LMFDB - Embedded newform 3969.2.a.o.1.2

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:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 567)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.523976$ 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)

$f(q)$	$=$	$q-0.523976 q^{2} -1.72545 q^{4} -2.20147 q^{5} +1.95205 q^{8} +1.15352 q^{10} -5.20147 q^{11} -3.15352 q^{13} +2.42807 q^{16} -3.24943 q^{17} -7.45090 q^{19} +3.79853 q^{20} +2.72545 q^{22} -4.40294 q^{23} -0.153520 q^{25} +1.65237 q^{26} +1.15352 q^{29} -2.00000 q^{31} -5.17635 q^{32} +1.70262 q^{34} +5.00000 q^{37} +3.90409 q^{38} -4.29738 q^{40} -11.4509 q^{41} +9.29738 q^{43} +8.97487 q^{44} +2.30704 q^{46} -1.04795 q^{47} +0.0804406 q^{50} +5.44124 q^{52} +0.249425 q^{53} +11.4509 q^{55} -0.604417 q^{58} -8.09591 q^{59} -8.60442 q^{61} +1.04795 q^{62} -2.14386 q^{64} +6.94239 q^{65} -7.60442 q^{67} +5.60672 q^{68} -9.60442 q^{71} -0.846480 q^{73} -2.61988 q^{74} +12.8561 q^{76} -7.60442 q^{79} -5.34533 q^{80} +6.00000 q^{82} +11.4509 q^{83} +7.15352 q^{85} -4.87161 q^{86} -10.1535 q^{88} +9.24943 q^{89} +7.59706 q^{92} +0.549103 q^{94} +16.4029 q^{95} +3.45090 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} + 3 q^{5} + 9 q^{8} - 3 q^{10} - 6 q^{11} - 3 q^{13} + 12 q^{16} + 3 q^{17} + 21 q^{20} - 3 q^{22} + 6 q^{23} + 6 q^{25} - 27 q^{26} - 3 q^{29} - 6 q^{31} + 18 q^{32} + 21 q^{34} + 15 q^{37}+ \cdots - 12 q^{97}+O(q^{100}) $ 3 * q + 6 * q^4 + 3 * q^5 + 9 * q^8 - 3 * q^10 - 6 * q^11 - 3 * q^13 + 12 * q^16 + 3 * q^17 + 21 * q^20 - 3 * q^22 + 6 * q^23 + 6 * q^25 - 27 * q^26 - 3 * q^29 - 6 * q^31 + 18 * q^32 + 21 * q^34 + 15 * q^37 + 18 * q^38 + 3 * q^40 - 12 * q^41 + 12 * q^43 + 3 * q^44 - 6 * q^46 - 27 * q^50 - 9 * q^52 - 12 * q^53 + 12 * q^55 + 27 * q^58 - 18 * q^59 + 3 * q^61 + 3 * q^64 + 21 * q^65 + 6 * q^67 + 39 * q^68 - 9 * q^73 + 48 * q^76 + 6 * q^79 + 3 * q^80 + 18 * q^82 + 12 * q^83 + 15 * q^85 - 45 * q^86 - 24 * q^88 + 15 * q^89 + 42 * q^92 + 24 * q^94 + 30 * q^95 - 12 * 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 $

$2$	−0.523976	−0.370507	−0.185254	−	0.982691i	$-0.559311\pi$
$2$	−0.523976	−0.370507	−0.185254	+	0.982691i	$0.559311\pi$
$3$	0	0
$3$	0	0
$4$	−1.72545	−0.862724
$5$	−2.20147	−0.984528	−0.492264	−	0.870446i	$-0.663831\pi$
$5$	−2.20147	−0.984528	−0.492264	+	0.870446i	$0.663831\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.95205	0.690153
$9$	0	0
$10$	1.15352	0.364775
$11$	−5.20147	−1.56830	−0.784151	−	0.620569i	$-0.786901\pi$
$11$	−5.20147	−1.56830	−0.784151	+	0.620569i	$0.786901\pi$
$12$	0	0
$13$	−3.15352	−0.874629	−0.437314	−	0.899309i	$-0.644070\pi$
$13$	−3.15352	−0.874629	−0.437314	+	0.899309i	$0.644070\pi$
$14$	0	0
$15$	0	0
$16$	2.42807	0.607018
$17$	−3.24943	−0.788101	−0.394051	−	0.919089i	$-0.628927\pi$
$17$	−3.24943	−0.788101	−0.394051	+	0.919089i	$0.628927\pi$
$18$	0	0
$19$	−7.45090	−1.70935	−0.854677	−	0.519161i	$-0.826245\pi$
$19$	−7.45090	−1.70935	−0.854677	+	0.519161i	$0.826245\pi$
$20$	3.79853	0.849377
$21$	0	0
$22$	2.72545	0.581068
$23$	−4.40294	−0.918077	−0.459039	−	0.888416i	$-0.651806\pi$
$23$	−4.40294	−0.918077	−0.459039	+	0.888416i	$0.651806\pi$
$24$	0	0
$25$	−0.153520	−0.0307039
$26$	1.65237	0.324056
$27$	0	0
$28$	0	0
$29$	1.15352	0.214203	0.107102	−	0.994248i	$-0.465843\pi$
$29$	1.15352	0.214203	0.107102	+	0.994248i	$0.465843\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−5.17635	−0.915057
$33$	0	0
$34$	1.70262	0.291997
$35$	0	0
$36$	0	0
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	3.90409	0.633328
$39$	0	0
$40$	−4.29738	−0.679475
$41$	−11.4509	−1.78833	−0.894165	−	0.447738i	$-0.852230\pi$
$41$	−11.4509	−1.78833	−0.894165	+	0.447738i	$0.852230\pi$
$42$	0	0
$43$	9.29738	1.41784	0.708918	−	0.705290i	$-0.249183\pi$
$43$	9.29738	1.41784	0.708918	+	0.705290i	$0.249183\pi$
$44$	8.97487	1.35301
$45$	0	0
$46$	2.30704	0.340154
$47$	−1.04795	−0.152860	−0.0764298	−	0.997075i	$-0.524352\pi$
$47$	−1.04795	−0.152860	−0.0764298	+	0.997075i	$0.524352\pi$
$48$	0	0
$49$	0	0
$50$	0.0804406	0.0113760
$51$	0	0
$52$	5.44124	0.754564
$53$	0.249425	0.0342612	0.0171306	−	0.999853i	$-0.494547\pi$
$53$	0.249425	0.0342612	0.0171306	+	0.999853i	$0.494547\pi$
$54$	0	0
$55$	11.4509	1.54404
$56$	0	0
$57$	0	0
$58$	−0.604417	−0.0793638
$59$	−8.09591	−1.05400	−0.526999	−	0.849866i	$-0.676683\pi$
$59$	−8.09591	−1.05400	−0.526999	+	0.849866i	$0.676683\pi$
$60$	0	0
$61$	−8.60442	−1.10168	−0.550841	−	0.834610i	$-0.685693\pi$
$61$	−8.60442	−1.10168	−0.550841	+	0.834610i	$0.685693\pi$
$62$	1.04795	0.133090
$63$	0	0
$64$	−2.14386	−0.267982
$65$	6.94239	0.861097
$66$	0	0
$67$	−7.60442	−0.929027	−0.464514	−	0.885566i	$-0.653771\pi$
$67$	−7.60442	−0.929027	−0.464514	+	0.885566i	$0.653771\pi$
$68$	5.60672	0.679914
$69$	0	0
$70$	0	0
