LMFDB - Embedded newform 3969.2.a.ba.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3969,2,Mod(1,3969)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3969.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-2,0,4,4,0,0,-3,0,-7,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$31.6926245622$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.574857.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 9x^{2} + 3x - 3 $ x^5 - 2x^4 - 5x^3 + 9x^2 + 3x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.05365$ 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)

$f(q)$	$=$	$q+2.05365 q^{2} +2.21746 q^{4} +0.146246 q^{5} +0.446582 q^{8} +0.300337 q^{10} +1.66404 q^{11} +0.199891 q^{13} -3.51780 q^{16} +6.27110 q^{17} +6.91758 q^{19} +0.324294 q^{20} +3.41735 q^{22} -6.18184 q^{23} -4.97861 q^{25} +0.410505 q^{26} +4.93514 q^{29} +2.51780 q^{31} -8.11747 q^{32} +12.8786 q^{34} +7.00046 q^{37} +14.2062 q^{38} +0.0653107 q^{40} +2.31790 q^{41} +1.88199 q^{43} +3.68994 q^{44} -12.6953 q^{46} +1.81177 q^{47} -10.2243 q^{50} +0.443250 q^{52} +5.34614 q^{53} +0.243359 q^{55} +10.1350 q^{58} +4.57099 q^{59} +0.678276 q^{61} +5.17066 q^{62} -9.63481 q^{64} +0.0292332 q^{65} -6.18684 q^{67} +13.9059 q^{68} +1.27749 q^{71} -1.55721 q^{73} +14.3765 q^{74} +15.3394 q^{76} +12.7957 q^{79} -0.514462 q^{80} +4.76015 q^{82} +7.51374 q^{83} +0.917122 q^{85} +3.86493 q^{86} +0.743131 q^{88} +9.06788 q^{89} -13.7080 q^{92} +3.72074 q^{94} +1.01167 q^{95} -7.97028 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 4 q^{4} + 4 q^{5} - 3 q^{8} - 7 q^{10} - 4 q^{11} - 8 q^{13} - 2 q^{16} + 12 q^{17} + q^{19} + 5 q^{20} + q^{22} - 3 q^{23} + q^{25} + 11 q^{26} - 7 q^{29} - 3 q^{31} + 2 q^{32} + 3 q^{34}+ \cdots - 12 q^{97}+O(q^{100}) $ 5 * q - 2 * q^2 + 4 * q^4 + 4 * q^5 - 3 * q^8 - 7 * q^10 - 4 * q^11 - 8 * q^13 - 2 * q^16 + 12 * q^17 + q^19 + 5 * q^20 + q^22 - 3 * q^23 + q^25 + 11 * q^26 - 7 * q^29 - 3 * q^31 + 2 * q^32 + 3 * q^34 + 20 * q^38 - 3 * q^40 + 5 * q^41 + 7 * q^43 + 10 * q^44 - 3 * q^46 + 27 * q^47 - 19 * q^50 - 10 * q^52 + 21 * q^53 - 2 * q^55 + 10 * q^58 + 30 * q^59 - 14 * q^61 + 6 * q^62 - 25 * q^64 + 11 * q^65 + 2 * q^67 + 27 * q^68 - 3 * q^71 + 15 * q^73 + 36 * q^74 + 5 * q^76 + 4 * q^79 + 20 * q^80 - 5 * q^82 + 9 * q^83 + 6 * q^85 + 8 * q^86 + 18 * q^88 + 28 * q^89 - 27 * q^92 - 3 * q^94 + 14 * 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 $

$2$	2.05365	1.45215	0.726073	−	0.687617i	$-0.241343\pi$
$2$	2.05365	1.45215	0.726073	+	0.687617i	$0.241343\pi$
$3$	0	0
$3$	0	0
$4$	2.21746	1.10873
$5$	0.146246	0.0654030	0.0327015	−	0.999465i	$-0.489589\pi$
$5$	0.146246	0.0654030	0.0327015	+	0.999465i	$0.489589\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0.446582	0.157891
$9$	0	0
$10$	0.300337	0.0949748
$11$	1.66404	0.501727	0.250864	−	0.968022i	$-0.419285\pi$
$11$	1.66404	0.501727	0.250864	+	0.968022i	$0.419285\pi$
$12$	0	0
$13$	0.199891	0.0554397	0.0277199	−	0.999616i	$-0.491175\pi$
$13$	0.199891	0.0554397	0.0277199	+	0.999616i	$0.491175\pi$
$14$	0	0
$15$	0	0
$16$	−3.51780	−0.879449
$17$	6.27110	1.52097	0.760483	−	0.649358i	$-0.224962\pi$
$17$	6.27110	1.52097	0.760483	+	0.649358i	$0.224962\pi$
$18$	0	0
$19$	6.91758	1.58700	0.793500	−	0.608570i	$-0.208256\pi$
$19$	6.91758	1.58700	0.793500	+	0.608570i	$0.208256\pi$
$20$	0.324294	0.0725143
$21$	0	0
$22$	3.41735	0.728581
$23$	−6.18184	−1.28900	−0.644501	−	0.764604i	$-0.722935\pi$
$23$	−6.18184	−1.28900	−0.644501	+	0.764604i	$0.722935\pi$
$24$	0	0
$25$	−4.97861	−0.995722
$26$	0.410505	0.0805066
$27$	0	0
$28$	0	0
$29$	4.93514	0.916433	0.458217	−	0.888841i	$-0.348488\pi$
$29$	4.93514	0.916433	0.458217	+	0.888841i	$0.348488\pi$
$30$	0	0
$31$	2.51780	0.452209	0.226105	−	0.974103i	$-0.427401\pi$
$31$	2.51780	0.452209	0.226105	+	0.974103i	$0.427401\pi$
$32$	−8.11747	−1.43498
$33$	0	0
$34$	12.8786	2.20867
$35$	0	0
$36$	0	0
$37$	7.00046	1.15087	0.575434	−	0.817848i	$-0.304833\pi$
$37$	7.00046	1.15087	0.575434	+	0.817848i	$0.304833\pi$
$38$	14.2062	2.30456
$39$	0	0
$40$	0.0653107	0.0103265
$41$	2.31790	0.361996	0.180998	−	0.983483i	$-0.442067\pi$
$41$	2.31790	0.361996	0.180998	+	0.983483i	$0.442067\pi$
$42$	0	0
$43$	1.88199	0.287000	0.143500	−	0.989650i	$-0.454164\pi$
$43$	1.88199	0.287000	0.143500	+	0.989650i	$0.454164\pi$
$44$	3.68994	0.556280
$45$	0	0
$46$	−12.6953	−1.87182
$47$	1.81177	0.264275	0.132137	−	0.991231i	$-0.457816\pi$
$47$	1.81177	0.264275	0.132137	+	0.991231i	$0.457816\pi$
$48$	0	0
$49$	0	0
$50$	−10.2243	−1.44593
$51$	0	0
$52$	0.443250	0.0614677
$53$	5.34614	0.734348	0.367174	−	0.930152i	$-0.380325\pi$
$53$	5.34614	0.734348	0.367174	+	0.930152i	$0.380325\pi$
$54$	0	0
$55$	0.243359	0.0328145
$56$	0	0
$57$	0	0
$58$	10.1350	1.33080
$59$	4.57099	0.595092	0.297546	−	0.954708i	$-0.403832\pi$
$59$	4.57099	0.595092	0.297546	+	0.954708i	$0.403832\pi$
$60$	0	0
$61$	0.678276	0.0868443	0.0434221	−	0.999057i	$-0.486174\pi$
$61$	0.678276	0.0868443	0.0434221	+	0.999057i	$0.486174\pi$
$62$	5.17066	0.656674
$63$	0	0
$64$	−9.63481	−1.20435
$65$	0.0292332	0.00362593
$66$	0	0
$67$	−6.18684	−0.755842	−0.377921	−	0.925838i	$-0.623361\pi$
