	0.902735i	$-0.358444\pi$
$103$	8.73204	0.860394	0.430197	+	0.902735i	$0.358444\pi$
$104$	7.61347	0.746563
$105$	0	0
$106$	11.0025	1.06866
$107$	18.1463	1.75427	0.877135	−	0.480244i	$-0.159452\pi$
$107$	18.1463	1.75427	0.877135	+	0.480244i	$0.159452\pi$
$108$	0	0
$109$	−4.22248	−0.404440	−0.202220	−	0.979340i	$-0.564816\pi$
$109$	−4.22248	−0.404440	−0.202220	+	0.979340i	$0.564816\pi$
$110$	−0.388161	−0.0370097
$111$	0	0
$112$	0	0
$113$	2.05648	0.193457	0.0967285	−	0.995311i	$-0.469162\pi$
$113$	2.05648	0.193457	0.0967285	+	0.995311i	$0.469162\pi$
$114$	0	0
$115$	0.200089	0.0186584
$116$	−7.28186	−0.676103
$117$	0	0
$118$	4.91840	0.452775
$119$	0	0
$120$	0	0
$121$	0.137670	0.0125155
$122$	6.25405	0.566215
$123$	0	0
$124$	7.25928	0.651902
$125$	−1.05349	−0.0942268
$126$	0	0
$127$	0.317159	0.0281433	0.0140717	−	0.999901i	$-0.495521\pi$
$127$	0.317159	0.0281433	0.0140717	+	0.999901i	$0.495521\pi$
$128$	0.768871	0.0679592
$129$	0	0
$130$	−0.288441	−0.0252980
$131$	−14.9563	−1.30674	−0.653370	−	0.757039i	$-0.726645\pi$
$131$	−14.9563	−1.30674	−0.653370	+	0.757039i	$0.726645\pi$
$132$	0	0
$133$	0	0
$134$	10.9966	0.949962
$135$	0	0
$136$	−4.95251	−0.424674
$137$	15.2473	1.30267	0.651334	−	0.758791i	$-0.274210\pi$
$137$	15.2473	1.30267	0.651334	+	0.758791i	$0.274210\pi$
$138$	0	0
$139$	−8.11886	−0.688633	−0.344316	−	0.938854i	$-0.611889\pi$
$139$	−8.11886	−0.688633	−0.344316	+	0.938854i	$0.611889\pi$
$140$	0	0
$141$	0	0
$142$	−3.62872	−0.304516
$143$	8.27636	0.692104
$144$	0	0
$145$	0.979825	0.0813700
$146$	5.20945	0.431137
$147$	0	0
$148$	1.55392	0.127731
$149$	11.1486	0.913329	0.456664	−	0.889639i	$-0.349044\pi$
$149$	11.1486	0.913329	0.456664	+	0.889639i	$0.349044\pi$
$150$	0	0
$151$	−11.2735	−0.917425	−0.458713	−	0.888585i	$-0.651689\pi$
$151$	−11.2735	−0.917425	−0.458713	+	0.888585i	$0.651689\pi$
$152$	−23.5859	−1.91307
$153$	0	0
$154$	0	0
$155$	−0.976786	−0.0784574
$156$	0	0
$157$	12.2064	0.974173	0.487087	−	0.873354i	$-0.338060\pi$
$157$	12.2064	0.974173	0.487087	+	0.873354i	$0.338060\pi$
$158$	8.48515	0.675042
$159$	0	0
$160$	0.436109	0.0344774
$161$	0	0
$162$	0	0
$163$	8.96264	0.702008	0.351004	−	0.936374i	$-0.385840\pi$
$163$	8.96264	0.702008	0.351004	+	0.936374i	$0.385840\pi$
$164$	5.86705	0.458139
$165$	0	0
$166$	1.28903	0.100048
$167$	−17.4167	−1.34774	−0.673871	−	0.738849i	$-0.735370\pi$
$167$	−17.4167	−1.34774	−0.673871	+	0.738849i	$0.735370\pi$
$168$	0	0
$169$	−6.84986	−0.526913
$170$	0.187629	0.0143905
$171$	0	0
$172$	−5.91595	−0.451087
$173$	−2.82933	−0.215110	−0.107555	−	0.994199i	$-0.534302\pi$
$173$	−2.82933	−0.215110	−0.107555	+	0.994199i	$0.534302\pi$
$174$	0	0
$175$	0	0
$176$	6.06740	0.457348
$177$	0	0
$178$	6.64946	0.498398
$179$	10.1627	0.759595	0.379798	−	0.925070i	$-0.375994\pi$
$179$	10.1627	0.759595	0.379798	+	0.925070i	$0.375994\pi$
$180$	0	0
$181$	−17.0870	−1.27006	−0.635032	−	0.772486i	$-0.719013\pi$
$181$	−17.0870	−1.27006	−0.635032	+	0.772486i	$0.719013\pi$
$182$	0	0
$183$	0	0
$184$	−5.82439	−0.429380
$185$	−0.209090	−0.0153726
$186$	0	0
$187$	−5.38371	−0.393696
$188$	2.50472	0.182676
$189$	0	0
$190$	0.893566	0.0648261
$191$	22.4000	1.62081	0.810404	−	0.585872i	$-0.199248\pi$
$191$	22.4000	1.62081	0.810404	+	0.585872i	$0.199248\pi$
$192$	0	0
$193$	−0.256786	−0.0184839	−0.00924194	−	0.999957i	$-0.502942\pi$
$193$	−0.256786	−0.0184839	−0.00924194	+	0.999957i	$0.502942\pi$
$194$	−4.19350	−0.301076
$195$	0	0
$196$	0	0
$197$	0.763370	0.0543878	0.0271939	−	0.999630i	$-0.491343\pi$
$197$	0.763370	0.0543878	0.0271939	+	0.999630i	$0.491343\pi$
$198$	0	0
$199$	−5.03121	−0.356653	−0.178327	−	0.983971i	$-0.557068\pi$
$199$	−5.03121	−0.356653	−0.178327	+	0.983971i	$0.557068\pi$
$200$	15.3159	1.08300
$201$	0	0
$202$	19.2664	1.35558
$203$	0	0
$204$	0	0
$205$	−0.789452	−0.0551377
$206$	9.62981	0.670941
$207$	0	0
$208$	4.50867	0.312620
$209$	−25.6395	−1.77352
$210$	0	0
$211$	7.21074	0.496407	0.248204	−	0.968708i	$-0.420160\pi$
$211$	7.21074	0.496407	0.248204	+	0.968708i	$0.420160\pi$
$212$	−7.81983	−0.537068
$213$	0	0
$214$	20.0120	1.36799
$215$	0.796033	0.0542890
$216$	0	0
$217$	0	0
$218$	−4.65661	−0.315385
$219$	0	0
$220$	0.275878	0.0185997
$221$	−4.00062	−0.269111
$222$	0	0
$223$	−11.1821	−0.748810	−0.374405	−	0.927265i	$-0.622153\pi$