$71$	−9.60442	−1.13983	−0.569917	−	0.821702i	$-0.693025\pi$
$71$	−9.60442	−1.13983	−0.569917	+	0.821702i	$0.693025\pi$
$72$	0	0
$73$	−0.846480	−0.0990730	−0.0495365	−	0.998772i	$-0.515774\pi$
$73$	−0.846480	−0.0990730	−0.0495365	+	0.998772i	$0.515774\pi$
$74$	−2.61988	−0.304555
$75$	0	0
$76$	12.8561	1.47470
$77$	0	0
$78$	0	0
$79$	−7.60442	−0.855564	−0.427782	−	0.903882i	$-0.640705\pi$
$79$	−7.60442	−0.855564	−0.427782	+	0.903882i	$0.640705\pi$
$80$	−5.34533	−0.597626
$81$	0	0
$82$	6.00000	0.662589
$83$	11.4509	1.25690	0.628450	−	0.777850i	$-0.283690\pi$
$83$	11.4509	1.25690	0.628450	+	0.777850i	$0.283690\pi$
$84$	0	0
$85$	7.15352	0.775908
$86$	−4.87161	−0.525319
$87$	0	0
$88$	−10.1535	−1.08237
$89$	9.24943	0.980437	0.490219	−	0.871600i	$-0.336917\pi$
$89$	9.24943	0.980437	0.490219	+	0.871600i	$0.336917\pi$
$90$	0	0
$91$	0	0
$92$	7.59706	0.792048
$93$	0	0
$94$	0.549103	0.0566356
$95$	16.4029	1.68291
$96$	0	0
$97$	3.45090	0.350386	0.175193	−	0.984534i	$-0.443945\pi$
$97$	3.45090	0.350386	0.175193	+	0.984534i	$0.443945\pi$
$98$	0	0
$99$	0	0
$100$	0.264890	0.0264890
$101$	−16.4029	−1.63215	−0.816077	−	0.577943i	$-0.803856\pi$
$101$	−16.4029	−1.63215	−0.816077	+	0.577943i	$0.803856\pi$
$102$	0	0
$103$	1.14386	0.112708	0.0563539	−	0.998411i	$-0.482053\pi$
$103$	1.14386	0.112708	0.0563539	+	0.998411i	$0.482053\pi$
$104$	−6.15582	−0.603628
$105$	0	0
$106$	−0.130693	−0.0126940
$107$	−9.10557	−0.880268	−0.440134	−	0.897932i	$-0.645069\pi$
$107$	−9.10557	−0.880268	−0.440134	+	0.897932i	$0.645069\pi$
$108$	0	0
$109$	−13.7483	−1.31685	−0.658423	−	0.752648i	$-0.728776\pi$
$109$	−13.7483	−1.31685	−0.658423	+	0.752648i	$0.728776\pi$
$110$	−6.00000	−0.572078
$111$	0	0
$112$	0	0
$113$	−19.9018	−1.87220	−0.936102	−	0.351729i	$-0.885594\pi$
$113$	−19.9018	−1.87220	−0.936102	+	0.351729i	$0.885594\pi$
$114$	0	0
$115$	9.69296	0.903873
$116$	−1.99034	−0.184798
$117$	0	0
$118$	4.24206	0.390514
$119$	0	0
$120$	0	0
$121$	16.0553	1.45957
$122$	4.50851	0.408181
$123$	0	0
$124$	3.45090	0.309900
$125$	11.3453	1.01476
$126$	0	0
$127$	9.29738	0.825009	0.412504	−	0.910956i	$-0.364654\pi$
$127$	9.29738	0.825009	0.412504	+	0.910956i	$0.364654\pi$
$128$	11.4760	1.01435
$129$	0	0
$130$	−3.63765	−0.319043
$131$	7.54680	0.659367	0.329684	−	0.944091i	$-0.393058\pi$
$131$	7.54680	0.659367	0.329684	+	0.944091i	$0.393058\pi$
$132$	0	0
$133$	0	0
$134$	3.98454	0.344211
$135$	0	0
$136$	−6.34303	−0.543910
$137$	1.40294	0.119862	0.0599308	−	0.998203i	$-0.480912\pi$
$137$	1.40294	0.119862	0.0599308	+	0.998203i	$0.480912\pi$
$138$	0	0
$139$	15.4509	1.31053	0.655264	−	0.755400i	$-0.272557\pi$
$139$	15.4509	1.31053	0.655264	+	0.755400i	$0.272557\pi$
$140$	0	0
$141$	0	0
$142$	5.03249	0.422317
$143$	16.4029	1.37168
$144$	0	0
$145$	−2.53944	−0.210889
$146$	0.443536	0.0367073
$147$	0	0
$148$	−8.62724	−0.709155
$149$	−13.4029	−1.09801	−0.549006	−	0.835818i	$-0.684994\pi$
$149$	−13.4029	−1.09801	−0.549006	+	0.835818i	$0.684994\pi$
$150$	0	0
$151$	11.6044	0.944354	0.472177	−	0.881504i	$-0.343468\pi$
$151$	11.6044	0.944354	0.472177	+	0.881504i	$0.343468\pi$
$152$	−14.5445	−1.17972
$153$	0	0
$154$	0	0
$155$	4.40294	0.353653
$156$	0	0
$157$	24.3624	1.94433	0.972164	−	0.234302i	$-0.0752806\pi$
$157$	24.3624	1.94433	0.972164	+	0.234302i	$0.0752806\pi$
$158$	3.98454	0.316993
$159$	0	0
$160$	11.3956	0.900900
$161$	0	0
$162$	0	0
$163$	5.60442	0.438972	0.219486	−	0.975616i	$-0.429562\pi$
$163$	5.60442	0.438972	0.219486	+	0.975616i	$0.429562\pi$
$164$	19.7579	1.54284
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−23.4509	−1.81468	−0.907342	−	0.420392i	$-0.861892\pi$
$167$	−23.4509	−1.81468	−0.907342	+	0.420392i	$0.861892\pi$
$168$	0	0
$169$	−3.05531	−0.235024
$170$	−3.74828	−0.287480
$171$	0	0
$172$	−16.0421	−1.22320
$173$	−22.5085	−1.71129	−0.855645	−	0.517563i	$-0.826839\pi$
$173$	−22.5085	−1.71129	−0.855645	+	0.517563i	$0.826839\pi$
$174$	0	0
$175$	0	0
$176$	−12.6295	−0.951988
$177$	0	0
$178$	−4.84648	−0.363259
$179$	−6.49885	−0.485747	−0.242873	−	0.970058i	$-0.578090\pi$
$179$	−6.49885	−0.485747	−0.242873	+	0.970058i	$0.578090\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	0	0
$184$	−8.59476	−0.633614
$185$	−11.0074	−0.809277
$186$	0	0
$187$	16.9018	1.23598
$188$	1.80819	0.131876
$189$	0	0
$190$	−8.59476	−0.623529
$191$	13.5085	0.977442	0.488721	−	0.872440i	$-0.337464\pi$
$191$	13.5085	0.977442	0.488721	+	0.872440i	$0.337464\pi$
$192$	0	0
$193$	17.7483	1.27755	0.638774	−	0.769394i	$-0.279442\pi$
$193$	17.7483	1.27755	0.638774	+	0.769394i	$0.279442\pi$
$194$	−1.80819	−0.129820
$195$	0	0
$196$	0	0
$197$	16.2088	1.15483	0.577416	−	0.816450i	$-0.304061\pi$
$197$	16.2088	1.15483	0.577416	+	0.816450i	$0.304061\pi$
$198$	0	0
$199$	7.14386	0.506415	0.253207	−	0.967412i	$-0.418515\pi$
$199$	7.14386	0.506415	0.253207	+	0.967412i	$0.418515\pi$