$67$	−6.18684	−0.755842	−0.377921	+	0.925838i	$0.623361\pi$
$68$	13.9059	1.68634
$69$	0	0
$70$	0	0
$71$	1.27749	0.151611	0.0758053	−	0.997123i	$-0.475847\pi$
$71$	1.27749	0.151611	0.0758053	+	0.997123i	$0.475847\pi$
$72$	0	0
$73$	−1.55721	−0.182257	−0.0911286	−	0.995839i	$-0.529047\pi$
$73$	−1.55721	−0.182257	−0.0911286	+	0.995839i	$0.529047\pi$
$74$	14.3765	1.67123
$75$	0	0
$76$	15.3394	1.75955
$77$	0	0
$78$	0	0
$79$	12.7957	1.43963	0.719817	−	0.694164i	$-0.244226\pi$
$79$	12.7957	1.43963	0.719817	+	0.694164i	$0.244226\pi$
$80$	−0.514462	−0.0575186
$81$	0	0
$82$	4.76015	0.525671
$83$	7.51374	0.824740	0.412370	−	0.911016i	$-0.364701\pi$
$83$	7.51374	0.824740	0.412370	+	0.911016i	$0.364701\pi$
$84$	0	0
$85$	0.917122	0.0994758
$86$	3.86493	0.416766
$87$	0	0
$88$	0.743131	0.0792181
$89$	9.06788	0.961193	0.480597	−	0.876942i	$-0.340420\pi$
$89$	9.06788	0.961193	0.480597	+	0.876942i	$0.340420\pi$
$90$	0	0
$91$	0	0
$92$	−13.7080	−1.42915
$93$	0	0
$94$	3.72074	0.383765
$95$	1.01167	0.103795
$96$	0	0
$97$	−7.97028	−0.809259	−0.404630	−	0.914481i	$-0.632600\pi$
$97$	−7.97028	−0.809259	−0.404630	+	0.914481i	$0.632600\pi$
$98$	0	0
$99$	0	0
$100$	−11.0399	−1.10399
$101$	−14.8430	−1.47693	−0.738467	−	0.674290i	$-0.764450\pi$
$101$	−14.8430	−1.47693	−0.738467	+	0.674290i	$0.764450\pi$
$102$	0	0
$103$	0.203948	0.0200956	0.0100478	−	0.999950i	$-0.496802\pi$
$103$	0.203948	0.0200956	0.0100478	+	0.999950i	$0.496802\pi$
$104$	0.0892677	0.00875342
$105$	0	0
$106$	10.9791	1.06638
$107$	−6.96889	−0.673708	−0.336854	−	0.941557i	$-0.609363\pi$
$107$	−6.96889	−0.673708	−0.336854	+	0.941557i	$0.609363\pi$
$108$	0	0
$109$	−6.66116	−0.638024	−0.319012	−	0.947751i	$-0.603351\pi$
$109$	−6.66116	−0.638024	−0.319012	+	0.947751i	$0.603351\pi$
$110$	0.499772	0.0476514
$111$	0	0
$112$	0	0
$113$	0.0386468	0.00363558	0.00181779	−	0.999998i	$-0.499421\pi$
$113$	0.0386468	0.00363558	0.00181779	+	0.999998i	$0.499421\pi$
$114$	0	0
$115$	−0.904067	−0.0843047
$116$	10.9435	1.01608
$117$	0	0
$118$	9.38718	0.864160
$119$	0	0
$120$	0	0
$121$	−8.23097	−0.748270
$122$	1.39294	0.126111
$123$	0	0
$124$	5.58311	0.501378
$125$	−1.45933	−0.130526
$126$	0	0
$127$	13.4788	1.19605	0.598027	−	0.801476i	$-0.295952\pi$
$127$	13.4788	1.19605	0.598027	+	0.801476i	$0.295952\pi$
$128$	−3.55154	−0.313915
$129$	0	0
$130$	0.0600345	0.00526538
$131$	19.8333	1.73284	0.866422	−	0.499312i	$-0.166414\pi$
$131$	19.8333	1.73284	0.866422	+	0.499312i	$0.166414\pi$
$132$	0	0
$133$	0	0
$134$	−12.7056	−1.09759
$135$	0	0
$136$	2.80056	0.240146
$137$	−6.44509	−0.550642	−0.275321	−	0.961352i	$-0.588784\pi$
$137$	−6.44509	−0.550642	−0.275321	+	0.961352i	$0.588784\pi$
$138$	0	0
$139$	12.5305	1.06283	0.531413	−	0.847113i	$-0.321661\pi$
$139$	12.5305	1.06283	0.531413	+	0.847113i	$0.321661\pi$
$140$	0	0
$141$	0	0
$142$	2.62352	0.220161
$143$	0.332626	0.0278156
$144$	0	0
$145$	0.721743	0.0599375
$146$	−3.19795	−0.264664
$147$	0	0
$148$	15.5232	1.27600
$149$	17.7673	1.45555	0.727776	−	0.685815i	$-0.240554\pi$
$149$	17.7673	1.45555	0.727776	+	0.685815i	$0.240554\pi$
$150$	0	0
$151$	8.46599	0.688953	0.344476	−	0.938795i	$-0.388056\pi$
$151$	8.46599	0.688953	0.344476	+	0.938795i	$0.388056\pi$
$152$	3.08927	0.250573
$153$	0	0
$154$	0	0
$155$	0.368217	0.0295759
$156$	0	0
$157$	−5.69935	−0.454858	−0.227429	−	0.973795i	$-0.573032\pi$
$157$	−5.69935	−0.454858	−0.227429	+	0.973795i	$0.573032\pi$
$158$	26.2779	2.09056
$159$	0	0
$160$	−1.18714	−0.0938520
$161$	0	0
$162$	0	0
$163$	2.12535	0.166470	0.0832349	−	0.996530i	$-0.473475\pi$
$163$	2.12535	0.166470	0.0832349	+	0.996530i	$0.473475\pi$
$164$	5.13986	0.401355
$165$	0	0
$166$	15.4306	1.19764
$167$	−11.5745	−0.895659	−0.447829	−	0.894119i	$-0.647803\pi$
$167$	−11.5745	−0.895659	−0.447829	+	0.894119i	$0.647803\pi$
$168$	0	0
$169$	−12.9600	−0.996926
$170$	1.88344	0.144453
$171$	0	0
$172$	4.17323	0.318206
$173$	15.9109	1.20968	0.604842	−	0.796345i	$-0.293236\pi$
$173$	15.9109	1.20968	0.604842	+	0.796345i	$0.293236\pi$
$174$	0	0
$175$	0	0
$176$	−5.85375	−0.441243
$177$	0	0
$178$	18.6222	1.39579
$179$	−7.75331	−0.579509	−0.289755	−	0.957101i	$-0.593574\pi$
$179$	−7.75331	−0.579509	−0.289755	+	0.957101i	$0.593574\pi$
$180$	0	0
$181$	12.1618	0.903982	0.451991	−	0.892022i	$-0.350714\pi$
$181$	12.1618	0.903982	0.451991	+	0.892022i	$0.350714\pi$
$182$	0	0
$183$	0	0
$184$	−2.76070	−0.203521
$185$	1.02379	0.0752703
$186$	0	0
$187$	10.4354	0.763110
$188$	4.01754	0.293009
$189$	0	0
$190$	2.07760	0.150725
$191$	−4.96765	−0.359447	−0.179723	−	0.983717i	$-0.557520\pi$
$191$	−4.96765	−0.359447	−0.179723	+	0.983717i	$0.557520\pi$
$192$	0	0
$193$	−14.9044	−1.07284	−0.536422	−	0.843950i	$-0.680224\pi$
$193$	−14.9044	−1.07284	−0.536422	+	0.843950i	$0.680224\pi$
$194$	−16.3681	−1.17516
$195$	0	0
$196$	0	0
$197$	−21.2608	−1.51477	−0.757386	−	0.652968i	$-0.773524\pi$
$197$	−21.2608	−1.51477	−0.757386	+	0.652968i	$0.773524\pi$
$198$	0	0
$199$	−19.9442	−1.41380	−0.706902	−	0.707311i	$-0.749908\pi$
$199$	−19.9442	−1.41380	−0.706902	+	0.707311i	$0.749908\pi$
$200$	−2.22336	−0.157215
$201$	0	0
$202$	−30.4823	−2.14472
$203$	0	0
$204$	0	0
$205$	0.338983	0.0236756
$206$	0.418838	0.0291818
$207$	0	0