$223$	−11.1821	−0.748810	−0.374405	+	0.927265i	$0.622153\pi$
$224$	0	0
$225$	0	0
$226$	2.26791	0.150859
$227$	−23.7706	−1.57771	−0.788857	−	0.614577i	$-0.789327\pi$
$227$	−23.7706	−1.57771	−0.788857	+	0.614577i	$0.789327\pi$
$228$	0	0
$229$	−1.90547	−0.125917	−0.0629586	−	0.998016i	$-0.520054\pi$
$229$	−1.90547	−0.125917	−0.0629586	+	0.998016i	$0.520054\pi$
$230$	0.220661	0.0145499
$231$	0	0
$232$	−28.5217	−1.87254
$233$	−6.54184	−0.428570	−0.214285	−	0.976771i	$-0.568742\pi$
$233$	−6.54184	−0.428570	−0.214285	+	0.976771i	$0.568742\pi$
$234$	0	0
$235$	−0.337028	−0.0219853
$236$	−3.49565	−0.227547
$237$	0	0
$238$	0	0
$239$	21.3469	1.38082	0.690409	−	0.723419i	$-0.257431\pi$
$239$	21.3469	1.38082	0.690409	+	0.723419i	$0.257431\pi$
$240$	0	0
$241$	−20.0662	−1.29258	−0.646288	−	0.763094i	$-0.723679\pi$
$241$	−20.0662	−1.29258	−0.646288	+	0.763094i	$0.723679\pi$
$242$	0.151825	0.00975967
$243$	0	0
$244$	−4.44494	−0.284558
$245$	0	0
$246$	0	0
$247$	−19.0526	−1.21229
$248$	28.4333	1.80552
$249$	0	0
$250$	−1.16180	−0.0734787
$251$	6.81467	0.430138	0.215069	−	0.976599i	$-0.431002\pi$
$251$	6.81467	0.430138	0.215069	+	0.976599i	$0.431002\pi$
$252$	0	0
$253$	−6.33151	−0.398059
$254$	0.349767	0.0219464
$255$	0	0
$256$	−15.5447	−0.971542
$257$	14.3883	0.897518	0.448759	−	0.893653i	$-0.351866\pi$
$257$	14.3883	0.897518	0.448759	+	0.893653i	$0.351866\pi$
$258$	0	0
$259$	0	0
$260$	0.205004	0.0127138
$261$	0	0
$262$	−16.4940	−1.01900
$263$	1.53901	0.0948992	0.0474496	−	0.998874i	$-0.484891\pi$
$263$	1.53901	0.0948992	0.0474496	+	0.998874i	$0.484891\pi$
$264$	0	0
$265$	1.05221	0.0646369
$266$	0	0
$267$	0	0
$268$	−7.81562	−0.477415
$269$	26.2571	1.60092	0.800461	−	0.599385i	$-0.204588\pi$
$269$	26.2571	1.60092	0.800461	+	0.599385i	$0.204588\pi$
$270$	0	0
$271$	17.9335	1.08938	0.544690	−	0.838637i	$-0.316647\pi$
$271$	17.9335	1.08938	0.544690	+	0.838637i	$0.316647\pi$
$272$	−2.93286	−0.177830
$273$	0	0
$274$	16.8150	1.01583
$275$	16.6495	1.00400
$276$	0	0
$277$	−18.8713	−1.13386	−0.566932	−	0.823764i	$-0.691870\pi$
$277$	−18.8713	−1.13386	−0.566932	+	0.823764i	$0.691870\pi$
$278$	−8.95359	−0.537001
$279$	0	0
$280$	0	0
$281$	−4.99157	−0.297772	−0.148886	−	0.988854i	$-0.547569\pi$
$281$	−4.99157	−0.297772	−0.148886	+	0.988854i	$0.547569\pi$
$282$	0	0
$283$	15.3927	0.915000	0.457500	−	0.889210i	$-0.348745\pi$
$283$	15.3927	0.915000	0.457500	+	0.889210i	$0.348745\pi$
$284$	2.57904	0.153038
$285$	0	0
$286$	9.12729	0.539708
$287$	0	0
$288$	0	0
$289$	−14.3976	−0.846919
$290$	1.08056	0.0634529
$291$	0	0
$292$	−3.70251	−0.216673
$293$	25.8025	1.50740	0.753700	−	0.657219i	$-0.228267\pi$
$293$	25.8025	1.50740	0.753700	+	0.657219i	$0.228267\pi$
$294$	0	0
$295$	0.470364	0.0273857
$296$	6.08642	0.353766
$297$	0	0
$298$	12.2948	0.712220
$299$	−4.70492	−0.272093
$300$	0	0
$301$	0	0
$302$	−12.4326	−0.715415
$303$	0	0
$304$	−13.9675	−0.801089
$305$	0.598098	0.0342470
$306$	0	0
$307$	22.2914	1.27224	0.636120	−	0.771590i	$-0.280538\pi$
$307$	22.2914	1.27224	0.636120	+	0.771590i	$0.280538\pi$
$308$	0	0
$309$	0	0
$310$	−1.07721	−0.0611816
$311$	−1.30986	−0.0742755	−0.0371377	−	0.999310i	$-0.511824\pi$
$311$	−1.30986	−0.0742755	−0.0371377	+	0.999310i	$0.511824\pi$
$312$	0	0
$313$	−21.5770	−1.21960	−0.609802	−	0.792554i	$-0.708751\pi$
$313$	−21.5770	−1.21960	−0.609802	+	0.792554i	$0.708751\pi$
$314$	13.4613	0.759667
$315$	0	0
$316$	−6.03065	−0.339250
$317$	24.7819	1.39189	0.695946	−	0.718094i	$-0.254985\pi$
$317$	24.7819	1.39189	0.695946	+	0.718094i	$0.254985\pi$
$318$	0	0
$319$	−31.0051	−1.73595
$320$	0.864432	0.0483232
$321$	0	0
$322$	0	0
$323$	12.3936	0.689597
$324$	0	0
$325$	12.3722	0.686283
$326$	9.88412	0.547431
$327$	0	0
$328$	22.9802	1.26887
$329$	0	0
$330$	0	0
$331$	13.8451	0.760996	0.380498	−	0.924782i	$-0.375753\pi$
$331$	13.8451	0.760996	0.380498	+	0.924782i	$0.375753\pi$
$332$	−0.916151	−0.0502803
$333$	0	0
$334$	−19.2073	−1.05098
$335$	1.05165	0.0574576
$336$	0	0
$337$	−3.38887	−0.184604	−0.0923018	−	0.995731i	$-0.529422\pi$
$337$	−3.38887	−0.184604	−0.0923018	+	0.995731i	$0.529422\pi$
$338$	−7.55412	−0.410890
$339$	0	0
$340$	−0.133353	−0.00723210
$341$	30.9089	1.67381
$342$	0	0
$343$	0	0
$344$	−23.1717	−1.24934
$345$	0	0
$346$	−3.12022	−0.167744