$200$	−0.299677	−0.0211904
$201$	0	0
$202$	8.59476	0.604725
$203$	0	0
$204$	0	0
$205$	25.2088	1.76066
$206$	−0.599355	−0.0417590
$207$	0	0
$208$	−7.65697	−0.530915
$209$	38.7556	2.68078
$210$	0	0
$211$	3.29738	0.227001	0.113500	−	0.993538i	$-0.463794\pi$
$211$	3.29738	0.227001	0.113500	+	0.993538i	$0.463794\pi$
$212$	−0.430370	−0.0295580
$213$	0	0
$214$	4.77110	0.326146
$215$	−20.4679	−1.39590
$216$	0	0
$217$	0	0
$218$	7.20377	0.487901
$219$	0	0
$220$	−19.7579	−1.33208
$221$	10.2471	0.689296
$222$	0	0
$223$	23.2088	1.55418	0.777089	−	0.629390i	$-0.216695\pi$
$223$	23.2088	1.55418	0.777089	+	0.629390i	$0.216695\pi$
$224$	0	0
$225$	0	0
$226$	10.4281	0.693665
$227$	11.6620	0.774036	0.387018	−	0.922072i	$-0.373505\pi$
$227$	11.6620	0.774036	0.387018	+	0.922072i	$0.373505\pi$
$228$	0	0
$229$	−1.39558	−0.0922227	−0.0461114	−	0.998936i	$-0.514683\pi$
$229$	−1.39558	−0.0922227	−0.0461114	+	0.998936i	$0.514683\pi$
$230$	−5.07888	−0.334892
$231$	0	0
$232$	2.25172	0.147833
$233$	8.75057	0.573269	0.286635	−	0.958040i	$-0.407463\pi$
$233$	8.75057	0.573269	0.286635	+	0.958040i	$0.407463\pi$
$234$	0	0
$235$	2.30704	0.150495
$236$	13.9691	0.909309
$237$	0	0
$238$	0	0
$239$	4.70262	0.304187	0.152094	−	0.988366i	$-0.451398\pi$
$239$	4.70262	0.304187	0.152094	+	0.988366i	$0.451398\pi$
$240$	0	0
$241$	4.60442	0.296597	0.148298	−	0.988943i	$-0.452620\pi$
$241$	4.60442	0.296597	0.148298	+	0.988943i	$0.452620\pi$
$242$	−8.41261	−0.540783
$243$	0	0
$244$	14.8465	0.950449
$245$	0	0
$246$	0	0
$247$	23.4966	1.49505
$248$	−3.90409	−0.247910
$249$	0	0
$250$	−5.94469	−0.375975
$251$	−3.14386	−0.198439	−0.0992193	−	0.995066i	$-0.531635\pi$
$251$	−3.14386	−0.198439	−0.0992193	+	0.995066i	$0.531635\pi$
$252$	0	0
$253$	22.9018	1.43982
$254$	−4.87161	−0.305672
$255$	0	0
$256$	−1.72545	−0.107841
$257$	20.1512	1.25700	0.628499	−	0.777810i	$-0.283670\pi$
$257$	20.1512	1.25700	0.628499	+	0.777810i	$0.283670\pi$
$258$	0	0
$259$	0	0
$260$	−11.9787	−0.742889
$261$	0	0
$262$	−3.95435	−0.244300
$263$	−17.2015	−1.06069	−0.530344	−	0.847782i	$-0.677937\pi$
$263$	−17.2015	−1.06069	−0.530344	+	0.847782i	$0.677937\pi$
$264$	0	0
$265$	−0.549103	−0.0337311
$266$	0	0
$267$	0	0
$268$	13.1210	0.801495
$269$	2.75057	0.167706	0.0838528	−	0.996478i	$-0.473277\pi$
$269$	2.75057	0.167706	0.0838528	+	0.996478i	$0.473277\pi$
$270$	0	0
$271$	12.8561	0.780955	0.390477	−	0.920612i	$-0.372310\pi$
$271$	12.8561	0.780955	0.390477	+	0.920612i	$0.372310\pi$
$272$	−7.88983	−0.478391
$273$	0	0
$274$	−0.735110	−0.0444096
$275$	0.798528	0.0481530
$276$	0	0
$277$	−16.7483	−1.00631	−0.503153	−	0.864197i	$-0.667827\pi$
$277$	−16.7483	−1.00631	−0.503153	+	0.864197i	$0.667827\pi$
$278$	−8.09591	−0.485560
$279$	0	0
$280$	0	0
$281$	−5.30704	−0.316591	−0.158296	−	0.987392i	$-0.550600\pi$
$281$	−5.30704	−0.316591	−0.158296	+	0.987392i	$0.550600\pi$
$282$	0	0
$283$	−12.3527	−0.734291	−0.367146	−	0.930163i	$-0.619665\pi$
$283$	−12.3527	−0.734291	−0.367146	+	0.930163i	$0.619665\pi$
$284$	16.5719	0.983363
$285$	0	0
$286$	−8.59476	−0.508219
$287$	0	0
$288$	0	0
$289$	−6.44124	−0.378896
$290$	1.33061	0.0781360
$291$	0	0
$292$	1.46056	0.0854727
$293$	10.2974	0.601579	0.300790	−	0.953691i	$-0.402750\pi$
$293$	10.2974	0.601579	0.300790	+	0.953691i	$0.402750\pi$
$294$	0	0
$295$	17.8229	1.03769
$296$	9.76024	0.567302
$297$	0	0
$298$	7.02283	0.406821
$299$	13.8848	0.802977
$300$	0	0
$301$	0	0
$302$	−6.08044	−0.349890
$303$	0	0
$304$	−18.0913	−1.03761
$305$	18.9424	1.08464
$306$	0	0
$307$	0.307039	0.0175236	0.00876182	−	0.999962i	$-0.497211\pi$
$307$	0.307039	0.0175236	0.00876182	+	0.999962i	$0.497211\pi$
$308$	0	0
$309$	0	0
$310$	−2.30704	−0.131031
$311$	1.04795	0.0594240	0.0297120	−	0.999559i	$-0.490541\pi$
$311$	1.04795	0.0594240	0.0297120	+	0.999559i	$0.490541\pi$
$312$	0	0
$313$	−32.0553	−1.81187	−0.905937	−	0.423413i	$-0.860832\pi$
$313$	−32.0553	−1.81187	−0.905937	+	0.423413i	$0.860832\pi$
$314$	−12.7653	−0.720387
$315$	0	0
$316$	13.1210	0.738116
$317$	6.44354	0.361905	0.180953	−	0.983492i	$-0.442082\pi$
$317$	6.44354	0.361905	0.180953	+	0.983492i	$0.442082\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	4.71964	0.263836
$321$	0	0
$322$	0	0
$323$	24.2111	1.34714
$324$	0	0
$325$	0.484127	0.0268545
$326$	−2.93658	−0.162642
$327$	0	0
$328$	−22.3527	−1.23422
$329$	0	0
$330$	0	0
$331$	−20.6141	−1.13305	−0.566526	−	0.824044i	$-0.691713\pi$
$331$	−20.6141	−1.13305	−0.566526	+	0.824044i	$0.691713\pi$
$332$	−19.7579	−1.08436
$333$	0	0
$334$	12.2877	0.672354
$335$	16.7409	0.914654
$336$	0	0
$337$	26.4606	1.44140	0.720699	−	0.693248i	$-0.243821\pi$
$337$	26.4606	1.44140	0.720699	+	0.693248i	$0.243821\pi$
$338$	1.60091	0.0870782
$339$	0	0
$340$	−12.3430	−0.669395
$341$	10.4029	0.563351
$342$	0	0
$343$	0	0
$344$	18.1489	0.978524
$345$	0	0
$346$	11.7939	0.634046