$208$	−0.703175	−0.0487564
$209$	11.5111	0.796241
$210$	0	0
$211$	−23.5139	−1.61876	−0.809381	−	0.587284i	$-0.800197\pi$
$211$	−23.5139	−1.61876	−0.809381	+	0.587284i	$0.800197\pi$
$212$	11.8548	0.814193
$213$	0	0
$214$	−14.3116	−0.978323
$215$	0.275232	0.0187707
$216$	0	0
$217$	0	0
$218$	−13.6797	−0.926504
$219$	0	0
$220$	0.539638	0.0363824
$221$	1.25354	0.0843220
$222$	0	0
$223$	4.06104	0.271947	0.135974	−	0.990712i	$-0.456584\pi$
$223$	4.06104	0.271947	0.135974	+	0.990712i	$0.456584\pi$
$224$	0	0
$225$	0	0
$226$	0.0793667	0.00527940
$227$	3.85285	0.255723	0.127861	−	0.991792i	$-0.459189\pi$
$227$	3.85285	0.255723	0.127861	+	0.991792i	$0.459189\pi$
$228$	0	0
$229$	−13.1162	−0.866746	−0.433373	−	0.901215i	$-0.642677\pi$
$229$	−13.1162	−0.866746	−0.433373	+	0.901215i	$0.642677\pi$
$230$	−1.85663	−0.122423
$231$	0	0
$232$	2.20395	0.144696
$233$	17.5023	1.14661	0.573307	−	0.819340i	$-0.305660\pi$
$233$	17.5023	1.14661	0.573307	+	0.819340i	$0.305660\pi$
$234$	0	0
$235$	0.264964	0.0172844
$236$	10.1360	0.659795
$237$	0	0
$238$	0	0
$239$	−7.31714	−0.473306	−0.236653	−	0.971594i	$-0.576051\pi$
$239$	−7.31714	−0.473306	−0.236653	+	0.971594i	$0.576051\pi$
$240$	0	0
$241$	−6.23107	−0.401378	−0.200689	−	0.979655i	$-0.564318\pi$
$241$	−6.23107	−0.401378	−0.200689	+	0.979655i	$0.564318\pi$
$242$	−16.9035	−1.08660
$243$	0	0
$244$	1.50405	0.0962868
$245$	0	0
$246$	0	0
$247$	1.38276	0.0879829
$248$	1.12440	0.0713997
$249$	0	0
$250$	−2.99694	−0.189543
$251$	−5.65283	−0.356803	−0.178402	−	0.983958i	$-0.557093\pi$
$251$	−5.65283	−0.356803	−0.178402	+	0.983958i	$0.557093\pi$
$252$	0	0
$253$	−10.2868	−0.646727
$254$	27.6808	1.73684
$255$	0	0
$256$	11.9760	0.748501
$257$	−11.8016	−0.736166	−0.368083	−	0.929793i	$-0.619986\pi$
$257$	−11.8016	−0.736166	−0.368083	+	0.929793i	$0.619986\pi$
$258$	0	0
$259$	0	0
$260$	0.0648233	0.00402017
$261$	0	0
$262$	40.7306	2.51634
$263$	−22.2401	−1.37138	−0.685691	−	0.727893i	$-0.740500\pi$
$263$	−22.2401	−1.37138	−0.685691	+	0.727893i	$0.740500\pi$
$264$	0	0
$265$	0.781849	0.0480286
$266$	0	0
$267$	0	0
$268$	−13.7191	−0.838024
$269$	−2.38884	−0.145650	−0.0728251	−	0.997345i	$-0.523201\pi$
$269$	−2.38884	−0.145650	−0.0728251	+	0.997345i	$0.523201\pi$
$270$	0	0
$271$	−23.2258	−1.41087	−0.705435	−	0.708775i	$-0.749248\pi$
$271$	−23.2258	−1.41087	−0.705435	+	0.708775i	$0.749248\pi$
$272$	−22.0605	−1.33761
$273$	0	0
$274$	−13.2359	−0.799612
$275$	−8.28461	−0.499581
$276$	0	0
$277$	−4.61800	−0.277469	−0.138734	−	0.990330i	$-0.544303\pi$
$277$	−4.61800	−0.277469	−0.138734	+	0.990330i	$0.544303\pi$
$278$	25.7333	1.54338
$279$	0	0
$280$	0	0
$281$	−11.8168	−0.704933	−0.352466	−	0.935825i	$-0.614657\pi$
$281$	−11.8168	−0.704933	−0.352466	+	0.935825i	$0.614657\pi$
$282$	0	0
$283$	−15.8497	−0.942165	−0.471082	−	0.882089i	$-0.656137\pi$
$283$	−15.8497	−0.942165	−0.471082	+	0.882089i	$0.656137\pi$
$284$	2.83279	0.168095
$285$	0	0
$286$	0.683097	0.0403924
$287$	0	0
$288$	0	0
$289$	22.3267	1.31334
$290$	1.48220	0.0870381
$291$	0	0
$292$	−3.45304	−0.202074
$293$	14.0961	0.823502	0.411751	−	0.911296i	$-0.364917\pi$
$293$	14.0961	0.823502	0.411751	+	0.911296i	$0.364917\pi$
$294$	0	0
$295$	0.668487	0.0389208
$296$	3.12628	0.181711
$297$	0	0
$298$	36.4877	2.11367
$299$	−1.23569	−0.0714619
$300$	0	0
$301$	0	0
$302$	17.3861	1.00046
$303$	0	0
$304$	−24.3346	−1.39569
$305$	0.0991949	0.00567988
$306$	0	0
$307$	−27.3916	−1.56332	−0.781660	−	0.623704i	$-0.785627\pi$
$307$	−27.3916	−1.56332	−0.781660	+	0.623704i	$0.785627\pi$
$308$	0	0
$309$	0	0
$310$	0.756186	0.0429485
$311$	14.0557	0.797026	0.398513	−	0.917163i	$-0.369526\pi$
$311$	14.0557	0.797026	0.398513	+	0.917163i	$0.369526\pi$
$312$	0	0
$313$	−21.7446	−1.22908	−0.614540	−	0.788886i	$-0.710658\pi$
$313$	−21.7446	−1.22908	−0.614540	+	0.788886i	$0.710658\pi$
$314$	−11.7045	−0.660520
$315$	0	0
$316$	28.3740	1.59616
$317$	8.56297	0.480944	0.240472	−	0.970656i	$-0.422698\pi$
$317$	8.56297	0.480944	0.240472	+	0.970656i	$0.422698\pi$
$318$	0	0
$319$	8.21228	0.459799
$320$	−1.40905	−0.0787682
$321$	0	0
$322$	0	0
$323$	43.3808	2.41377
$324$	0	0
$325$	−0.995179	−0.0552026
$326$	4.36471	0.241739
$327$	0	0
$328$	1.03514	0.0571558
$329$	0	0
$330$	0	0
$331$	10.8472	0.596216	0.298108	−	0.954532i	$-0.403644\pi$
$331$	10.8472	0.596216	0.298108	+	0.954532i	$0.403644\pi$
$332$	16.6614	0.914413
$333$	0	0
$334$	−23.7698	−1.30063
$335$	−0.904798	−0.0494344
$336$	0	0
$337$	−3.34822	−0.182389	−0.0911945	−	0.995833i	$-0.529069\pi$
$337$	−3.34822	−0.182389	−0.0911945	+	0.995833i	$0.529069\pi$
$338$	−26.6153	−1.44768
$339$	0	0
$340$	2.03368	0.110292
$341$	4.18971	0.226886
$342$	0	0
$343$	0	0
$344$	0.840462	0.0453147
$345$	0	0
$346$	32.6754	1.75664
$347$	−11.5330	−0.619126	−0.309563	−	0.950879i	$-0.600183\pi$
$347$	−11.5330	−0.619126	−0.309563	+	0.950879i	$0.600183\pi$
$348$	0	0
$349$	−8.89834	−0.476317	−0.238159	−	0.971226i	$-0.576544\pi$
$349$	−8.89834	−0.476317	−0.238159	+	0.971226i	$0.576544\pi$
$350$	0	0
$351$	0	0
$352$	−13.5078	−0.719968
$353$	2.64699	0.140885	0.0704424	−	0.997516i	$-0.477559\pi$
$353$	2.64699	0.140885	0.0704424	+	0.997516i	$0.477559\pi$
$354$	0	0
$355$	0.186828	0.00991579