$347$	14.5148	0.779195	0.389597	−	0.920985i	$-0.372614\pi$
$347$	14.5148	0.779195	0.389597	+	0.920985i	$0.372614\pi$
$348$	0	0
$349$	15.7282	0.841914	0.420957	−	0.907081i	$-0.361694\pi$
$349$	15.7282	0.841914	0.420957	+	0.907081i	$0.361694\pi$
$350$	0	0
$351$	0	0
$352$	−13.8000	−0.735542
$353$	−4.14423	−0.220575	−0.110287	−	0.993900i	$-0.535177\pi$
$353$	−4.14423	−0.220575	−0.110287	+	0.993900i	$0.535177\pi$
$354$	0	0
$355$	−0.347028	−0.0184183
$356$	−4.72597	−0.250476
$357$	0	0
$358$	11.2076	0.592338
$359$	−7.93988	−0.419051	−0.209525	−	0.977803i	$-0.567192\pi$
$359$	−7.93988	−0.419051	−0.209525	+	0.977803i	$0.567192\pi$
$360$	0	0
$361$	40.0233	2.10649
$362$	−18.8437	−0.990405
$363$	0	0
$364$	0	0
$365$	0.498199	0.0260769
$366$	0	0
$367$	13.1491	0.686377	0.343189	−	0.939266i	$-0.388493\pi$
$367$	13.1491	0.686377	0.343189	+	0.939266i	$0.388493\pi$
$368$	−3.44918	−0.179801
$369$	0	0
$370$	−0.230588	−0.0119877
$371$	0	0
$372$	0	0
$373$	7.81086	0.404431	0.202216	−	0.979341i	$-0.435186\pi$
$373$	7.81086	0.404431	0.202216	+	0.979341i	$0.435186\pi$
$374$	−5.93723	−0.307007
$375$	0	0
$376$	9.81055	0.505940
$377$	−23.0398	−1.18661
$378$	0	0
$379$	−31.6147	−1.62394	−0.811968	−	0.583702i	$-0.801604\pi$
$379$	−31.6147	−1.62394	−0.811968	+	0.583702i	$0.801604\pi$
$380$	−0.635084	−0.0325791
$381$	0	0
$382$	24.7030	1.26392
$383$	−10.7319	−0.548373	−0.274186	−	0.961677i	$-0.588408\pi$
$383$	−10.7319	−0.548373	−0.274186	+	0.961677i	$0.588408\pi$
$384$	0	0
$385$	0	0
$386$	−0.283187	−0.0144139
$387$	0	0
$388$	2.98045	0.151309
$389$	−24.1468	−1.22429	−0.612147	−	0.790744i	$-0.709694\pi$
$389$	−24.1468	−1.22429	−0.612147	+	0.790744i	$0.709694\pi$
$390$	0	0
$391$	3.06052	0.154777
$392$	0	0
$393$	0	0
$394$	0.841854	0.0424120
$395$	0.811466	0.0408293
$396$	0	0
$397$	−24.0569	−1.20738	−0.603691	−	0.797218i	$-0.706304\pi$
$397$	−24.0569	−1.20738	−0.603691	+	0.797218i	$0.706304\pi$
$398$	−5.54849	−0.278121
$399$	0	0
$400$	9.07003	0.453501
$401$	1.56232	0.0780183	0.0390092	−	0.999239i	$-0.487580\pi$
$401$	1.56232	0.0780183	0.0390092	+	0.999239i	$0.487580\pi$
$402$	0	0
$403$	22.9683	1.14413
$404$	−13.6932	−0.681263
$405$	0	0
$406$	0	0
$407$	6.61635	0.327960
$408$	0	0
$409$	22.3456	1.10492	0.552460	−	0.833539i	$-0.313689\pi$
$409$	22.3456	1.10492	0.552460	+	0.833539i	$0.313689\pi$
$410$	−0.870619	−0.0429968
$411$	0	0
$412$	−6.84419	−0.337189
$413$	0	0
$414$	0	0
$415$	0.123275	0.00605131
$416$	−10.2547	−0.502779
$417$	0	0
$418$	−28.2755	−1.38300
$419$	−5.97295	−0.291798	−0.145899	−	0.989299i	$-0.546607\pi$
$419$	−5.97295	−0.291798	−0.145899	+	0.989299i	$0.546607\pi$
$420$	0	0
$421$	−14.6319	−0.713114	−0.356557	−	0.934273i	$-0.616050\pi$
$421$	−14.6319	−0.713114	−0.356557	+	0.934273i	$0.616050\pi$
$422$	7.95210	0.387102
$423$	0	0
$424$	−30.6289	−1.48747
$425$	−8.04799	−0.390385
$426$	0	0
$427$	0	0
$428$	−14.2231	−0.687500
$429$	0	0
$430$	0.877876	0.0423349
$431$	19.4034	0.934628	0.467314	−	0.884091i	$-0.345222\pi$
$431$	19.4034	0.934628	0.467314	+	0.884091i	$0.345222\pi$
$432$	0	0
$433$	−1.35217	−0.0649810	−0.0324905	−	0.999472i	$-0.510344\pi$
$433$	−1.35217	−0.0649810	−0.0324905	+	0.999472i	$0.510344\pi$
$434$	0	0
$435$	0	0
$436$	3.30959	0.158501
$437$	14.5754	0.697238
$438$	0	0
$439$	−17.3412	−0.827650	−0.413825	−	0.910356i	$-0.635807\pi$
$439$	−17.3412	−0.827650	−0.413825	+	0.910356i	$0.635807\pi$
$440$	1.08056	0.0515139
$441$	0	0
$442$	−4.41194	−0.209854
$443$	−19.6100	−0.931698	−0.465849	−	0.884864i	$-0.654251\pi$
$443$	−19.6100	−0.931698	−0.465849	+	0.884864i	$0.654251\pi$
$444$	0	0
$445$	0.635912	0.0301451
$446$	−12.3318	−0.583927
$447$	0	0
$448$	0	0
$449$	17.7345	0.836942	0.418471	−	0.908230i	$-0.362566\pi$
$449$	17.7345	0.836942	0.418471	+	0.908230i	$0.362566\pi$
$450$	0	0
$451$	24.9810	1.17631
$452$	−1.61187	−0.0758160
$453$	0	0
$454$	−26.2146	−1.23031
$455$	0	0
$456$	0	0
$457$	0.485451	0.0227084	0.0113542	−	0.999936i	$-0.496386\pi$
$457$	0.485451	0.0227084	0.0113542	+	0.999936i	$0.496386\pi$
$458$	−2.10138	−0.0981912
$459$	0	0
$460$	−0.156830	−0.00731224
$461$	−7.99375	−0.372306	−0.186153	−	0.982521i	$-0.559602\pi$
$461$	−7.99375	−0.372306	−0.186153	+	0.982521i	$0.559602\pi$
$462$	0	0
$463$	−10.4856	−0.487308	−0.243654	−	0.969862i	$-0.578346\pi$