$347$	4.49149	0.241116	0.120558	−	0.992706i	$-0.461532\pi$
$347$	4.49149	0.241116	0.120558	+	0.992706i	$0.461532\pi$
$348$	0	0
$349$	−3.75794	−0.201158	−0.100579	−	0.994929i	$-0.532069\pi$
$349$	−3.75794	−0.201158	−0.100579	+	0.994929i	$0.532069\pi$
$350$	0	0
$351$	0	0
$352$	26.9246	1.43509
$353$	−19.2591	−1.02506	−0.512529	−	0.858670i	$-0.671291\pi$
$353$	−19.2591	−1.02506	−0.512529	+	0.858670i	$0.671291\pi$
$354$	0	0
$355$	21.1439	1.12220
$356$	−15.9594	−0.845847
$357$	0	0
$358$	3.40524	0.179973
$359$	−9.10557	−0.480573	−0.240287	−	0.970702i	$-0.577241\pi$
$359$	−9.10557	−0.480573	−0.240287	+	0.970702i	$0.577241\pi$
$360$	0	0
$361$	36.5159	1.92189
$362$	1.04795	0.0550792
$363$	0	0
$364$	0	0
$365$	1.86350	0.0975402
$366$	0	0
$367$	−4.59476	−0.239844	−0.119922	−	0.992783i	$-0.538264\pi$
$367$	−4.59476	−0.239844	−0.119922	+	0.992783i	$0.538264\pi$
$368$	−10.6907	−0.557289
$369$	0	0
$370$	5.76760	0.299843
$371$	0	0
$372$	0	0
$373$	−21.3624	−1.10610	−0.553050	−	0.833148i	$-0.686536\pi$
$373$	−21.3624	−1.10610	−0.553050	+	0.833148i	$0.686536\pi$
$374$	−8.85614	−0.457940
$375$	0	0
$376$	−2.04565	−0.105497
$377$	−3.63765	−0.187348
$378$	0	0
$379$	−35.1203	−1.80401	−0.902004	−	0.431728i	$-0.857904\pi$
$379$	−35.1203	−1.80401	−0.902004	+	0.431728i	$0.857904\pi$
$380$	−28.3024	−1.45188
$381$	0	0
$382$	−7.07814	−0.362149
$383$	3.19411	0.163211	0.0816057	−	0.996665i	$-0.473995\pi$
$383$	3.19411	0.163211	0.0816057	+	0.996665i	$0.473995\pi$
$384$	0	0
$385$	0	0
$386$	−9.29968	−0.473341
$387$	0	0
$388$	−5.95435	−0.302286
$389$	−14.8059	−0.750688	−0.375344	−	0.926886i	$-0.622475\pi$
$389$	−14.8059	−0.750688	−0.375344	+	0.926886i	$0.622475\pi$
$390$	0	0
$391$	14.3070	0.723538
$392$	0	0
$393$	0	0
$394$	−8.49305	−0.427873
$395$	16.7409	0.842327
$396$	0	0
$397$	−26.8921	−1.34968	−0.674839	−	0.737965i	$-0.735787\pi$
$397$	−26.8921	−1.34968	−0.674839	+	0.737965i	$0.735787\pi$
$398$	−3.74321	−0.187630
$399$	0	0
$400$	−0.372756	−0.0186378
$401$	−19.4029	−0.968937	−0.484468	−	0.874809i	$-0.660987\pi$
$401$	−19.4029	−0.968937	−0.484468	+	0.874809i	$0.660987\pi$
$402$	0	0
$403$	6.30704	0.314176
$404$	28.3024	1.40810
$405$	0	0
$406$	0	0
$407$	−26.0074	−1.28914
$408$	0	0
$409$	9.39558	0.464582	0.232291	−	0.972646i	$-0.425378\pi$
$409$	9.39558	0.464582	0.232291	+	0.972646i	$0.425378\pi$
$410$	−13.2088	−0.652338
$411$	0	0
$412$	−1.97367	−0.0972357
$413$	0	0
$414$	0	0
$415$	−25.2088	−1.23745
$416$	16.3237	0.800336
$417$	0	0
$418$	−20.3070	−0.993250
$419$	−24.6597	−1.20471	−0.602353	−	0.798230i	$-0.705770\pi$
$419$	−24.6597	−1.20471	−0.602353	+	0.798230i	$0.705770\pi$
$420$	0	0
$421$	−14.2088	−0.692496	−0.346248	−	0.938143i	$-0.612544\pi$
$421$	−14.2088	−0.692496	−0.346248	+	0.938143i	$0.612544\pi$
$422$	−1.72775	−0.0841055
$423$	0	0
$424$	0.486890	0.0236455
$425$	0.498850	0.0241978
$426$	0	0
$427$	0	0
$428$	15.7112	0.759429
$429$	0	0
$430$	10.7247	0.517191
$431$	12.9977	0.626077	0.313039	−	0.949740i	$-0.398653\pi$
$431$	12.9977	0.626077	0.313039	+	0.949740i	$0.398653\pi$
$432$	0	0
$433$	0.911456	0.0438018	0.0219009	−	0.999760i	$-0.493028\pi$
$433$	0.911456	0.0438018	0.0219009	+	0.999760i	$0.493028\pi$
$434$	0	0
$435$	0	0
$436$	23.7219	1.13608
$437$	32.8059	1.56932
$438$	0	0
$439$	10.5491	0.503481	0.251741	−	0.967795i	$-0.418997\pi$
$439$	10.5491	0.503481	0.251741	+	0.967795i	$0.418997\pi$
$440$	22.3527	1.06562
$441$	0	0
$442$	−5.36925	−0.255389
$443$	17.4006	0.826730	0.413365	−	0.910566i	$-0.364353\pi$
$443$	17.4006	0.826730	0.413365	+	0.910566i	$0.364353\pi$
$444$	0	0
$445$	−20.3624	−0.965268
$446$	−12.1609	−0.575834
$447$	0	0
$448$	0	0
$449$	−0.748275	−0.0353133	−0.0176566	−	0.999844i	$-0.505621\pi$
$449$	−0.748275	−0.0353133	−0.0176566	+	0.999844i	$0.505621\pi$
$450$	0	0
$451$	59.5615	2.80464
$452$	34.3395	1.61520
$453$	0	0
$454$	−6.11063	−0.286786
$455$	0	0
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	0.731253	0.0341692
$459$	0	0
$460$	−16.7247	−0.779793
$461$	−38.0457	−1.77196	−0.885981	−	0.463721i	$-0.846514\pi$
$461$	−38.0457	−1.77196	−0.885981	+	0.463721i	$0.846514\pi$
$462$	0	0
$463$	−25.8921	−1.20331	−0.601655	−	0.798756i	$-0.705492\pi$
$463$	−25.8921	−1.20331	−0.601655	+	0.798756i	$0.705492\pi$
$464$	2.80083	0.130025
$465$	0	0
$466$	−4.58509	−0.212400
$467$	17.0627	0.789566	0.394783	−	0.918774i	$-0.370820\pi$
$467$	17.0627	0.789566	0.394783	+	0.918774i	$0.370820\pi$
$468$	0	0
$469$	0	0
$470$	−1.20883	−0.0557594
$471$	0	0
$472$	−15.8036	−0.727419
$473$	−48.3601	−2.22360
$474$	0	0
$475$	1.14386	0.0524838
$476$	0	0
$477$	0	0
$478$	−2.46406	−0.112704
$479$	2.80589	0.128204	0.0641022	−	0.997943i	$-0.479582\pi$
$479$	2.80589	0.128204	0.0641022	+	0.997943i	$0.479582\pi$
$480$	0	0
$481$	−15.7676	−0.718941
$482$	−2.41261	−0.109891
$483$	0	0
$484$	−27.7026	−1.25921
$485$	−7.59706	−0.344965