$356$	20.1076	1.06570
$357$	0	0
$358$	−15.9225	−0.841533
$359$	25.9671	1.37049	0.685245	−	0.728312i	$-0.259695\pi$
$359$	25.9671	1.37049	0.685245	+	0.728312i	$0.259695\pi$
$360$	0	0
$361$	28.8529	1.51857
$362$	24.9761	1.31271
$363$	0	0
$364$	0	0
$365$	−0.227735	−0.0119202
$366$	0	0
$367$	−17.5874	−0.918056	−0.459028	−	0.888422i	$-0.651802\pi$
$367$	−17.5874	−0.918056	−0.459028	+	0.888422i	$0.651802\pi$
$368$	21.7464	1.13361
$369$	0	0
$370$	2.10249	0.109303
$371$	0	0
$372$	0	0
$373$	0.815075	0.0422030	0.0211015	−	0.999777i	$-0.493283\pi$
$373$	0.815075	0.0422030	0.0211015	+	0.999777i	$0.493283\pi$
$374$	21.4306	1.10815
$375$	0	0
$376$	0.809107	0.0417265
$377$	0.986490	0.0508068
$378$	0	0
$379$	−20.4312	−1.04948	−0.524741	−	0.851262i	$-0.675838\pi$
$379$	−20.4312	−1.04948	−0.524741	+	0.851262i	$0.675838\pi$
$380$	2.24333	0.115080
$381$	0	0
$382$	−10.2018	−0.521969
$383$	−17.8928	−0.914278	−0.457139	−	0.889395i	$-0.651126\pi$
$383$	−17.8928	−0.914278	−0.457139	+	0.889395i	$0.651126\pi$
$384$	0	0
$385$	0	0
$386$	−30.6084	−1.55793
$387$	0	0
$388$	−17.6738	−0.897249
$389$	15.6278	0.792363	0.396181	−	0.918172i	$-0.370335\pi$
$389$	15.6278	0.792363	0.396181	+	0.918172i	$0.370335\pi$
$390$	0	0
$391$	−38.7669	−1.96053
$392$	0	0
$393$	0	0
$394$	−43.6622	−2.19967
$395$	1.87132	0.0941564
$396$	0	0
$397$	19.2613	0.966696	0.483348	−	0.875428i	$-0.339421\pi$
$397$	19.2613	0.966696	0.483348	+	0.875428i	$0.339421\pi$
$398$	−40.9582	−2.05305
$399$	0	0
$400$	17.5137	0.875687
$401$	14.3013	0.714172	0.357086	−	0.934072i	$-0.383770\pi$
$401$	14.3013	0.714172	0.357086	+	0.934072i	$0.383770\pi$
$402$	0	0
$403$	0.503284	0.0250704
$404$	−32.9137	−1.63752
$405$	0	0
$406$	0	0
$407$	11.6490	0.577422
$408$	0	0
$409$	−31.8610	−1.57542	−0.787712	−	0.616044i	$-0.788734\pi$
$409$	−31.8610	−1.57542	−0.787712	+	0.616044i	$0.788734\pi$
$410$	0.696152	0.0343805
$411$	0	0
$412$	0.452247	0.0222806
$413$	0	0
$414$	0	0
$415$	1.09885	0.0539405
$416$	−1.62261	−0.0795549
$417$	0	0
$418$	23.6398	1.15626
$419$	23.8960	1.16739	0.583697	−	0.811971i	$-0.301605\pi$
$419$	23.8960	1.16739	0.583697	+	0.811971i	$0.301605\pi$
$420$	0	0
$421$	2.44501	0.119163	0.0595813	−	0.998223i	$-0.481023\pi$
$421$	2.44501	0.119163	0.0595813	+	0.998223i	$0.481023\pi$
$422$	−48.2892	−2.35068
$423$	0	0
$424$	2.38749	0.115947
$425$	−31.2214	−1.51446
$426$	0	0
$427$	0	0
$428$	−15.4532	−0.746960
$429$	0	0
$430$	0.565230	0.0272578
$431$	−4.92764	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$431$	−4.92764	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$432$	0	0
$433$	−30.8539	−1.48274	−0.741371	−	0.671095i	$-0.765824\pi$
$433$	−30.8539	−1.48274	−0.741371	+	0.671095i	$0.765824\pi$
$434$	0	0
$435$	0	0
$436$	−14.7709	−0.707395
$437$	−42.7633	−2.04565
$438$	0	0
$439$	−2.44822	−0.116847	−0.0584235	−	0.998292i	$-0.518607\pi$
$439$	−2.44822	−0.116847	−0.0584235	+	0.998292i	$0.518607\pi$
$440$	0.108680	0.00518110
$441$	0	0
$442$	2.57432	0.122448
$443$	−26.2950	−1.24931	−0.624657	−	0.780899i	$-0.714761\pi$
$443$	−26.2950	−1.24931	−0.624657	+	0.780899i	$0.714761\pi$
$444$	0	0
$445$	1.32614	0.0628650
$446$	8.33993	0.394907
$447$	0	0
$448$	0	0
$449$	−38.7077	−1.82673	−0.913365	−	0.407141i	$-0.866526\pi$
$449$	−38.7077	−1.82673	−0.913365	+	0.407141i	$0.866526\pi$
$450$	0	0
$451$	3.85709	0.181623
$452$	0.0856976	0.00403087
$453$	0	0
$454$	7.91239	0.371347
$455$	0	0
$456$	0	0
$457$	−9.15511	−0.428258	−0.214129	−	0.976805i	$-0.568691\pi$
$457$	−9.15511	−0.428258	−0.214129	+	0.976805i	$0.568691\pi$
$458$	−26.9361	−1.25864
$459$	0	0
$460$	−2.00473	−0.0934710
$461$	29.2304	1.36140	0.680698	−	0.732564i	$-0.261676\pi$
$461$	29.2304	1.36140	0.680698	+	0.732564i	$0.261676\pi$
$462$	0	0
$463$	16.4206	0.763131	0.381565	−	0.924342i	$-0.375385\pi$
$463$	16.4206	0.763131	0.381565	+	0.924342i	$0.375385\pi$
$464$	−17.3608	−0.805956
$465$	0	0
$466$	35.9435	1.66505
$467$	15.3726	0.711361	0.355680	−	0.934608i	$-0.384249\pi$
$467$	15.3726	0.711361	0.355680	+	0.934608i	$0.384249\pi$
$468$	0	0
$469$	0	0
$470$	0.544142	0.0250994
$471$	0	0
$472$	2.04132	0.0939594
$473$	3.13170	0.143996
$474$	0	0
$475$	−34.4399	−1.58021
$476$	0	0
$477$	0	0
$478$	−15.0268	−0.687310
$479$	37.9291	1.73303	0.866513	−	0.499155i	$-0.166356\pi$
$479$	37.9291	1.73303	0.866513	+	0.499155i	$0.166356\pi$
$480$	0	0
$481$	1.39933	0.0638038
$482$	−12.7964	−0.582860
$483$	0	0
$484$	−18.2518	−0.829629
$485$	−1.16562	−0.0529280
$486$	0	0
$487$	−4.60495	−0.208670	−0.104335	−	0.994542i	$-0.533271\pi$
$487$	−4.60495	−0.208670	−0.104335	+	0.994542i	$0.533271\pi$
$488$	0.302906	0.0137119
$489$	0	0
$490$	0	0
$491$	30.3751	1.37081	0.685405	−	0.728162i	$-0.259625\pi$
$491$	30.3751	1.37081	0.685405	+	0.728162i	$0.259625\pi$
$492$	0	0
$493$	30.9488	1.39386
$494$	2.83970	0.127764
$495$	0	0
$496$	−8.85709	−0.397695
$497$	0	0
$498$	0	0
$499$	9.26871	0.414925	0.207462	−	0.978243i	$-0.433480\pi$
$499$	9.26871	0.414925	0.207462	+	0.978243i	$0.433480\pi$
$500$	−3.23600	−0.144718
$501$	0	0
$502$	−11.6089	−0.518131
$503$	22.4230	0.999791	0.499896	−	0.866086i	$-0.333372\pi$
$503$	22.4230	0.999791	0.499896	+	0.866086i	$0.333372\pi$
$504$	0	0
$505$	−2.17072	−0.0965960
