LMFDB - Embedded newform 3969.2.a.bb.1.1

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:	$1$
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			1.1
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)

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

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 $

$2$	−2.05365	−1.45215	−0.726073	−	0.687617i	$-0.758657\pi$
$2$	−2.05365	−1.45215	−0.726073	+	0.687617i	$0.758657\pi$
$3$	0	0
$3$	0	0
$4$	2.21746	1.10873
$5$	−0.146246	−0.0654030	−0.0327015	−	0.999465i	$-0.510411\pi$
$5$	−0.146246	−0.0654030	−0.0327015	+	0.999465i	$0.510411\pi$
$6$	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	$-0.580715\pi$
$11$	−1.66404	−0.501727	−0.250864	+	0.968022i	$0.580715\pi$
$12$	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	$-0.775038\pi$
$17$	−6.27110	−1.52097	−0.760483	+	0.649358i	$0.775038\pi$
$18$	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	$-0.277065\pi$
$23$	6.18184	1.28900	0.644501	+	0.764604i	$0.277065\pi$
$24$	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	$-0.651512\pi$
$29$	−4.93514	−0.916433	−0.458217	+	0.888841i	$0.651512\pi$
$30$	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	$-0.557933\pi$
$41$	−2.31790	−0.361996	−0.180998	+	0.983483i	$0.557933\pi$
$42$	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	$-0.542184\pi$
$47$	−1.81177	−0.264275	−0.132137	+	0.991231i	$0.542184\pi$
$48$	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	$-0.619675\pi$
$53$	−5.34614	−0.734348	−0.367174	+	0.930152i	$0.619675\pi$
$54$	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	$-0.596168\pi$
$59$	−4.57099	−0.595092	−0.297546	+	0.954708i	$0.596168\pi$
$60$	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	$-0.524153\pi$
$71$	−1.27749	−0.151611	−0.0758053	+	0.997123i	$0.524153\pi$
$72$	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	$-0.635299\pi$
$83$	−7.51374	−0.824740	−0.412370	+	0.911016i	$0.635299\pi$
$84$	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	$-0.659580\pi$
$89$	−9.06788	−0.961193	−0.480597	+	0.876942i	$0.659580\pi$
$90$	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	$-0.235550\pi$
$101$	14.8430	1.47693	0.738467	+	0.674290i	$0.235550\pi$
$102$	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	$-0.390637\pi$
$107$	6.96889	0.673708	0.336854	+	0.941557i	$0.390637\pi$
$108$	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	$-0.500579\pi$
$113$	−0.0386468	−0.00363558	−0.00181779	+	0.999998i	$0.500579\pi$
$114$	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	$-0.833586\pi$
$131$	−19.8333	−1.73284	−0.866422	+	0.499312i	$0.833586\pi$
$132$	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	$-0.411216\pi$
$137$	6.44509	0.550642	0.275321	+	0.961352i	$0.411216\pi$
$138$	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	$-0.759446\pi$
$149$	−17.7673	−1.45555	−0.727776	+	0.685815i	$0.759446\pi$
$150$	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	$-0.352197\pi$
$167$	11.5745	0.895659	0.447829	+	0.894119i	$0.352197\pi$
$168$	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	$-0.706764\pi$
$173$	−15.9109	−1.20968	−0.604842	+	0.796345i	$0.706764\pi$
$174$	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	$-0.406426\pi$
$179$	7.75331	0.579509	0.289755	+	0.957101i	$0.406426\pi$
$180$	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	$-0.442480\pi$
$191$	4.96765	0.359447	0.179723	+	0.983717i	$0.442480\pi$
$192$	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	$-0.226476\pi$
$197$	21.2608	1.51477	0.757386	+	0.652968i	$0.226476\pi$
$198$	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	$-0.540811\pi$
$227$	−3.85285	−0.255723	−0.127861	+	0.991792i	$0.540811\pi$
$228$	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	$-0.694340\pi$
$233$	−17.5023	−1.14661	−0.573307	+	0.819340i	$0.694340\pi$
$234$	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	$-0.423949\pi$
$239$	7.31714	0.473306	0.236653	+	0.971594i	$0.423949\pi$
$240$	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	$-0.442907\pi$
$251$	5.65283	0.356803	0.178402	+	0.983958i	$0.442907\pi$