$463$	−10.4856	−0.487308	−0.243654	+	0.969862i	$0.578346\pi$
$464$	−16.8905	−0.784120
$465$	0	0
$466$	−7.21443	−0.334202
$467$	21.8977	1.01331	0.506653	−	0.862150i	$-0.330883\pi$
$467$	21.8977	1.01331	0.506653	+	0.862150i	$0.330883\pi$
$468$	0	0
$469$	0	0
$470$	−0.371679	−0.0171443
$471$	0	0
$472$	−13.6918	−0.630218
$473$	−25.1893	−1.15820
$474$	0	0
$475$	−38.3278	−1.75860
$476$	0	0
$477$	0	0
$478$	23.5417	1.07677
$479$	4.00169	0.182842	0.0914210	−	0.995812i	$-0.470859\pi$
$479$	4.00169	0.182842	0.0914210	+	0.995812i	$0.470859\pi$
$480$	0	0
$481$	4.91658	0.224177
$482$	−22.1292	−1.00796
$483$	0	0
$484$	−0.107906	−0.00490483
$485$	−0.401040	−0.0182103
$486$	0	0
$487$	−26.4755	−1.19972	−0.599859	−	0.800106i	$-0.704777\pi$
$487$	−26.4755	−1.19972	−0.599859	+	0.800106i	$0.704777\pi$
$488$	−17.4100	−0.788116
$489$	0	0
$490$	0	0
$491$	28.4299	1.28302	0.641511	−	0.767114i	$-0.278308\pi$
$491$	28.4299	1.28302	0.641511	+	0.767114i	$0.278308\pi$
$492$	0	0
$493$	14.9872	0.674989
$494$	−21.0115	−0.945350
$495$	0	0
$496$	16.8381	0.756052
$497$	0	0
$498$	0	0
$499$	−7.43118	−0.332665	−0.166333	−	0.986070i	$-0.553193\pi$
$499$	−7.43118	−0.332665	−0.166333	+	0.986070i	$0.553193\pi$
$500$	0.825726	0.0369276
$501$	0	0
$502$	7.51531	0.335425
$503$	−10.1610	−0.453057	−0.226529	−	0.974004i	$-0.572738\pi$
$503$	−10.1610	−0.453057	−0.226529	+	0.974004i	$0.572738\pi$
$504$	0	0
$505$	1.84252	0.0819910
$506$	−6.98247	−0.310409
$507$	0	0
$508$	−0.248590	−0.0110294
$509$	28.9063	1.28125	0.640625	−	0.767854i	$-0.278675\pi$
$509$	28.9063	1.28125	0.640625	+	0.767854i	$0.278675\pi$
$510$	0	0
$511$	0	0
$512$	−18.6806	−0.825575
$513$	0	0
$514$	15.8676	0.699891
$515$	0.920934	0.0405812
$516$	0	0
$517$	10.6647	0.469034
$518$	0	0
$519$	0	0
$520$	0.802963	0.0352123
$521$	−33.7990	−1.48076	−0.740381	−	0.672187i	$-0.765355\pi$
$521$	−33.7990	−1.48076	−0.740381	+	0.672187i	$0.765355\pi$
$522$	0	0
$523$	−14.3779	−0.628701	−0.314351	−	0.949307i	$-0.601787\pi$
$523$	−14.3779	−0.628701	−0.314351	+	0.949307i	$0.601787\pi$
$524$	11.7228	0.512113
$525$	0	0
$526$	1.69724	0.0740030
$527$	−14.9407	−0.650828
$528$	0	0
$529$	−19.4007	−0.843508
$530$	1.16039	0.0504043
$531$	0	0
$532$	0	0
$533$	18.5633	0.804066
$534$	0	0
$535$	1.91382	0.0827417
$536$	−30.6124	−1.32225
$537$	0	0
$538$	28.9567	1.24841
$539$	0	0
$540$	0	0
$541$	−25.1764	−1.08242	−0.541210	−	0.840888i	$-0.682034\pi$
$541$	−25.1764	−1.08242	−0.541210	+	0.840888i	$0.682034\pi$
$542$	19.7773	0.849506
$543$	0	0
$544$	6.67063	0.286001
$545$	−0.445329	−0.0190758
$546$	0	0
$547$	−3.18023	−0.135977	−0.0679883	−	0.997686i	$-0.521658\pi$
$547$	−3.18023	−0.135977	−0.0679883	+	0.997686i	$0.521658\pi$
$548$	−11.9509	−0.510517
$549$	0	0
$550$	18.3612	0.782926
$551$	71.3752	3.04068
$552$	0	0
$553$	0	0
$554$	−20.8115	−0.884195
$555$	0	0
$556$	6.36358	0.269876
$557$	−20.0459	−0.849371	−0.424686	−	0.905341i	$-0.639615\pi$
$557$	−20.0459	−0.849371	−0.424686	+	0.905341i	$0.639615\pi$
$558$	0	0
$559$	−18.7181	−0.791689
$560$	0	0
$561$	0	0
$562$	−5.50477	−0.232205
$563$	−39.8013	−1.67743	−0.838713	−	0.544574i	$-0.816691\pi$
$563$	−39.8013	−1.67743	−0.838713	+	0.544574i	$0.816691\pi$
$564$	0	0
$565$	0.216888	0.00912457
$566$	16.9753	0.713523
$567$	0	0
$568$	10.1017	0.423856
$569$	−13.8159	−0.579194	−0.289597	−	0.957149i	$-0.593521\pi$
$569$	−13.8159	−0.579194	−0.289597	+	0.957149i	$0.593521\pi$
$570$	0	0
$571$	10.4387	0.436846	0.218423	−	0.975854i	$-0.429909\pi$
$571$	10.4387	0.436846	0.218423	+	0.975854i	$0.429909\pi$
$572$	−6.48703	−0.271236
$573$	0	0
$574$	0	0
$575$	−9.46483	−0.394711
$576$	0	0
$577$	−25.4923	−1.06126	−0.530628	−	0.847605i	$-0.678044\pi$
$577$	−25.4923	−1.06126	−0.530628	+	0.847605i	$0.678044\pi$
$578$	−15.8779	−0.660433
$579$	0	0
$580$	−0.767989	−0.0318890
$581$	0	0
$582$	0	0
$583$	−33.2957	−1.37897
$584$	−14.5021	−0.600100
$585$	0	0
$586$	28.4554	1.17548
$587$	35.0337	1.44600	0.722998	−	0.690850i	$-0.242764\pi$
$587$	35.0337	1.44600	0.722998	+	0.690850i	$0.242764\pi$
$588$	0	0
$589$	−71.1539	−2.93184
$590$	0.518724	0.0213555
$591$	0	0
$592$	3.60435	0.148138
$593$	−36.1292	−1.48365	−0.741824	−	0.670594i	$-0.766039\pi$
$593$	−36.1292	−1.48365	−0.741824	+	0.670594i	$0.766039\pi$
$594$	0	0
$595$	0	0
$596$	−8.73830	−0.357935