$486$	0	0
$487$	−6.50621	−0.294825	−0.147412	−	0.989075i	$-0.547094\pi$
$487$	−6.50621	−0.294825	−0.147412	+	0.989075i	$0.547094\pi$
$488$	−16.7962	−0.760330
$489$	0	0
$490$	0	0
$491$	−4.61408	−0.208230	−0.104115	−	0.994565i	$-0.533201\pi$
$491$	−4.61408	−0.208230	−0.104115	+	0.994565i	$0.533201\pi$
$492$	0	0
$493$	−3.74828	−0.168814
$494$	−12.3116	−0.553927
$495$	0	0
$496$	−4.85614	−0.218047
$497$	0	0
$498$	0	0
$499$	−35.4966	−1.58904	−0.794522	−	0.607235i	$-0.792278\pi$
$499$	−35.4966	−1.58904	−0.794522	+	0.607235i	$0.792278\pi$
$500$	−19.5758	−0.875456
$501$	0	0
$502$	1.64731	0.0735229
$503$	−18.9211	−0.843651	−0.421825	−	0.906677i	$-0.638611\pi$
$503$	−18.9211	−0.843651	−0.421825	+	0.906677i	$0.638611\pi$
$504$	0	0
$505$	36.1106	1.60690
$506$	−12.0000	−0.533465
$507$	0	0
$508$	−16.0421	−0.711755
$509$	−14.8059	−0.656260	−0.328130	−	0.944633i	$-0.606418\pi$
$509$	−14.8059	−0.656260	−0.328130	+	0.944633i	$0.606418\pi$
$510$	0	0
$511$	0	0
$512$	−22.0480	−0.974391
$513$	0	0
$514$	−10.5588	−0.465727
$515$	−2.51817	−0.110964
$516$	0	0
$517$	5.45090	0.239730
$518$	0	0
$519$	0	0
$520$	13.5519	0.594289
$521$	−12.7100	−0.556834	−0.278417	−	0.960460i	$-0.589810\pi$
$521$	−12.7100	−0.556834	−0.278417	+	0.960460i	$0.589810\pi$
$522$	0	0
$523$	−0.352692	−0.0154222	−0.00771108	−	0.999970i	$-0.502455\pi$
$523$	−0.352692	−0.0154222	−0.00771108	+	0.999970i	$0.502455\pi$
$524$	−13.0216	−0.568852
$525$	0	0
$526$	9.01317	0.392993
$527$	6.49885	0.283094
$528$	0	0
$529$	−3.61408	−0.157134
$530$	0.287717	0.0124976
$531$	0	0
$532$	0	0
$533$	36.1106	1.56412
$534$	0	0
$535$	20.0457	0.866649
$536$	−14.8442	−0.641171
$537$	0	0
$538$	−1.44124	−0.0621361
$539$	0	0
$540$	0	0
$541$	−22.6930	−0.975647	−0.487823	−	0.872942i	$-0.662209\pi$
$541$	−22.6930	−0.975647	−0.487823	+	0.872942i	$0.662209\pi$
$542$	−6.73631	−0.289349
$543$	0	0
$544$	16.8201	0.721158
$545$	30.2664	1.29647
$546$	0	0
$547$	−1.49379	−0.0638698	−0.0319349	−	0.999490i	$-0.510167\pi$
$547$	−1.49379	−0.0638698	−0.0319349	+	0.999490i	$0.510167\pi$
$548$	−2.42071	−0.103408
$549$	0	0
$550$	−0.418410	−0.0178410
$551$	−8.59476	−0.366149
$552$	0	0
$553$	0	0
$554$	8.77570	0.372844
$555$	0	0
$556$	−26.6597	−1.13062
$557$	8.50115	0.360205	0.180103	−	0.983648i	$-0.442357\pi$
$557$	8.50115	0.360205	0.180103	+	0.983648i	$0.442357\pi$
$558$	0	0
$559$	−29.3195	−1.24008
$560$	0	0
$561$	0	0
$562$	2.78076	0.117299
$563$	23.4006	0.986220	0.493110	−	0.869967i	$-0.335860\pi$
$563$	23.4006	0.986220	0.493110	+	0.869967i	$0.335860\pi$
$564$	0	0
$565$	43.8133	1.84324
$566$	6.47252	0.272060
$567$	0	0
$568$	−18.7483	−0.786660
$569$	8.11293	0.340112	0.170056	−	0.985434i	$-0.445605\pi$
$569$	8.11293	0.340112	0.170056	+	0.985434i	$0.445605\pi$
$570$	0	0
$571$	16.5948	0.694469	0.347234	−	0.937778i	$-0.387121\pi$
$571$	16.5948	0.694469	0.347234	+	0.937778i	$0.387121\pi$
$572$	−28.3024	−1.18338
$573$	0	0
$574$	0	0
$575$	0.675938	0.0281886
$576$	0	0
$577$	10.8921	0.453445	0.226723	−	0.973959i	$-0.427199\pi$
$577$	10.8921	0.453445	0.226723	+	0.973959i	$0.427199\pi$
$578$	3.37506	0.140384
$579$	0	0
$580$	4.38168	0.181939
$581$	0	0
$582$	0	0
$583$	−1.29738	−0.0537319
$584$	−1.65237	−0.0683755
$585$	0	0
$586$	−5.39558	−0.222889
$587$	−43.3195	−1.78799	−0.893993	−	0.448081i	$-0.852107\pi$
$587$	−43.3195	−1.78799	−0.893993	+	0.448081i	$0.852107\pi$
$588$	0	0
$589$	14.9018	0.614018
$590$	−9.33879	−0.384472
$591$	0	0
$592$	12.1404	0.498965
$593$	28.4583	1.16864	0.584320	−	0.811523i	$-0.301361\pi$
$593$	28.4583	1.16864	0.584320	+	0.811523i	$0.301361\pi$
$594$	0	0
$595$	0	0
$596$	23.1261	0.947282
$597$	0	0
$598$	−7.27529	−0.297509
$599$	29.7003	1.21352	0.606761	−	0.794884i	$-0.292468\pi$
$599$	29.7003	1.21352	0.606761	+	0.794884i	$0.292468\pi$
$600$	0	0
$601$	−37.5062	−1.52991	−0.764955	−	0.644084i	$-0.777239\pi$
$601$	−37.5062	−1.52991	−0.764955	+	0.644084i	$0.777239\pi$
$602$	0	0
$603$	0	0
$604$	−20.0228	−0.814717
$605$	−35.3453	−1.43699
$606$	0	0
$607$	−2.65973	−0.107955	−0.0539776	−	0.998542i	$-0.517190\pi$
$607$	−2.65973	−0.107955	−0.0539776	+	0.998542i	$0.517190\pi$
$608$	38.5684	1.56416
$609$	0	0
$610$	−9.92536	−0.401866
$611$	3.30474	0.133695
$612$	0	0
$613$	−41.6694	−1.68301	−0.841505	−	0.540249i	$-0.818330\pi$
$613$	−41.6694	−1.68301	−0.841505	+	0.540249i	$0.818330\pi$
$614$	−0.160881	−0.00649264
$615$	0	0
$616$	0	0
$617$	43.5519	1.75333	0.876666	−	0.481099i	$-0.159762\pi$
$617$	43.5519	1.75333	0.876666	+	0.481099i	$0.159762\pi$
$618$	0	0
$619$	14.0650	0.565319	0.282660	−	0.959220i	$-0.408783\pi$
$619$	14.0650	0.565319	0.282660	+	0.959220i	$0.408783\pi$
$620$	−7.59706	−0.305105
$621$	0	0
$622$	−0.549103	−0.0220170
$623$	0	0
$624$	0	0
$625$	−24.2088	−0.968353
$626$	16.7962	0.671312
$627$	0	0
$628$	−42.0360	−1.67742
$629$	−16.2471	−0.647815
$630$	0	0