$506$	−21.1255	−0.939143
$507$	0	0
$508$	29.8888	1.32610
$509$	37.6414	1.66843	0.834213	−	0.551443i	$-0.185923\pi$
$509$	37.6414	1.66843	0.834213	+	0.551443i	$0.185923\pi$
$510$	0	0
$511$	0	0
$512$	31.6976	1.40085
$513$	0	0
$514$	−24.2364	−1.06902
$515$	0.0298266	0.00131432
$516$	0	0
$517$	3.01487	0.132594
$518$	0	0
$519$	0	0
$520$	0.0130550	0.000572500 0
$521$	34.9283	1.53023	0.765117	−	0.643891i	$-0.222681\pi$
$521$	34.9283	1.53023	0.765117	+	0.643891i	$0.222681\pi$
$522$	0	0
$523$	−23.7471	−1.03839	−0.519194	−	0.854656i	$-0.673768\pi$
$523$	−23.7471	−1.03839	−0.519194	+	0.854656i	$0.673768\pi$
$524$	43.9795	1.92126
$525$	0	0
$526$	−45.6732	−1.99145
$527$	15.7894	0.687795
$528$	0	0
$529$	15.2151	0.661526
$530$	1.60564	0.0697446
$531$	0	0
$532$	0	0
$533$	0.463328	0.0200690
$534$	0	0
$535$	−1.01917	−0.0440626
$536$	−2.76293	−0.119340
$537$	0	0
$538$	−4.90583	−0.211505
$539$	0	0
$540$	0	0
$541$	−17.1708	−0.738232	−0.369116	−	0.929383i	$-0.620340\pi$
$541$	−17.1708	−0.738232	−0.369116	+	0.929383i	$0.620340\pi$
$542$	−47.6976	−2.04879
$543$	0	0
$544$	−50.9055	−2.18255
$545$	−0.974166	−0.0417287
$546$	0	0
$547$	20.0091	0.855529	0.427765	−	0.903890i	$-0.359301\pi$
$547$	20.0091	0.855529	0.427765	+	0.903890i	$0.359301\pi$
$548$	−14.2917	−0.610512
$549$	0	0
$550$	−17.0137	−0.725465
$551$	34.1392	1.45438
$552$	0	0
$553$	0	0
$554$	−9.48374	−0.402926
$555$	0	0
$556$	27.7860	1.17839
$557$	0.245481	0.0104014	0.00520068	−	0.999986i	$-0.498345\pi$
$557$	0.245481	0.0104014	0.00520068	+	0.999986i	$0.498345\pi$
$558$	0	0
$559$	0.376192	0.0159112
$560$	0	0
$561$	0	0
$562$	−24.2676	−1.02367
$563$	44.2509	1.86495	0.932477	−	0.361230i	$-0.117643\pi$
$563$	44.2509	1.86495	0.932477	+	0.361230i	$0.117643\pi$
$564$	0	0
$565$	0.00565192	0.000237778 0
$566$	−32.5496	−1.36816
$567$	0	0
$568$	0.570506	0.0239379
$569$	−5.53533	−0.232053	−0.116027	−	0.993246i	$-0.537016\pi$
$569$	−5.53533	−0.232053	−0.116027	+	0.993246i	$0.537016\pi$
$570$	0	0
$571$	−4.10381	−0.171739	−0.0858696	−	0.996306i	$-0.527367\pi$
$571$	−4.10381	−0.171739	−0.0858696	+	0.996306i	$0.527367\pi$
$572$	0.737585	0.0308400
$573$	0	0
$574$	0	0
$575$	30.7770	1.28349
$576$	0	0
$577$	−5.64550	−0.235025	−0.117513	−	0.993071i	$-0.537492\pi$
$577$	−5.64550	−0.235025	−0.117513	+	0.993071i	$0.537492\pi$
$578$	45.8512	1.90716
$579$	0	0
$580$	1.60044	0.0664545
$581$	0	0
$582$	0	0
$583$	8.89619	0.368442
$584$	−0.695420	−0.0287767
$585$	0	0
$586$	28.9483	1.19585
$587$	18.7329	0.773189	0.386595	−	0.922250i	$-0.373651\pi$
$587$	18.7329	0.773189	0.386595	+	0.922250i	$0.373651\pi$
$588$	0	0
$589$	17.4170	0.717657
$590$	1.37283	0.0565187
$591$	0	0
$592$	−24.6262	−1.01213
$593$	−18.8703	−0.774912	−0.387456	−	0.921888i	$-0.626646\pi$
$593$	−18.8703	−0.774912	−0.387456	+	0.921888i	$0.626646\pi$
$594$	0	0
$595$	0	0
$596$	39.3982	1.61381
$597$	0	0
$598$	−2.53767	−0.103773
$599$	2.67451	0.109278	0.0546388	−	0.998506i	$-0.482599\pi$
$599$	2.67451	0.109278	0.0546388	+	0.998506i	$0.482599\pi$
$600$	0	0
$601$	−13.2143	−0.539023	−0.269511	−	0.962997i	$-0.586862\pi$
$601$	−13.2143	−0.539023	−0.269511	+	0.962997i	$0.586862\pi$
$602$	0	0
$603$	0	0
$604$	18.7730	0.763862
$605$	−1.20374	−0.0489391
$606$	0	0
$607$	−25.8052	−1.04740	−0.523701	−	0.851902i	$-0.675449\pi$
$607$	−25.8052	−1.04740	−0.523701	+	0.851902i	$0.675449\pi$
$608$	−56.1532	−2.27731
$609$	0	0
$610$	0.203711	0.00824802
$611$	0.362157	0.0146513
$612$	0	0
$613$	−26.9533	−1.08863	−0.544316	−	0.838880i	$-0.683211\pi$
$613$	−26.9533	−1.08863	−0.544316	+	0.838880i	$0.683211\pi$
$614$	−56.2526	−2.27017
$615$	0	0
$616$	0	0
$617$	9.53175	0.383734	0.191867	−	0.981421i	$-0.438546\pi$
$617$	9.53175	0.383734	0.191867	+	0.981421i	$0.438546\pi$
$618$	0	0
$619$	−34.7071	−1.39500	−0.697499	−	0.716586i	$-0.745704\pi$
$619$	−34.7071	−1.39500	−0.697499	+	0.716586i	$0.745704\pi$
$620$	0.816505	0.0327916
$621$	0	0
$622$	28.8654	1.15740
$623$	0	0
$624$	0	0
$625$	24.6796	0.987186
$626$	−44.6558	−1.78480
$627$	0	0
$628$	−12.6381	−0.504314
$629$	43.9006	1.75043
$630$	0	0
$631$	−36.7963	−1.46484	−0.732419	−	0.680854i	$-0.761609\pi$
$631$	−36.7963	−1.46484	−0.732419	+	0.680854i	$0.761609\pi$
$632$	5.71435	0.227305
$633$	0	0
$634$	17.5853	0.698401
$635$	1.97122	0.0782256
$636$	0	0
$637$	0	0
$638$	16.8651	0.667696
$639$	0	0
$640$	−0.519397	−0.0205310
$641$	−44.1844	−1.74518	−0.872590	−	0.488454i	$-0.837561\pi$
$641$	−44.1844	−1.74518	−0.872590	+	0.488454i	$0.837561\pi$
$642$	0	0
$643$	14.4813	0.571087	0.285543	−	0.958366i	$-0.407826\pi$
$643$	14.4813	0.571087	0.285543	+	0.958366i	$0.407826\pi$
$644$	0	0
$645$	0	0
$646$	89.0889	3.50515
$647$	−33.3071	−1.30944	−0.654719	−	0.755872i	$-0.727213\pi$
$647$	−33.3071	−1.30944	−0.654719	+	0.755872i	$0.727213\pi$
$648$	0	0
$649$	7.60631	0.298574
$650$	−2.04374	−0.0801622
$651$	0	0
$652$	4.71286	0.184570
$653$	−9.06643	−0.354797	−0.177398	−	0.984139i	$-0.556768\pi$
$653$	−9.06643	−0.354797	−0.177398	+	0.984139i	$0.556768\pi$
$654$	0	0
$655$	2.90054	0.113333
$656$	−8.15391	−0.318357
$657$	0	0
$658$	0	0
$659$	−32.3611	−1.26061	−0.630305	−	0.776348i	$-0.717070\pi$
$659$	−32.3611	−1.26061	−0.630305	+	0.776348i	$0.717070\pi$
$660$	0	0