$597$	0	0
$598$	−5.18865	−0.212180
$599$	40.9484	1.67310	0.836552	−	0.547887i	$-0.184568\pi$
$599$	40.9484	1.67310	0.836552	+	0.547887i	$0.184568\pi$
$600$	0	0
$601$	25.7094	1.04871	0.524354	−	0.851501i	$-0.324307\pi$
$601$	25.7094	1.04871	0.524354	+	0.851501i	$0.324307\pi$
$602$	0	0
$603$	0	0
$604$	8.83620	0.359540
$605$	0.0145196	0.000590304 0
$606$	0	0
$607$	−6.84516	−0.277836	−0.138918	−	0.990304i	$-0.544362\pi$
$607$	−6.84516	−0.277836	−0.138918	+	0.990304i	$0.544362\pi$
$608$	31.7683	1.28837
$609$	0	0
$610$	0.659590	0.0267060
$611$	7.92493	0.320608
$612$	0	0
$613$	−29.1297	−1.17654	−0.588269	−	0.808666i	$-0.700190\pi$
$613$	−29.1297	−1.17654	−0.588269	+	0.808666i	$0.700190\pi$
$614$	24.5833	0.992101
$615$	0	0
$616$	0	0
$617$	−20.6789	−0.832503	−0.416252	−	0.909250i	$-0.636656\pi$
$617$	−20.6789	−0.832503	−0.416252	+	0.909250i	$0.636656\pi$
$618$	0	0
$619$	8.86355	0.356256	0.178128	−	0.984007i	$-0.442996\pi$
$619$	8.86355	0.356256	0.178128	+	0.984007i	$0.442996\pi$
$620$	0.765607	0.0307475
$621$	0	0
$622$	−1.44453	−0.0579205
$623$	0	0
$624$	0	0
$625$	24.8333	0.993331
$626$	−23.7954	−0.951055
$627$	0	0
$628$	−9.56737	−0.381779
$629$	−3.19820	−0.127521
$630$	0	0
$631$	26.4661	1.05360	0.526799	−	0.849990i	$-0.323392\pi$
$631$	26.4661	1.05360	0.526799	+	0.849990i	$0.323392\pi$
$632$	−23.6210	−0.939592
$633$	0	0
$634$	27.3299	1.08541
$635$	0.0334495	0.00132740
$636$	0	0
$637$	0	0
$638$	−34.1928	−1.35371
$639$	0	0
$640$	0.0810898	0.00320536
$641$	16.5319	0.652971	0.326486	−	0.945202i	$-0.394136\pi$
$641$	16.5319	0.652971	0.326486	+	0.945202i	$0.394136\pi$
$642$	0	0
$643$	−30.8920	−1.21826	−0.609130	−	0.793070i	$-0.708481\pi$
$643$	−30.8920	−1.21826	−0.609130	+	0.793070i	$0.708481\pi$
$644$	0	0
$645$	0	0
$646$	13.6678	0.537752
$647$	−1.29981	−0.0511007	−0.0255503	−	0.999674i	$-0.508134\pi$
$647$	−1.29981	−0.0511007	−0.0255503	+	0.999674i	$0.508134\pi$
$648$	0	0
$649$	−14.8840	−0.584247
$650$	13.6442	0.535169
$651$	0	0
$652$	−7.02494	−0.275118
$653$	44.8870	1.75656	0.878281	−	0.478144i	$-0.158690\pi$
$653$	44.8870	1.75656	0.878281	+	0.478144i	$0.158690\pi$
$654$	0	0
$655$	−1.57738	−0.0616335
$656$	13.6088	0.531333
$657$	0	0
$658$	0	0
$659$	17.9233	0.698195	0.349097	−	0.937086i	$-0.386488\pi$
$659$	17.9233	0.698195	0.349097	+	0.937086i	$0.386488\pi$
$660$	0	0
$661$	−33.0256	−1.28455	−0.642274	−	0.766475i	$-0.722009\pi$
$661$	−33.0256	−1.28455	−0.642274	+	0.766475i	$0.722009\pi$
$662$	15.2686	0.593430
$663$	0	0
$664$	−3.58840	−0.139257
$665$	0	0
$666$	0	0
$667$	17.6257	0.682469
$668$	13.6512	0.528182
$669$	0	0
$670$	1.15977	0.0448058
$671$	−18.9259	−0.730626
$672$	0	0
$673$	21.3515	0.823040	0.411520	−	0.911401i	$-0.364998\pi$
$673$	21.3515	0.823040	0.411520	+	0.911401i	$0.364998\pi$
$674$	−3.73729	−0.143955
$675$	0	0
$676$	5.36894	0.206498
$677$	8.30167	0.319059	0.159530	−	0.987193i	$-0.449002\pi$
$677$	8.30167	0.319059	0.159530	+	0.987193i	$0.449002\pi$
$678$	0	0
$679$	0	0
$680$	−0.522322	−0.0200301
$681$	0	0
$682$	34.0868	1.30525
$683$	−2.49456	−0.0954518	−0.0477259	−	0.998860i	$-0.515197\pi$
$683$	−2.49456	−0.0954518	−0.0477259	+	0.998860i	$0.515197\pi$
$684$	0	0
$685$	1.60808	0.0614414
$686$	0	0
$687$	0	0
$688$	−13.7222	−0.523154
$689$	−24.7419	−0.942591
$690$	0	0
$691$	16.8691	0.641731	0.320865	−	0.947125i	$-0.396026\pi$
$691$	16.8691	0.641731	0.320865	+	0.947125i	$0.396026\pi$
$692$	2.21763	0.0843017
$693$	0	0
$694$	16.0071	0.607621
$695$	−0.856265	−0.0324800
$696$	0	0
$697$	−12.0753	−0.457385
$698$	17.3453	0.656530
$699$	0	0
$700$	0	0
$701$	−16.4806	−0.622465	−0.311232	−	0.950334i	$-0.600742\pi$
$701$	−16.4806	−0.622465	−0.311232	+	0.950334i	$0.600742\pi$
$702$	0	0
$703$	−15.2312	−0.574454
$704$	−27.3536	−1.03093
$705$	0	0
$706$	−4.57031	−0.172006
$707$	0	0
$708$	0	0
$709$	−29.4925	−1.10761	−0.553807	−	0.832645i	$-0.686825\pi$
$709$	−29.4925	−1.10761	−0.553807	+	0.832645i	$0.686825\pi$
$710$	−0.382707	−0.0143628
$711$	0	0
$712$	−18.5108	−0.693721
$713$	−17.5710	−0.658040
$714$	0	0
$715$	0.872876	0.0326437
$716$	−7.96554	−0.297686
$717$	0	0
$718$	−8.75620	−0.326779
$719$	−0.434622	−0.0162087	−0.00810433	−	0.999967i	$-0.502580\pi$
$719$	−0.434622	−0.0162087	−0.00810433	+	0.999967i	$0.502580\pi$
$720$	0	0
$721$	0	0
$722$	44.1382	1.64265