$631$	−0.594756	−0.0236769	−0.0118384	−	0.999930i	$-0.503768\pi$
$631$	−0.594756	−0.0236769	−0.0118384	+	0.999930i	$0.503768\pi$
$632$	−14.8442	−0.590470
$633$	0	0
$634$	−3.37626	−0.134088
$635$	−20.4679	−0.812245
$636$	0	0
$637$	0	0
$638$	3.14386	0.124467
$639$	0	0
$640$	−25.2641	−0.998653
$641$	7.69066	0.303763	0.151881	−	0.988399i	$-0.451467\pi$
$641$	7.69066	0.303763	0.151881	+	0.988399i	$0.451467\pi$
$642$	0	0
$643$	−49.8229	−1.96482	−0.982412	−	0.186727i	$-0.940212\pi$
$643$	−49.8229	−1.96482	−0.982412	+	0.186727i	$0.940212\pi$
$644$	0	0
$645$	0	0
$646$	−12.6861	−0.499126
$647$	28.2761	1.11165	0.555824	−	0.831300i	$-0.312403\pi$
$647$	28.2761	1.11165	0.555824	+	0.831300i	$0.312403\pi$
$648$	0	0
$649$	42.1106	1.65299
$650$	−0.253671	−0.00994980
$651$	0	0
$652$	−9.67013	−0.378712
$653$	9.55416	0.373883	0.186942	−	0.982371i	$-0.440142\pi$
$653$	9.55416	0.373883	0.186942	+	0.982371i	$0.440142\pi$
$654$	0	0
$655$	−16.6141	−0.649166
$656$	−27.8036	−1.08555
$657$	0	0
$658$	0	0
$659$	−48.1992	−1.87757	−0.938787	−	0.344499i	$-0.888049\pi$
$659$	−48.1992	−1.87757	−0.938787	+	0.344499i	$0.888049\pi$
$660$	0	0
$661$	−2.05531	−0.0799425	−0.0399712	−	0.999201i	$-0.512727\pi$
$661$	−2.05531	−0.0799425	−0.0399712	+	0.999201i	$0.512727\pi$
$662$	10.8013	0.419804
$663$	0	0
$664$	22.3527	0.867453
$665$	0	0
$666$	0	0
$667$	−5.07888	−0.196655
$668$	40.4633	1.56557
$669$	0	0
$670$	−8.77184	−0.338886
$671$	44.7556	1.72777
$672$	0	0
$673$	3.79117	0.146139	0.0730694	−	0.997327i	$-0.476721\pi$
$673$	3.79117	0.146139	0.0730694	+	0.997327i	$0.476721\pi$
$674$	−13.8647	−0.534049
$675$	0	0
$676$	5.27179	0.202761
$677$	22.5136	0.865267	0.432633	−	0.901570i	$-0.357584\pi$
$677$	22.5136	0.865267	0.432633	+	0.901570i	$0.357584\pi$
$678$	0	0
$679$	0	0
$680$	13.9640	0.535495
$681$	0	0
$682$	−5.45090	−0.208726
$683$	15.3933	0.589008	0.294504	−	0.955650i	$-0.404846\pi$
$683$	15.3933	0.589008	0.294504	+	0.955650i	$0.404846\pi$
$684$	0	0
$685$	−3.08854	−0.118007
$686$	0	0
$687$	0	0
$688$	22.5747	0.860652
$689$	−0.786567	−0.0299658
$690$	0	0
$691$	23.2088	0.882906	0.441453	−	0.897284i	$-0.354463\pi$
$691$	23.2088	0.882906	0.441453	+	0.897284i	$0.354463\pi$
$692$	38.8373	1.47637
$693$	0	0
$694$	−2.35343	−0.0893351
$695$	−34.0147	−1.29025
$696$	0	0
$697$	37.2088	1.40939
$698$	1.96907	0.0745304
$699$	0	0
$700$	0	0
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	0	0
$703$	−37.2545	−1.40508
$704$	11.1512	0.420277
$705$	0	0
$706$	10.0913	0.379791
$707$	0	0
$708$	0	0
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	−11.0789	−0.415783
$711$	0	0
$712$	18.0553	0.676652
$713$	8.80589	0.329783
$714$	0	0
$715$	−36.1106	−1.35046
$716$	11.2134	0.419066
$717$	0	0
$718$	4.77110	0.178056
$719$	−18.2111	−0.679161	−0.339580	−	0.940577i	$-0.610285\pi$
$719$	−18.2111	−0.679161	−0.339580	+	0.940577i	$0.610285\pi$
$720$	0	0
$721$	0	0
$722$	−19.1335	−0.712073
$723$	0	0
$724$	3.45090	0.128252
$725$	−0.177088	−0.00657688
$726$	0	0
$727$	−8.83682	−0.327739	−0.163870	−	0.986482i	$-0.552398\pi$
$727$	−8.83682	−0.327739	−0.163870	+	0.986482i	$0.552398\pi$
$728$	0	0
$729$	0	0
$730$	−0.976432	−0.0361394
$731$	−30.2111	−1.11740
$732$	0	0
$733$	44.6404	1.64883	0.824416	−	0.565985i	$-0.191504\pi$
$733$	44.6404	1.64883	0.824416	+	0.565985i	$0.191504\pi$
$734$	2.40754	0.0888641
$735$	0	0
$736$	22.7912	0.840094
$737$	39.5542	1.45700
$738$	0	0
$739$	11.6044	0.426875	0.213438	−	0.976957i	$-0.431534\pi$
$739$	11.6044	0.426875	0.213438	+	0.976957i	$0.431534\pi$
$740$	18.9926	0.698183
$741$	0	0
$742$	0	0
$743$	−26.7939	−0.982974	−0.491487	−	0.870885i	$-0.663546\pi$
$743$	−26.7939	−0.982974	−0.491487	+	0.870885i	$0.663546\pi$
$744$	0	0
$745$	29.5062	1.08102
$746$	11.1934	0.409818
$747$	0	0
$748$	−29.1632	−1.06631
$749$	0	0
$750$	0	0
$751$	−18.2185	−0.664802	−0.332401	−	0.943138i	$-0.607859\pi$
$751$	−18.2185	−0.664802	−0.332401	+	0.943138i	$0.607859\pi$
$752$	−2.54450	−0.0927885
$753$	0	0
$754$	1.90604	0.0694139
$755$	−25.5468	−0.929743
$756$	0	0
$757$	12.9018	0.468924	0.234462	−	0.972125i	$-0.424667\pi$
$757$	12.9018	0.468924	0.234462	+	0.972125i	$0.424667\pi$
$758$	18.4022	0.668398
$759$	0	0
$760$	32.0193	1.16146
$761$	18.2162	0.660337	0.330168	−	0.943922i	$-0.392894\pi$
$761$	18.2162	0.660337	0.330168	+	0.943922i	$0.392894\pi$
$762$	0	0
$763$	0	0
$764$	−23.3082	−0.843263
$765$	0	0
$766$	−1.67364	−0.0604710
$767$	25.5306	0.921856
$768$	0	0
$769$	−38.6044	−1.39211	−0.696055	−	0.717988i	$-0.745063\pi$
$769$	−38.6044	−1.39211	−0.696055	+	0.717988i	$0.745063\pi$
$770$	0	0
$771$	0	0
$772$	−30.6237	−1.10217
$773$	33.9594	1.22144	0.610718	−	0.791849i	$-0.290881\pi$
$773$	33.9594	1.22144	0.610718	+	0.791849i	$0.290881\pi$
$774$	0	0
$775$	0.307039	0.0110292
$776$	6.73631	0.241820
$777$	0	0
$778$	7.75794	0.278136
$779$	85.3195	3.05689
$780$	0	0
$781$	49.9571	1.78761