$661$	8.65915	0.336802	0.168401	−	0.985719i	$-0.446140\pi$
$661$	8.65915	0.336802	0.168401	+	0.985719i	$0.446140\pi$
$662$	22.2763	0.865794
$663$	0	0
$664$	3.35550	0.130219
$665$	0	0
$666$	0	0
$667$	−30.5083	−1.18128
$668$	−25.6659	−0.993043
$669$	0	0
$670$	−1.85813	−0.0717860
$671$	1.12868	0.0435721
$672$	0	0
$673$	−14.4968	−0.558812	−0.279406	−	0.960173i	$-0.590138\pi$
$673$	−14.4968	−0.558812	−0.279406	+	0.960173i	$0.590138\pi$
$674$	−6.87605	−0.264856
$675$	0	0
$676$	−28.7384	−1.10532
$677$	−38.3315	−1.47320	−0.736600	−	0.676329i	$-0.763570\pi$
$677$	−38.3315	−1.47320	−0.736600	+	0.676329i	$0.763570\pi$
$678$	0	0
$679$	0	0
$680$	0.409570	0.0157063
$681$	0	0
$682$	8.60418	0.329471
$683$	6.63318	0.253812	0.126906	−	0.991915i	$-0.459495\pi$
$683$	6.63318	0.253812	0.126906	+	0.991915i	$0.459495\pi$
$684$	0	0
$685$	−0.942567	−0.0360136
$686$	0	0
$687$	0	0
$688$	−6.62044	−0.252402
$689$	1.06864	0.0407121
$690$	0	0
$691$	23.3875	0.889704	0.444852	−	0.895604i	$-0.353256\pi$
$691$	23.3875	0.889704	0.444852	+	0.895604i	$0.353256\pi$
$692$	35.2818	1.34121
$693$	0	0
$694$	−23.6848	−0.899061
$695$	1.83254	0.0695121
$696$	0	0
$697$	14.5358	0.550583
$698$	−18.2740	−0.691683
$699$	0	0
$700$	0	0
$701$	9.26736	0.350023	0.175012	−	0.984566i	$-0.444004\pi$
$701$	9.26736	0.350023	0.175012	+	0.984566i	$0.444004\pi$
$702$	0	0
$703$	48.4262	1.82643
$704$	−16.0327	−0.604256
$705$	0	0
$706$	5.43597	0.204585
$707$	0	0
$708$	0	0
$709$	14.2355	0.534626	0.267313	−	0.963610i	$-0.413864\pi$
$709$	14.2355	0.534626	0.267313	+	0.963610i	$0.413864\pi$
$710$	0.383678	0.0143992
$711$	0	0
$712$	4.04956	0.151763
$713$	−15.5646	−0.582899
$714$	0	0
$715$	0.0486452	0.00181923
$716$	−17.1926	−0.642519
$717$	0	0
$718$	53.3272	1.99015
$719$	13.8570	0.516777	0.258389	−	0.966041i	$-0.416808\pi$
$719$	13.8570	0.516777	0.258389	+	0.966041i	$0.416808\pi$
$720$	0	0
$721$	0	0
$722$	59.2536	2.20519
$723$	0	0
$724$	26.9684	1.00227
$725$	−24.5702	−0.912513
$726$	0	0
$727$	31.4000	1.16456	0.582280	−	0.812988i	$-0.302161\pi$
$727$	31.4000	1.16456	0.582280	+	0.812988i	$0.302161\pi$
$728$	0	0
$729$	0	0
$730$	−0.467686	−0.0173098
$731$	11.8021	0.436518
$732$	0	0
$733$	26.6006	0.982515	0.491257	−	0.871014i	$-0.336537\pi$
$733$	26.6006	0.982515	0.491257	+	0.871014i	$0.336537\pi$
$734$	−36.1183	−1.33315
$735$	0	0
$736$	50.1809	1.84969
$737$	−10.2951	−0.379227
$738$	0	0
$739$	−33.0039	−1.21407	−0.607034	−	0.794676i	$-0.707641\pi$
$739$	−33.0039	−1.21407	−0.607034	+	0.794676i	$0.707641\pi$
$740$	2.27020	0.0834543
$741$	0	0
$742$	0	0
$743$	−38.6015	−1.41615	−0.708076	−	0.706136i	$-0.750437\pi$
$743$	−38.6015	−1.41615	−0.708076	+	0.706136i	$0.750437\pi$
$744$	0	0
$745$	2.59839	0.0951975
$746$	1.67388	0.0612849
$747$	0	0
$748$	23.1400	0.846082
$749$	0	0
$750$	0	0
$751$	−37.8996	−1.38297	−0.691487	−	0.722389i	$-0.743044\pi$
$751$	−37.8996	−1.38297	−0.691487	+	0.722389i	$0.743044\pi$
$752$	−6.37345	−0.232416
$753$	0	0
$754$	2.02590	0.0737789
$755$	1.23811	0.0450596
$756$	0	0
$757$	22.5927	0.821147	0.410573	−	0.911828i	$-0.365329\pi$
$757$	22.5927	0.821147	0.410573	+	0.911828i	$0.365329\pi$
$758$	−41.9585	−1.52400
$759$	0	0
$760$	0.451792	0.0163882
$761$	−27.7470	−1.00583	−0.502913	−	0.864337i	$-0.667739\pi$
$761$	−27.7470	−1.00583	−0.502913	+	0.864337i	$0.667739\pi$
$762$	0	0
$763$	0	0
$764$	−11.0156	−0.398529
$765$	0	0
$766$	−36.7454	−1.32766
$767$	0.913698	0.0329917
$768$	0	0
$769$	−12.1534	−0.438262	−0.219131	−	0.975695i	$-0.570322\pi$
$769$	−12.1534	−0.438262	−0.219131	+	0.975695i	$0.570322\pi$
$770$	0	0
$771$	0	0
$772$	−33.0499	−1.18949
$773$	−41.5591	−1.49478	−0.747388	−	0.664388i	$-0.768692\pi$
$773$	−41.5591	−1.49478	−0.747388	+	0.664388i	$0.768692\pi$
$774$	0	0
$775$	−12.5351	−0.450275
$776$	−3.55939	−0.127775
$777$	0	0
$778$	32.0940	1.15063
$779$	16.0343	0.574488
$780$	0	0
$781$	2.12580	0.0760671
$782$	−79.6135	−2.84697
$783$	0	0
$784$	0	0
$785$	−0.833506	−0.0297491
$786$	0	0
$787$	20.8969	0.744893	0.372446	−	0.928054i	$-0.378519\pi$
$787$	20.8969	0.744893	0.372446	+	0.928054i	$0.378519\pi$
$788$	−47.1450	−1.67947
$789$	0	0
$790$	3.84303	0.136729
$791$	0	0
$792$	0	0
$793$	0.135581	0.00481462
$794$	39.5558	1.40378
$795$	0	0
$796$	−44.2254	−1.56753
$797$	−0.638766	−0.0226263	−0.0113131	−	0.999936i	$-0.503601\pi$
$797$	−0.638766	−0.0226263	−0.0113131	+	0.999936i	$0.503601\pi$
$798$	0	0
$799$	11.3618	0.401953
$800$	40.4137	1.42884
$801$	0	0
$802$	29.3698	1.03708
$803$	−2.59125	−0.0914433
$804$	0	0
$805$	0	0
$806$	1.03357	0.0364058
$807$	0	0
$808$	−6.62862	−0.233194
$809$	−50.5592	−1.77757	−0.888783	−	0.458327i	$-0.848449\pi$
$809$	−50.5592	−1.77757	−0.888783	+	0.458327i	$0.848449\pi$
$810$	0	0
$811$	0.784071	0.0275325	0.0137662	−	0.999905i	$-0.495618\pi$
$811$	0.784071	0.0275325	0.0137662	+	0.999905i	$0.495618\pi$
$812$	0	0
$813$	0	0
$814$	23.9230	0.838501
$815$	0.310823	0.0108876
$816$	0	0
$817$	13.0188	0.455470
$818$	−65.4311	−2.28775
$819$	0	0
$820$	0.751682	0.0262499
$821$	43.4413	1.51611	0.758056	−	0.652189i	$-0.226149\pi$
$821$	43.4413	1.51611	0.758056	+	0.652189i	$0.226149\pi$
$822$	0	0
$823$	3.96546	0.138227	0.0691136	−	0.997609i	$-0.477983\pi$