$723$	0	0
$724$	13.3928	0.497740
$725$	−46.3488	−1.72135
$726$	0	0
$727$	27.1678	1.00760	0.503799	−	0.863821i	$-0.331935\pi$
$727$	27.1678	1.00760	0.503799	+	0.863821i	$0.331935\pi$
$728$	0	0
$729$	0	0
$730$	0.549420	0.0203350
$731$	12.1760	0.450344
$732$	0	0
$733$	5.66614	0.209284	0.104642	−	0.994510i	$-0.466630\pi$
$733$	5.66614	0.209284	0.104642	+	0.994510i	$0.466630\pi$
$734$	14.5010	0.535242
$735$	0	0
$736$	7.84498	0.289170
$737$	−33.2777	−1.22580
$738$	0	0
$739$	−13.6108	−0.500681	−0.250341	−	0.968158i	$-0.580543\pi$
$739$	−13.6108	−0.500681	−0.250341	+	0.968158i	$0.580543\pi$
$740$	0.163885	0.00602455
$741$	0	0
$742$	0	0
$743$	−12.6684	−0.464759	−0.232380	−	0.972625i	$-0.574651\pi$
$743$	−12.6684	−0.464759	−0.232380	+	0.972625i	$0.574651\pi$
$744$	0	0
$745$	1.17580	0.0430779
$746$	8.61393	0.315378
$747$	0	0
$748$	4.21977	0.154290
$749$	0	0
$750$	0	0
$751$	−7.14538	−0.260739	−0.130369	−	0.991465i	$-0.541616\pi$
$751$	−7.14538	−0.260739	−0.130369	+	0.991465i	$0.541616\pi$
$752$	5.80977	0.211860
$753$	0	0
$754$	−25.4086	−0.925325
$755$	−1.18897	−0.0432712
$756$	0	0
$757$	37.6446	1.36822	0.684108	−	0.729381i	$-0.260192\pi$
$757$	37.6446	1.36822	0.684108	+	0.729381i	$0.260192\pi$
$758$	−34.8651	−1.26636
$759$	0	0
$760$	−2.48751	−0.0902315
$761$	10.0472	0.364209	0.182104	−	0.983279i	$-0.441709\pi$
$761$	10.0472	0.364209	0.182104	+	0.983279i	$0.441709\pi$
$762$	0	0
$763$	0	0
$764$	−17.5572	−0.635196
$765$	0	0
$766$	−11.8352	−0.427625
$767$	−11.0602	−0.399361
$768$	0	0
$769$	32.2927	1.16450	0.582252	−	0.813008i	$-0.302172\pi$
$769$	32.2927	1.16450	0.582252	+	0.813008i	$0.302172\pi$
$770$	0	0
$771$	0	0
$772$	0.201270	0.00724385
$773$	−48.5878	−1.74758	−0.873792	−	0.486300i	$-0.838346\pi$
$773$	−48.5878	−1.74758	−0.873792	+	0.486300i	$0.838346\pi$
$774$	0	0
$775$	46.2051	1.65973
$776$	11.6739	0.419068
$777$	0	0
$778$	−26.6295	−0.954713
$779$	−57.5075	−2.06042
$780$	0	0
$781$	10.9812	0.392938
$782$	3.37518	0.120696
$783$	0	0
$784$	0	0
$785$	1.28736	0.0459477
$786$	0	0
$787$	−48.9551	−1.74506	−0.872531	−	0.488560i	$-0.837522\pi$
$787$	−48.9551	−1.74506	−0.872531	+	0.488560i	$0.837522\pi$
$788$	−0.598331	−0.0213147
$789$	0	0
$790$	0.894896	0.0318390
$791$	0	0
$792$	0	0
$793$	−14.0638	−0.499419
$794$	−26.5303	−0.941525
$795$	0	0
$796$	3.94348	0.139773
$797$	−2.88833	−0.102310	−0.0511550	−	0.998691i	$-0.516290\pi$
$797$	−2.88833	−0.102310	−0.0511550	+	0.998691i	$0.516290\pi$
$798$	0	0
$799$	−5.15511	−0.182375
$800$	−20.6293	−0.729356
$801$	0	0
$802$	1.72294	0.0608393
$803$	−15.7647	−0.556326
$804$	0	0
$805$	0	0
$806$	25.3298	0.892203
$807$	0	0
$808$	−53.6339	−1.88683
$809$	−11.6974	−0.411258	−0.205629	−	0.978630i	$-0.565924\pi$
$809$	−11.6974	−0.411258	−0.205629	+	0.978630i	$0.565924\pi$
$810$	0	0
$811$	17.1780	0.603199	0.301600	−	0.953435i	$-0.402479\pi$
$811$	17.1780	0.603199	0.301600	+	0.953435i	$0.402479\pi$
$812$	0	0
$813$	0	0
$814$	7.29660	0.255746
$815$	0.945254	0.0331108
$816$	0	0
$817$	57.9869	2.02870
$818$	24.6431	0.861624
$819$	0	0
$820$	0.618775	0.0216085
$821$	−34.0137	−1.18709	−0.593543	−	0.804803i	$-0.702271\pi$
$821$	−34.0137	−1.18709	−0.593543	+	0.804803i	$0.702271\pi$
$822$	0	0
$823$	43.3780	1.51206	0.756031	−	0.654536i	$-0.227136\pi$
$823$	43.3780	1.51206	0.756031	+	0.654536i	$0.227136\pi$
$824$	−26.8075	−0.933883
$825$	0	0
$826$	0	0
$827$	34.0909	1.18546	0.592728	−	0.805403i	$-0.298051\pi$
$827$	34.0909	1.18546	0.592728	+	0.805403i	$0.298051\pi$
$828$	0	0
$829$	16.9167	0.587540	0.293770	−	0.955876i	$-0.405090\pi$
$829$	16.9167	0.587540	0.293770	+	0.955876i	$0.405090\pi$
$830$	0.135949	0.00471885
$831$	0	0
$832$	−20.3264	−0.704691
$833$	0	0
$834$	0	0
$835$	−1.83687	−0.0635674
$836$	20.0963	0.695044
$837$	0	0
$838$	−6.58705	−0.227546
$839$	16.3249	0.563597	0.281799	−	0.959474i	$-0.409069\pi$
$839$	16.3249	0.563597	0.281799	+	0.959474i	$0.409069\pi$
$840$	0	0
$841$	57.3120	1.97628
$842$	−16.1362	−0.556092
$843$	0	0
$844$	−5.65179	−0.194543
$845$	−0.722428	−0.0248523
$846$	0	0
$847$	0	0
$848$	−18.1383	−0.622872
$849$	0	0
$850$	−8.87544	−0.304425
$851$	−3.76124	−0.128934
$852$	0	0
$853$	−28.9048	−0.989681	−0.494841	−	0.868984i	$-0.664774\pi$
$853$	−28.9048	−0.989681	−0.494841	+	0.868984i	$0.664774\pi$
$854$	0	0