$782$	−7.49655	−0.268076
$783$	0	0
$784$	0	0
$785$	−53.6330	−1.91425
$786$	0	0
$787$	6.85614	0.244395	0.122198	−	0.992506i	$-0.461006\pi$
$787$	6.85614	0.244395	0.122198	+	0.992506i	$0.461006\pi$
$788$	−27.9675	−0.996301
$789$	0	0
$790$	−8.77184	−0.312088
$791$	0	0
$792$	0	0
$793$	27.1342	0.963564
$794$	14.0908	0.500065
$795$	0	0
$796$	−12.3264	−0.436896
$797$	26.6501	0.943994	0.471997	−	0.881600i	$-0.343533\pi$
$797$	26.6501	0.943994	0.471997	+	0.881600i	$0.343533\pi$
$798$	0	0
$799$	3.40524	0.120469
$800$	0.794670	0.0280958
$801$	0	0
$802$	10.1667	0.358998
$803$	4.40294	0.155377
$804$	0	0
$805$	0	0
$806$	−3.30474	−0.116404
$807$	0	0
$808$	−32.0193	−1.12644
$809$	45.7100	1.60708	0.803539	−	0.595252i	$-0.202948\pi$
$809$	45.7100	1.60708	0.803539	+	0.595252i	$0.202948\pi$
$810$	0	0
$811$	−33.2088	−1.16612	−0.583060	−	0.812429i	$-0.698145\pi$
$811$	−33.2088	−1.16612	−0.583060	+	0.812429i	$0.698145\pi$
$812$	0	0
$813$	0	0
$814$	13.6272	0.477635
$815$	−12.3380	−0.432180
$816$	0	0
$817$	−69.2738	−2.42358
$818$	−4.92306	−0.172131
$819$	0	0
$820$	−43.4966	−1.51897
$821$	−0.304740	−0.0106355	−0.00531774	−	0.999986i	$-0.501693\pi$
$821$	−0.304740	−0.0106355	−0.00531774	+	0.999986i	$0.501693\pi$
$822$	0	0
$823$	46.4177	1.61802	0.809009	−	0.587796i	$-0.200004\pi$
$823$	46.4177	1.61802	0.809009	+	0.587796i	$0.200004\pi$
$824$	2.23287	0.0777856
$825$	0	0
$826$	0	0
$827$	40.8133	1.41922	0.709608	−	0.704597i	$-0.248872\pi$
$827$	40.8133	1.41922	0.709608	+	0.704597i	$0.248872\pi$
$828$	0	0
$829$	7.40524	0.257195	0.128597	−	0.991697i	$-0.458953\pi$
$829$	7.40524	0.257195	0.128597	+	0.991697i	$0.458953\pi$
$830$	13.2088	0.458485
$831$	0	0
$832$	6.76070	0.234385
$833$	0	0
$834$	0	0
$835$	51.6265	1.78661
$836$	−66.8709	−2.31278
$837$	0	0
$838$	12.9211	0.446353
$839$	18.3720	0.634272	0.317136	−	0.948380i	$-0.397279\pi$
$839$	18.3720	0.634272	0.317136	+	0.948380i	$0.397279\pi$
$840$	0	0
$841$	−27.6694	−0.954117
$842$	7.44509	0.256575
$843$	0	0
$844$	−5.68946	−0.195839
$845$	6.72619	0.231388
$846$	0	0
$847$	0	0
$848$	0.605622	0.0207971
$849$	0	0
$850$	−0.261386	−0.00896546
$851$	−22.0147	−0.754655
$852$	0	0
$853$	−37.5615	−1.28608	−0.643041	−	0.765832i	$-0.722328\pi$
$853$	−37.5615	−1.28608	−0.643041	+	0.765832i	$0.722328\pi$
$854$	0	0
$855$	0	0
$856$	−17.7745	−0.607520
$857$	−2.15122	−0.0734843	−0.0367421	−	0.999325i	$-0.511698\pi$
$857$	−2.15122	−0.0734843	−0.0367421	+	0.999325i	$0.511698\pi$
$858$	0	0
$859$	−14.2877	−0.487491	−0.243745	−	0.969839i	$-0.578376\pi$
$859$	−14.2877	−0.487491	−0.243745	+	0.969839i	$0.578376\pi$
$860$	35.3163	1.20428
$861$	0	0
$862$	−6.81049	−0.231966
$863$	−27.6810	−0.942272	−0.471136	−	0.882061i	$-0.656156\pi$
$863$	−27.6810	−0.942272	−0.471136	+	0.882061i	$0.656156\pi$
$864$	0	0
$865$	49.5519	1.68481
$866$	−0.477581	−0.0162289
$867$	0	0
$868$	0	0
$869$	39.5542	1.34178
$870$	0	0
$871$	23.9807	0.812554
$872$	−26.8373	−0.908825
$873$	0	0
$874$	−17.1895	−0.581444
$875$	0	0
$876$	0	0
$877$	−19.0000	−0.641584	−0.320792	−	0.947150i	$-0.603949\pi$
$877$	−19.0000	−0.641584	−0.320792	+	0.947150i	$0.603949\pi$
$878$	−5.52748	−0.186543
$879$	0	0
$880$	27.8036	0.937259
$881$	−4.06498	−0.136953	−0.0684763	−	0.997653i	$-0.521814\pi$
$881$	−4.06498	−0.136953	−0.0684763	+	0.997653i	$0.521814\pi$
$882$	0	0
$883$	−20.5255	−0.690739	−0.345370	−	0.938467i	$-0.612246\pi$
$883$	−20.5255	−0.690739	−0.345370	+	0.938467i	$0.612246\pi$
$884$	−17.6809	−0.594673
$885$	0	0
$886$	−9.11753	−0.306309
$887$	10.1152	0.339636	0.169818	−	0.985475i	$-0.445682\pi$
$887$	10.1152	0.339636	0.169818	+	0.985475i	$0.445682\pi$
$888$	0	0
$889$	0	0
$890$	10.6694	0.357639
$891$	0	0
$892$	−40.0457	−1.34083
$893$	7.80819	0.261291
$894$	0	0
$895$	14.3070	0.478232
$896$	0	0
$897$	0	0
$898$	0.392079	0.0130838
$899$	−2.30704	−0.0769441
$900$	0	0
$901$	−0.810488	−0.0270013
$902$	−31.2088	−1.03914
$903$	0	0
$904$	−38.8492	−1.29211
$905$	4.40294	0.146359
$906$	0	0
$907$	22.3956	0.743633	0.371817	−	0.928306i	$-0.378735\pi$
$907$	22.3956	0.743633	0.371817	+	0.928306i	$0.378735\pi$
$908$	−20.1222	−0.667780
$909$	0	0
$910$	0	0
$911$	−33.8036	−1.11996	−0.559981	−	0.828505i	$-0.689192\pi$
$911$	−33.8036	−1.11996	−0.559981	+	0.828505i	$0.689192\pi$
$912$	0	0
$913$	−59.5615	−1.97120
$914$	0.523976	0.0173316
$915$	0	0
$916$	2.40801	0.0795628
$917$	0	0
$918$	0	0
$919$	35.6044	1.17448	0.587241	−	0.809412i	$-0.300214\pi$
$919$	35.6044	1.17448	0.587241	+	0.809412i	$0.300214\pi$
$920$	18.9211	0.623811
$921$	0	0
$922$	19.9350	0.656525
$923$	30.2877	0.996932
$924$	0	0
$925$	−0.767598	−0.0252385
$926$	13.5669	0.445835
$927$	0	0
$928$	−5.97102	−0.196008
$929$	32.4126	1.06342	0.531712	−	0.846926i	$-0.321549\pi$
$929$	32.4126	1.06342	0.531712	+	0.846926i	$0.321549\pi$
$930$	0	0
$931$	0	0
$932$	−15.0987	−0.494573
$933$	0	0