$823$	3.96546	0.138227	0.0691136	+	0.997609i	$0.477983\pi$
$824$	0.0910797	0.00317291
$825$	0	0
$826$	0	0
$827$	29.3159	1.01941	0.509707	−	0.860348i	$-0.329754\pi$
$827$	29.3159	1.01941	0.509707	+	0.860348i	$0.329754\pi$
$828$	0	0
$829$	−35.0427	−1.21708	−0.608541	−	0.793522i	$-0.708245\pi$
$829$	−35.0427	−1.21708	−0.608541	+	0.793522i	$0.708245\pi$
$830$	2.25665	0.0783295
$831$	0	0
$832$	−1.92591	−0.0667689
$833$	0	0
$834$	0	0
$835$	−1.69272	−0.0585788
$836$	25.5255	0.882816
$837$	0	0
$838$	49.0738	1.69523
$839$	−37.5843	−1.29755	−0.648777	−	0.760979i	$-0.724719\pi$
$839$	−37.5843	−1.29755	−0.648777	+	0.760979i	$0.724719\pi$
$840$	0	0
$841$	−4.64435	−0.160150
$842$	5.02119	0.173042
$843$	0	0
$844$	−52.1411	−1.79477
$845$	−1.89535	−0.0652020
$846$	0	0
$847$	0	0
$848$	−18.8066	−0.645822
$849$	0	0
$850$	−64.1177	−2.19922
$851$	−43.2757	−1.48347
$852$	0	0
$853$	32.7699	1.12202	0.561009	−	0.827810i	$-0.310413\pi$
$853$	32.7699	1.12202	0.561009	+	0.827810i	$0.310413\pi$
$854$	0	0
$855$	0	0
$856$	−3.11218	−0.106372
$857$	−27.5347	−0.940566	−0.470283	−	0.882516i	$-0.655848\pi$
$857$	−27.5347	−0.940566	−0.470283	+	0.882516i	$0.655848\pi$
$858$	0	0
$859$	46.5101	1.58690	0.793451	−	0.608634i	$-0.208282\pi$
$859$	46.5101	1.58690	0.793451	+	0.608634i	$0.208282\pi$
$860$	0.610316	0.0208116
$861$	0	0
$862$	−10.1196	−0.344675
$863$	−4.88014	−0.166122	−0.0830610	−	0.996544i	$-0.526470\pi$
$863$	−4.88014	−0.166122	−0.0830610	+	0.996544i	$0.526470\pi$
$864$	0	0
$865$	2.32690	0.0791170
$866$	−63.3629	−2.15316
$867$	0	0
$868$	0	0
$869$	21.2926	0.722303
$870$	0	0
$871$	−1.23669	−0.0419037
$872$	−2.97476	−0.100738
$873$	0	0
$874$	−87.8207	−2.97058
$875$	0	0
$876$	0	0
$877$	39.2892	1.32670	0.663352	−	0.748308i	$-0.269133\pi$
$877$	39.2892	1.32670	0.663352	+	0.748308i	$0.269133\pi$
$878$	−5.02777	−0.169679
$879$	0	0
$880$	−0.856086	−0.0288587
$881$	−47.3713	−1.59598	−0.797990	−	0.602670i	$-0.794103\pi$
$881$	−47.3713	−1.59598	−0.797990	+	0.602670i	$0.794103\pi$
$882$	0	0
$883$	−2.67206	−0.0899221	−0.0449610	−	0.998989i	$-0.514316\pi$
$883$	−2.67206	−0.0899221	−0.0449610	+	0.998989i	$0.514316\pi$
$884$	2.77966	0.0934902
$885$	0	0
$886$	−54.0007	−1.81419
$887$	22.9600	0.770922	0.385461	−	0.922724i	$-0.374042\pi$
$887$	22.9600	0.770922	0.385461	+	0.922724i	$0.374042\pi$
$888$	0	0
$889$	0	0
$890$	2.72342	0.0912891
$891$	0	0
$892$	9.00518	0.301516
$893$	12.5331	0.419404
$894$	0	0
$895$	−1.13389	−0.0379017
$896$	0	0
$897$	0	0
$898$	−79.4920	−2.65268
$899$	12.4257	0.414420
$900$	0	0
$901$	33.5262	1.11692
$902$	7.92109	0.263743
$903$	0	0
$904$	0.0172590	0.000574024 0
$905$	1.77862	0.0591232
$906$	0	0
$907$	−27.8982	−0.926345	−0.463173	−	0.886268i	$-0.653289\pi$
$907$	−27.8982	−0.926345	−0.463173	+	0.886268i	$0.653289\pi$
$908$	8.54354	0.283527
$909$	0	0
$910$	0	0
$911$	37.4762	1.24164	0.620820	−	0.783953i	$-0.286800\pi$
$911$	37.4762	1.24164	0.620820	+	0.783953i	$0.286800\pi$
$912$	0	0
$913$	12.5032	0.413794
$914$	−18.8014	−0.621894
$915$	0	0
$916$	−29.0847	−0.960987
$917$	0	0
$918$	0	0
$919$	30.2147	0.996691	0.498345	−	0.866979i	$-0.333941\pi$
$919$	30.2147	0.996691	0.498345	+	0.866979i	$0.333941\pi$
$920$	−0.403740	−0.0133109
$921$	0	0
$922$	60.0289	1.97695
$923$	0.255359	0.00840525
$924$	0	0
$925$	−34.8526	−1.14594
$926$	33.7221	1.10818
$927$	0	0
$928$	−40.0609	−1.31506
$929$	45.9351	1.50708	0.753540	−	0.657402i	$-0.228344\pi$
$929$	45.9351	1.50708	0.753540	+	0.657402i	$0.228344\pi$
$930$	0	0
$931$	0	0
$932$	38.8106	1.27128
$933$	0	0
$934$	31.5699	1.03300
$935$	1.52613	0.0499097
$936$	0	0
$937$	45.3797	1.48249	0.741245	−	0.671235i	$-0.234236\pi$
$937$	45.3797	1.48249	0.741245	+	0.671235i	$0.234236\pi$
$938$	0	0
$939$	0	0
$940$	0.587547	0.0191637
$941$	−49.4003	−1.61040	−0.805202	−	0.593000i	$-0.797943\pi$
$941$	−49.4003	−1.61040	−0.805202	+	0.593000i	$0.797943\pi$
$942$	0	0
$943$	−14.3289	−0.466613
$944$	−16.0798	−0.523353
$945$	0	0
$946$	6.43141	0.209103
$947$	31.6505	1.02850	0.514252	−	0.857639i	$-0.328070\pi$
$947$	31.6505	1.02850	0.514252	+	0.857639i	$0.328070\pi$
$948$	0	0
$949$	−0.311271	−0.0101043
$950$	−70.7274	−2.29470
$951$	0	0
$952$	0	0
$953$	−19.1237	−0.619477	−0.309739	−	0.950822i	$-0.600242\pi$
$953$	−19.1237	−0.619477	−0.309739	+	0.950822i	$0.600242\pi$
$954$	0	0
$955$	−0.726498	−0.0235089
$956$	−16.2255	−0.524769
$957$	0	0
$958$	77.8929	2.51661
$959$	0	0
$960$	0	0
$961$	−24.6607	−0.795507
$962$	2.87372	0.0926525
$963$	0	0
$964$	−13.8171	−0.445020
$965$	−2.17971	−0.0701673
$966$	0	0
$967$	−9.97050	−0.320630	−0.160315	−	0.987066i	$-0.551251\pi$
$967$	−9.97050	−0.320630	−0.160315	+	0.987066i	$0.551251\pi$
$968$	−3.67581	−0.118145
$969$	0	0
$970$	−2.39377	−0.0768592
$971$	1.04511	0.0335391	0.0167695	−	0.999859i	$-0.494662\pi$
$971$	1.04511	0.0335391	0.0167695	+	0.999859i	$0.494662\pi$
$972$	0	0
$973$	0	0
$974$	−9.45693	−0.303020
$975$	0	0
$976$	−2.38603	−0.0763751
$977$	−18.8862	−0.604222	−0.302111	−	0.953273i	$-0.597691\pi$
$977$	−18.8862	−0.604222	−0.302111	+	0.953273i	$0.597691\pi$
$978$	0	0
$979$	15.0893	0.482257
$980$	0	0
$981$	0	0
$982$	62.3797	1.99062
$983$	−2.28891	−0.0730050	−0.0365025	−	0.999334i	$-0.511622\pi$