$855$	0	0
$856$	−55.7095	−1.90411
$857$	−29.0567	−0.992559	−0.496280	−	0.868163i	$-0.665301\pi$
$857$	−29.0567	−0.992559	−0.496280	+	0.868163i	$0.665301\pi$
$858$	0	0
$859$	12.5964	0.429783	0.214892	−	0.976638i	$-0.431060\pi$
$859$	12.5964	0.429783	0.214892	+	0.976638i	$0.431060\pi$
$860$	−0.623932	−0.0212759
$861$	0	0
$862$	21.3983	0.728829
$863$	14.6662	0.499243	0.249621	−	0.968344i	$-0.419694\pi$
$863$	14.6662	0.499243	0.249621	+	0.968344i	$0.419694\pi$
$864$	0	0
$865$	−0.298398	−0.0101458
$866$	−1.49119	−0.0506726
$867$	0	0
$868$	0	0
$869$	−25.6776	−0.871053
$870$	0	0
$871$	−24.7286	−0.837896
$872$	12.9631	0.438985
$873$	0	0
$874$	16.0740	0.543711
$875$	0	0
$876$	0	0
$877$	33.1902	1.12075	0.560376	−	0.828238i	$-0.310657\pi$
$877$	33.1902	1.12075	0.560376	+	0.828238i	$0.310657\pi$
$878$	−19.1241	−0.645407
$879$	0	0
$880$	0.639905	0.0215712
$881$	−31.7179	−1.06860	−0.534301	−	0.845294i	$-0.679425\pi$
$881$	−31.7179	−1.06860	−0.534301	+	0.845294i	$0.679425\pi$
$882$	0	0
$883$	−39.5231	−1.33006	−0.665029	−	0.746818i	$-0.731581\pi$
$883$	−39.5231	−1.33006	−0.665029	+	0.746818i	$0.731581\pi$
$884$	3.13569	0.105465
$885$	0	0
$886$	−21.6262	−0.726545
$887$	49.9026	1.67556	0.837782	−	0.546005i	$-0.183852\pi$
$887$	49.9026	1.67556	0.837782	+	0.546005i	$0.183852\pi$
$888$	0	0
$889$	0	0
$890$	0.701292	0.0235074
$891$	0	0
$892$	8.76457	0.293459
$893$	−24.5507	−0.821559
$894$	0	0
$895$	1.07182	0.0358270
$896$	0	0
$897$	0	0
$898$	19.5578	0.652653
$899$	−86.0443	−2.86974
$900$	0	0
$901$	16.0944	0.536183
$902$	27.5494	0.917295
$903$	0	0
$904$	−6.31341	−0.209981
$905$	−1.80210	−0.0599037
$906$	0	0
$907$	−13.9216	−0.462259	−0.231129	−	0.972923i	$-0.574242\pi$
$907$	−13.9216	−0.462259	−0.231129	+	0.972923i	$0.574242\pi$
$908$	18.6315	0.618308
$909$	0	0
$910$	0	0
$911$	5.40855	0.179193	0.0895967	−	0.995978i	$-0.471442\pi$
$911$	5.40855	0.179193	0.0895967	+	0.995978i	$0.471442\pi$
$912$	0	0
$913$	−3.90084	−0.129099
$914$	0.535362	0.0177082
$915$	0	0
$916$	1.49352	0.0493471
$917$	0	0
$918$	0	0
$919$	34.0283	1.12249	0.561245	−	0.827649i	$-0.310322\pi$
$919$	34.0283	1.12249	0.561245	+	0.827649i	$0.310322\pi$
$920$	−0.614276	−0.0202521
$921$	0	0
$922$	−8.81561	−0.290327
$923$	8.16008	0.268592
$924$	0	0
$925$	9.89063	0.325202
$926$	−11.5637	−0.380006
$927$	0	0
$928$	38.4165	1.26108
$929$	10.6329	0.348855	0.174427	−	0.984670i	$-0.444193\pi$
$929$	10.6329	0.348855	0.174427	+	0.984670i	$0.444193\pi$
$930$	0	0
$931$	0	0
$932$	5.12751	0.167957
$933$	0	0
$934$	24.1491	0.790183
$935$	−0.567799	−0.0185690
$936$	0	0
$937$	−52.6692	−1.72063	−0.860314	−	0.509765i	$-0.829732\pi$
$937$	−52.6692	−1.72063	−0.860314	+	0.509765i	$0.829732\pi$
$938$	0	0
$939$	0	0
$940$	0.264163	0.00861605
$941$	−34.3656	−1.12029	−0.560143	−	0.828396i	$-0.689254\pi$
$941$	−34.3656	−1.12029	−0.560143	+	0.828396i	$0.689254\pi$
$942$	0	0
$943$	−14.2011	−0.462453
$944$	−8.10825	−0.263901
$945$	0	0
$946$	−27.7791	−0.903175
$947$	40.5840	1.31880	0.659401	−	0.751791i	$-0.270810\pi$
$947$	40.5840	1.31880	0.659401	+	0.751791i	$0.270810\pi$
$948$	0	0
$949$	−11.7147	−0.380276
$950$	−42.2685	−1.37137
$951$	0	0
$952$	0	0
$953$	22.6904	0.735013	0.367507	−	0.930021i	$-0.380211\pi$
$953$	22.6904	0.735013	0.367507	+	0.930021i	$0.380211\pi$
$954$	0	0
$955$	2.36244	0.0764468
$956$	−16.7318	−0.541144
$957$	0	0
$958$	4.41312	0.142581
$959$	0	0
$960$	0	0
$961$	54.7775	1.76702
$962$	5.42208	0.174815
$963$	0	0
$964$	15.7279	0.506562
$965$	−0.0270822	−0.000871808 0
$966$	0	0
$967$	24.2776	0.780714	0.390357	−	0.920664i	$-0.372352\pi$
$967$	24.2776	0.780714	0.390357	+	0.920664i	$0.372352\pi$
$968$	−0.422650	−0.0135845
$969$	0	0
$970$	−0.442272	−0.0142005
$971$	45.5771	1.46264	0.731319	−	0.682035i	$-0.238905\pi$
$971$	45.5771	1.46264	0.731319	+	0.682035i	$0.238905\pi$
$972$	0	0
$973$	0	0
$974$	−29.1975	−0.935549
$975$	0	0
$976$	−10.3102	−0.330020
$977$	14.6896	0.469963	0.234981	−	0.972000i	$-0.424497\pi$
$977$	14.6896	0.469963	0.234981	+	0.972000i	$0.424497\pi$
$978$	0	0
$979$	−20.1225	−0.643117
$980$	0	0
$981$	0	0
$982$	31.3528	1.00051
$983$	−44.5909	−1.42223	−0.711115	−	0.703076i	$-0.751809\pi$
$983$	−44.5909	−1.42223	−0.711115	+	0.703076i	$0.751809\pi$
$984$	0	0
$985$	0.0805096	0.00256525
$986$	16.5281	0.526361