$934$	−8.94044	−0.292540
$935$	−37.2088	−1.21686
$936$	0	0
$937$	4.34303	0.141881	0.0709403	−	0.997481i	$-0.477400\pi$
$937$	4.34303	0.141881	0.0709403	+	0.997481i	$0.477400\pi$
$938$	0	0
$939$	0	0
$940$	−3.98068	−0.129835
$941$	31.9401	1.04122	0.520609	−	0.853796i	$-0.325705\pi$
$941$	31.9401	1.04122	0.520609	+	0.853796i	$0.325705\pi$
$942$	0	0
$943$	50.4177	1.64183
$944$	−19.6574	−0.639795
$945$	0	0
$946$	25.3395	0.823859
$947$	−40.5136	−1.31651	−0.658257	−	0.752793i	$-0.728706\pi$
$947$	−40.5136	−1.31651	−0.658257	+	0.752793i	$0.728706\pi$
$948$	0	0
$949$	2.66939	0.0866522
$950$	−0.599355	−0.0194456
$951$	0	0
$952$	0	0
$953$	9.74828	0.315778	0.157889	−	0.987457i	$-0.449531\pi$
$953$	9.74828	0.315778	0.157889	+	0.987457i	$0.449531\pi$
$954$	0	0
$955$	−29.7386	−0.962319
$956$	−8.11413	−0.262430
$957$	0	0
$958$	−1.47022	−0.0475006
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	8.26185	0.266373
$963$	0	0
$964$	−7.94469	−0.255881
$965$	−39.0723	−1.25778
$966$	0	0
$967$	29.6930	0.954861	0.477431	−	0.878669i	$-0.341568\pi$
$967$	29.6930	0.954861	0.477431	+	0.878669i	$0.341568\pi$
$968$	31.3407	1.00733
$969$	0	0
$970$	3.98068	0.127812
$971$	−31.9954	−1.02678	−0.513391	−	0.858155i	$-0.671611\pi$
$971$	−31.9954	−1.02678	−0.513391	+	0.858155i	$0.671611\pi$
$972$	0	0
$973$	0	0
$974$	3.40910	0.109235
$975$	0	0
$976$	−20.8921	−0.668741
$977$	1.80819	0.0578491	0.0289245	−	0.999582i	$-0.490792\pi$
$977$	1.80819	0.0578491	0.0289245	+	0.999582i	$0.490792\pi$
$978$	0	0
$979$	−48.1106	−1.53762
$980$	0	0
$981$	0	0
$982$	2.41767	0.0771509
$983$	−43.5468	−1.38893	−0.694464	−	0.719528i	$-0.744358\pi$
$983$	−43.5468	−1.38893	−0.694464	+	0.719528i	$0.744358\pi$
$984$	0	0
$985$	−35.6833	−1.13696
$986$	1.96401	0.0625468
$987$	0	0
$988$	−40.5421	−1.28982
$989$	−40.9358	−1.30168
$990$	0	0
$991$	−33.1010	−1.05149	−0.525743	−	0.850643i	$-0.676213\pi$
$991$	−33.1010	−1.05149	−0.525743	+	0.850643i	$0.676213\pi$
$992$	10.3527	0.328698
$993$	0	0
$994$	0	0
$995$	−15.7270	−0.498580
$996$	0	0
$997$	0.539441	0.0170843	0.00854214	−	0.999964i	$-0.497281\pi$
$997$	0.539441	0.0170843	0.00854214	+	0.999964i	$0.497281\pi$
$998$	18.5994	0.588752
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.o.1.2		3
3.2	odd	2		3969.2.a.n.1.2		3
7.6	odd	2		567.2.a.e.1.2	✓	3
21.20	even	2		567.2.a.f.1.2	yes	3
28.27	even	2		9072.2.a.bu.1.3		3
63.13	odd	6		567.2.f.m.379.2		6
63.20	even	6		567.2.f.l.190.2		6
63.34	odd	6		567.2.f.m.190.2		6
63.41	even	6		567.2.f.l.379.2		6
84.83	odd	2		9072.2.a.cb.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.a.e.1.2	✓	3	7.6	odd	2
567.2.a.f.1.2	yes	3	21.20	even	2
567.2.f.l.190.2		6	63.20	even	6
567.2.f.l.379.2		6	63.41	even	6
567.2.f.m.190.2		6	63.34	odd	6
567.2.f.m.379.2		6	63.13	odd	6
3969.2.a.n.1.2		3	3.2	odd	2
3969.2.a.o.1.2		3	1.1	even	1	trivial
9072.2.a.bu.1.3		3	28.27	even	2
9072.2.a.cb.1.1		3	84.83	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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q-0.523976 q^{2} -1.72545 q^{4} -2.20147 q^{5} +1.95205 q^{8} +1.15352 q^{10} -5.20147 q^{11} -3.15352 q^{13} +2.42807 q^{16} -3.24943 q^{17} -7.45090 q^{19} +3.79853 q^{20} +2.72545 q^{22} -4.40294 q^{23} -0.153520 q^{25} +1.65237 q^{26} +1.15352 q^{29} -2.00000 q^{31} -5.17635 q^{32} +1.70262 q^{34} +5.00000 q^{37} +3.90409 q^{38} -4.29738 q^{40} -11.4509 q^{41} +9.29738 q^{43} +8.97487 q^{44} +2.30704 q^{46} -1.04795 q^{47} +0.0804406 q^{50} +5.44124 q^{52} +0.249425 q^{53} +11.4509 q^{55} -0.604417 q^{58} -8.09591 q^{59} -8.60442 q^{61} +1.04795 q^{62} -2.14386 q^{64} +6.94239 q^{65} -7.60442 q^{67} +5.60672 q^{68} -9.60442 q^{71} -0.846480 q^{73} -2.61988 q^{74} +12.8561 q^{76} -7.60442 q^{79} -5.34533 q^{80} +6.00000 q^{82} +11.4509 q^{83} +7.15352 q^{85} -4.87161 q^{86} -10.1535 q^{88} +9.24943 q^{89} +7.59706 q^{92} +0.549103 q^{94} +16.4029 q^{95} +3.45090 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} + 3 q^{5} + 9 q^{8} - 3 q^{10} - 6 q^{11} - 3 q^{13} + 12 q^{16} + 3 q^{17} + 21 q^{20} - 3 q^{22} + 6 q^{23} + 6 q^{25} - 27 q^{26} - 3 q^{29} - 6 q^{31} + 18 q^{32} + 21 q^{34} + 15 q^{37}+ \cdots - 12 q^{97}+O(q^{100}) \) 3 * q + 6 * q^4 + 3 * q^5 + 9 * q^8 - 3 * q^10 - 6 * q^11 - 3 * q^13 + 12 * q^16 + 3 * q^17 + 21 * q^20 - 3 * q^22 + 6 * q^23 + 6 * q^25 - 27 * q^26 - 3 * q^29 - 6 * q^31 + 18 * q^32 + 21 * q^34 + 15 * q^37 + 18 * q^38 + 3 * q^40 - 12 * q^41 + 12 * q^43 + 3 * q^44 - 6 * q^46 - 27 * q^50 - 9 * q^52 - 12 * q^53 + 12 * q^55 + 27 * q^58 - 18 * q^59 + 3 * q^61 + 3 * q^64 + 21 * q^65 + 6 * q^67 + 39 * q^68 - 9 * q^73 + 48 * q^76 + 6 * q^79 + 3 * q^80 + 18 * q^82 + 12 * q^83 + 15 * q^85 - 45 * q^86 - 24 * q^88 + 15 * q^89 + 42 * q^92 + 24 * q^94 + 30 * q^95 - 12 * q^97

Introduction

L-functions

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