$983$	−2.28891	−0.0730050	−0.0365025	+	0.999334i	$0.511622\pi$
$984$	0	0
$985$	−3.10930	−0.0990707
$986$	63.5579	2.02409
$987$	0	0
$988$	3.06621	0.0975492
$989$	−11.6341	−0.369944
$990$	0	0
$991$	19.0698	0.605773	0.302886	−	0.953027i	$-0.402050\pi$
$991$	19.0698	0.605773	0.302886	+	0.953027i	$0.402050\pi$
$992$	−20.4381	−0.648911
$993$	0	0
$994$	0	0
$995$	−2.91675	−0.0924671
$996$	0	0
$997$	−37.0151	−1.17228	−0.586139	−	0.810210i	$-0.699353\pi$
$997$	−37.0151	−1.17228	−0.586139	+	0.810210i	$0.699353\pi$
$998$	19.0346	0.602531
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.ba.1.5		5
3.2	odd	2		3969.2.a.bb.1.1		5
7.3	odd	6		567.2.e.f.163.1		10
7.5	odd	6		567.2.e.f.487.1		10
7.6	odd	2		3969.2.a.z.1.5		5
9.2	odd	6		1323.2.f.f.442.5		10
9.4	even	3		441.2.f.f.295.1		10
9.5	odd	6		1323.2.f.f.883.5		10
9.7	even	3		441.2.f.f.148.1		10
21.5	even	6		567.2.e.e.487.5		10
21.17	even	6		567.2.e.e.163.5		10
21.20	even	2		3969.2.a.bc.1.1		5
63.2	odd	6		1323.2.g.f.361.5		10
63.4	even	3		441.2.g.f.79.1		10
63.5	even	6		189.2.h.b.46.1		10
63.11	odd	6		1323.2.h.f.226.1		10
63.13	odd	6		441.2.f.e.295.1		10
63.16	even	3		441.2.g.f.67.1		10
63.20	even	6		1323.2.f.e.442.5		10
63.23	odd	6		1323.2.h.f.802.1		10
63.25	even	3		441.2.h.f.373.5		10
63.31	odd	6		63.2.g.b.16.1	yes	10
63.32	odd	6		1323.2.g.f.667.5		10
63.34	odd	6		441.2.f.e.148.1		10
63.38	even	6		189.2.h.b.37.1		10
63.40	odd	6		63.2.h.b.25.5	yes	10
63.41	even	6		1323.2.f.e.883.5		10
63.47	even	6		189.2.g.b.172.5		10
63.52	odd	6		63.2.h.b.58.5	yes	10
63.58	even	3		441.2.h.f.214.5		10
63.59	even	6		189.2.g.b.100.5		10
63.61	odd	6		63.2.g.b.4.1	✓	10
252.31	even	6		1008.2.t.i.961.2		10
252.47	odd	6		3024.2.t.i.1873.3		10
252.59	odd	6		3024.2.t.i.289.3		10
252.103	even	6		1008.2.q.i.529.5		10
252.115	even	6		1008.2.q.i.625.5		10
252.131	odd	6		3024.2.q.i.2881.3		10
252.187	even	6		1008.2.t.i.193.2		10
252.227	odd	6		3024.2.q.i.2305.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.g.b.4.1	✓	10	63.61	odd	6
63.2.g.b.16.1	yes	10	63.31	odd	6
63.2.h.b.25.5	yes	10	63.40	odd	6
63.2.h.b.58.5	yes	10	63.52	odd	6
189.2.g.b.100.5		10	63.59	even	6
189.2.g.b.172.5		10	63.47	even	6
189.2.h.b.37.1		10	63.38	even	6
189.2.h.b.46.1		10	63.5	even	6
441.2.f.e.148.1		10	63.34	odd	6
441.2.f.e.295.1		10	63.13	odd	6
441.2.f.f.148.1		10	9.7	even	3
441.2.f.f.295.1		10	9.4	even	3
441.2.g.f.67.1		10	63.16	even	3
441.2.g.f.79.1		10	63.4	even	3
441.2.h.f.214.5		10	63.58	even	3
441.2.h.f.373.5		10	63.25	even	3
567.2.e.e.163.5		10	21.17	even	6
567.2.e.e.487.5		10	21.5	even	6
567.2.e.f.163.1		10	7.3	odd	6
567.2.e.f.487.1		10	7.5	odd	6
1008.2.q.i.529.5		10	252.103	even	6
1008.2.q.i.625.5		10	252.115	even	6
1008.2.t.i.193.2		10	252.187	even	6
1008.2.t.i.961.2		10	252.31	even	6
1323.2.f.e.442.5		10	63.20	even	6
1323.2.f.e.883.5		10	63.41	even	6
1323.2.f.f.442.5		10	9.2	odd	6
1323.2.f.f.883.5		10	9.5	odd	6
1323.2.g.f.361.5		10	63.2	odd	6
1323.2.g.f.667.5		10	63.32	odd	6
1323.2.h.f.226.1		10	63.11	odd	6
1323.2.h.f.802.1		10	63.23	odd	6
3024.2.q.i.2305.3		10	252.227	odd	6
3024.2.q.i.2881.3		10	252.131	odd	6
3024.2.t.i.289.3		10	252.59	odd	6
3024.2.t.i.1873.3		10	252.47	odd	6
3969.2.a.z.1.5		5	7.6	odd	2
3969.2.a.ba.1.5		5	1.1	even	1	trivial
3969.2.a.bb.1.1		5	3.2	odd	2
3969.2.a.bc.1.1		5	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q+2.05365 q^{2} +2.21746 q^{4} +0.146246 q^{5} +0.446582 q^{8} +0.300337 q^{10} +1.66404 q^{11} +0.199891 q^{13} -3.51780 q^{16} +6.27110 q^{17} +6.91758 q^{19} +0.324294 q^{20} +3.41735 q^{22} -6.18184 q^{23} -4.97861 q^{25} +0.410505 q^{26} +4.93514 q^{29} +2.51780 q^{31} -8.11747 q^{32} +12.8786 q^{34} +7.00046 q^{37} +14.2062 q^{38} +0.0653107 q^{40} +2.31790 q^{41} +1.88199 q^{43} +3.68994 q^{44} -12.6953 q^{46} +1.81177 q^{47} -10.2243 q^{50} +0.443250 q^{52} +5.34614 q^{53} +0.243359 q^{55} +10.1350 q^{58} +4.57099 q^{59} +0.678276 q^{61} +5.17066 q^{62} -9.63481 q^{64} +0.0292332 q^{65} -6.18684 q^{67} +13.9059 q^{68} +1.27749 q^{71} -1.55721 q^{73} +14.3765 q^{74} +15.3394 q^{76} +12.7957 q^{79} -0.514462 q^{80} +4.76015 q^{82} +7.51374 q^{83} +0.917122 q^{85} +3.86493 q^{86} +0.743131 q^{88} +9.06788 q^{89} -13.7080 q^{92} +3.72074 q^{94} +1.01167 q^{95} -7.97028 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 4 q^{4} + 4 q^{5} - 3 q^{8} - 7 q^{10} - 4 q^{11} - 8 q^{13} - 2 q^{16} + 12 q^{17} + q^{19} + 5 q^{20} + q^{22} - 3 q^{23} + q^{25} + 11 q^{26} - 7 q^{29} - 3 q^{31} + 2 q^{32} + 3 q^{34}+ \cdots - 12 q^{97}+O(q^{100}) \) 5 * q - 2 * q^2 + 4 * q^4 + 4 * q^5 - 3 * q^8 - 7 * q^10 - 4 * q^11 - 8 * q^13 - 2 * q^16 + 12 * q^17 + q^19 + 5 * q^20 + q^22 - 3 * q^23 + q^25 + 11 * q^26 - 7 * q^29 - 3 * q^31 + 2 * q^32 + 3 * q^34 + 20 * q^38 - 3 * q^40 + 5 * q^41 + 7 * q^43 + 10 * q^44 - 3 * q^46 + 27 * q^47 - 19 * q^50 - 10 * q^52 + 21 * q^53 - 2 * q^55 + 10 * q^58 + 30 * q^59 - 14 * q^61 + 6 * q^62 - 25 * q^64 + 11 * q^65 + 2 * q^67 + 27 * q^68 - 3 * q^71 + 15 * q^73 + 36 * q^74 + 5 * q^76 + 4 * q^79 + 20 * q^80 - 5 * q^82 + 9 * q^83 + 6 * q^85 + 8 * q^86 + 18 * q^88 + 28 * q^89 - 27 * q^92 - 3 * q^94 + 14 * q^95 - 12 * q^97

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