$987$	0	0
$988$	14.9335	0.475097
$989$	14.3195	0.455334
$990$	0	0
$991$	24.1829	0.768196	0.384098	−	0.923292i	$-0.374512\pi$
$991$	24.1829	0.768196	0.384098	+	0.923292i	$0.374512\pi$
$992$	−38.2973	−1.21594
$993$	0	0
$994$	0	0
$995$	−0.530622	−0.0168219
$996$	0	0
$997$	10.8652	0.344105	0.172053	−	0.985088i	$-0.444960\pi$
$997$	10.8652	0.344105	0.172053	+	0.985088i	$0.444960\pi$
$998$	−8.19521	−0.259415
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.bi.1.8		12
3.2	odd	2		3969.2.a.bh.1.5		12
7.6	odd	2	inner	3969.2.a.bi.1.7		12
9.2	odd	6		441.2.f.h.148.7	✓	24
9.4	even	3		1323.2.f.h.883.6		24
9.5	odd	6		441.2.f.h.295.7	yes	24
9.7	even	3		1323.2.f.h.442.6		24
21.20	even	2		3969.2.a.bh.1.6		12
63.2	odd	6		441.2.g.h.67.8		24
63.4	even	3		1323.2.g.h.667.6		24
63.5	even	6		441.2.h.h.214.5		24
63.11	odd	6		441.2.h.h.373.6		24
63.13	odd	6		1323.2.f.h.883.5		24
63.16	even	3		1323.2.g.h.361.6		24
63.20	even	6		441.2.f.h.148.8	yes	24
63.23	odd	6		441.2.h.h.214.6		24
63.25	even	3		1323.2.h.h.226.7		24
63.31	odd	6		1323.2.g.h.667.5		24
63.32	odd	6		441.2.g.h.79.8		24
63.34	odd	6		1323.2.f.h.442.5		24
63.38	even	6		441.2.h.h.373.5		24
63.40	odd	6		1323.2.h.h.802.8		24
63.41	even	6		441.2.f.h.295.8	yes	24
63.47	even	6		441.2.g.h.67.7		24
63.52	odd	6		1323.2.h.h.226.8		24
63.58	even	3		1323.2.h.h.802.7		24
63.59	even	6		441.2.g.h.79.7		24
63.61	odd	6		1323.2.g.h.361.5		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.f.h.148.7	✓	24	9.2	odd	6
441.2.f.h.148.8	yes	24	63.20	even	6
441.2.f.h.295.7	yes	24	9.5	odd	6
441.2.f.h.295.8	yes	24	63.41	even	6
441.2.g.h.67.7		24	63.47	even	6
441.2.g.h.67.8		24	63.2	odd	6
441.2.g.h.79.7		24	63.59	even	6
441.2.g.h.79.8		24	63.32	odd	6
441.2.h.h.214.5		24	63.5	even	6
441.2.h.h.214.6		24	63.23	odd	6
441.2.h.h.373.5		24	63.38	even	6
441.2.h.h.373.6		24	63.11	odd	6
1323.2.f.h.442.5		24	63.34	odd	6
1323.2.f.h.442.6		24	9.7	even	3
1323.2.f.h.883.5		24	63.13	odd	6
1323.2.f.h.883.6		24	9.4	even	3
1323.2.g.h.361.5		24	63.61	odd	6
1323.2.g.h.361.6		24	63.16	even	3
1323.2.g.h.667.5		24	63.31	odd	6
1323.2.g.h.667.6		24	63.4	even	3
1323.2.h.h.226.7		24	63.25	even	3
1323.2.h.h.226.8		24	63.52	odd	6
1323.2.h.h.802.7		24	63.58	even	3
1323.2.h.h.802.8		24	63.40	odd	6
3969.2.a.bh.1.5		12	3.2	odd	2
3969.2.a.bh.1.6		12	21.20	even	2
3969.2.a.bi.1.7		12	7.6	odd	2	inner
3969.2.a.bi.1.8		12	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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q+1.10281 q^{2} -0.783802 q^{4} +0.105466 q^{5} -3.07001 q^{8} +O(q^{10})\)	\(q+1.10281 q^{2} -0.783802 q^{4} +0.105466 q^{5} -3.07001 q^{8} +0.116309 q^{10} -3.33731 q^{11} -2.47995 q^{13} -1.81805 q^{16} +1.61319 q^{17} +7.68266 q^{19} -0.0826646 q^{20} -3.68044 q^{22} +1.89719 q^{23} -4.98888 q^{25} -2.73492 q^{26} +9.29042 q^{29} -9.26162 q^{31} +4.13506 q^{32} +1.77905 q^{34} -1.98254 q^{37} +8.47254 q^{38} -0.323782 q^{40} -7.48537 q^{41} +7.54776 q^{43} +2.61579 q^{44} +2.09224 q^{46} -3.19560 q^{47} -5.50180 q^{50} +1.94379 q^{52} +9.97679 q^{53} -0.351974 q^{55} +10.2456 q^{58} +4.45986 q^{59} +5.67100 q^{61} -10.2138 q^{62} +8.19630 q^{64} -0.261550 q^{65} +9.97141 q^{67} -1.26442 q^{68} -3.29042 q^{71} +4.72378 q^{73} -2.18637 q^{74} -6.02169 q^{76} +7.69409 q^{79} -0.191743 q^{80} -8.25496 q^{82} +1.16886 q^{83} +0.170137 q^{85} +8.32378 q^{86} +10.2456 q^{88} +6.02954 q^{89} -1.48702 q^{92} -3.52415 q^{94} +0.810260 q^{95} -3.80255 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8}+O(q^{10}) \) 12 * q + 4 * q^2 + 12 * q^4 + 12 * q^8	\( 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8} + 20 q^{11} + 12 q^{16} + 32 q^{23} + 12 q^{25} + 16 q^{29} + 48 q^{32} + 12 q^{37} + 56 q^{44} - 24 q^{46} - 4 q^{50} + 32 q^{53} + 48 q^{64} + 60 q^{65} + 12 q^{67} + 56 q^{71} + 68 q^{74} - 12 q^{79} - 12 q^{85} + 76 q^{86} + 16 q^{92} + 64 q^{95}+O(q^{100}) \) 12 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 + 20 * q^11 + 12 * q^16 + 32 * q^23 + 12 * q^25 + 16 * q^29 + 48 * q^32 + 12 * q^37 + 56 * q^44 - 24 * q^46 - 4 * q^50 + 32 * q^53 + 48 * q^64 + 60 * q^65 + 12 * q^67 + 56 * q^71 + 68 * q^74 - 12 * q^79 - 12 * q^85 + 76 * q^86 + 16 * q^92 + 64 * q^95

Introduction

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