$252$	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	$-0.380014\pi$
$257$	11.8016	0.736166	0.368083	+	0.929793i	$0.380014\pi$
$258$	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	$-0.259500\pi$
$263$	22.2401	1.37138	0.685691	+	0.727893i	$0.259500\pi$
$264$	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	$-0.476799\pi$
$269$	2.38884	0.145650	0.0728251	+	0.997345i	$0.476799\pi$
$270$	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	$-0.385343\pi$
$281$	11.8168	0.704933	0.352466	+	0.935825i	$0.385343\pi$
$282$	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	$-0.635083\pi$
$293$	−14.0961	−0.823502	−0.411751	+	0.911296i	$0.635083\pi$
$294$	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	$-0.630474\pi$
$311$	−14.0557	−0.797026	−0.398513	+	0.917163i	$0.630474\pi$
$312$	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	$-0.577302\pi$
$317$	−8.56297	−0.480944	−0.240472	+	0.970656i	$0.577302\pi$
$318$	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	$-0.399817\pi$
$347$	11.5330	0.619126	0.309563	+	0.950879i	$0.399817\pi$
$348$	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	$-0.522441\pi$
$353$	−2.64699	−0.140885	−0.0704424	+	0.997516i	$0.522441\pi$
$354$	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	$-0.740305\pi$
$359$	−25.9671	−1.37049	−0.685245	+	0.728312i	$0.740305\pi$
$360$	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	$-0.348874\pi$
$383$	17.8928	0.914278	0.457139	+	0.889395i	$0.348874\pi$
$384$	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	$-0.629665\pi$
$389$	−15.6278	−0.792363	−0.396181	+	0.918172i	$0.629665\pi$
$390$	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	$-0.616230\pi$
$401$	−14.3013	−0.714172	−0.357086	+	0.934072i	$0.616230\pi$
$402$	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	$-0.698395\pi$
$419$	−23.8960	−1.16739	−0.583697	+	0.811971i	$0.698395\pi$
$420$	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	$-0.462134\pi$
$431$	4.92764	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$432$	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	$-0.285239\pi$
$443$	26.2950	1.24931	0.624657	+	0.780899i	$0.285239\pi$
$444$	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	$-0.133474\pi$
$449$	38.7077	1.82673	0.913365	+	0.407141i	$0.133474\pi$
$450$	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	$-0.738324\pi$
$461$	−29.2304	−1.36140	−0.680698	+	0.732564i	$0.738324\pi$
$462$	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	$-0.615751\pi$
$467$	−15.3726	−0.711361	−0.355680	+	0.934608i	$0.615751\pi$
$468$	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	$-0.833644\pi$
$479$	−37.9291	−1.73303	−0.866513	+	0.499155i	$0.833644\pi$
$480$	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	$-0.740375\pi$
$491$	−30.3751	−1.37081	−0.685405	+	0.728162i	$0.740375\pi$
$492$	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	$-0.666628\pi$
$503$	−22.4230	−0.999791	−0.499896	+	0.866086i	$0.666628\pi$
$504$	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	$-0.814077\pi$
$509$	−37.6414	−1.66843	−0.834213	+	0.551443i	$0.814077\pi$
$510$	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	$-0.777319\pi$
$521$	−34.9283	−1.53023	−0.765117	+	0.643891i	$0.777319\pi$
$522$	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	$-0.501655\pi$
$557$	−0.245481	−0.0104014	−0.00520068	+	0.999986i	$0.501655\pi$
$558$	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	$-0.882357\pi$
$563$	−44.2509	−1.86495	−0.932477	+	0.361230i	$0.882357\pi$
$564$	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	$-0.462984\pi$
$569$	5.53533	0.232053	0.116027	+	0.993246i	$0.462984\pi$
$570$	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	$-0.626349\pi$
$587$	−18.7329	−0.773189	−0.386595	+	0.922250i	$0.626349\pi$
$588$	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	$-0.373354\pi$
$593$	18.8703	0.774912	0.387456	+	0.921888i	$0.373354\pi$
$594$	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	$-0.517401\pi$
$599$	−2.67451	−0.109278	−0.0546388	+	0.998506i	$0.517401\pi$
$600$	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	$-0.561454\pi$
$617$	−9.53175	−0.383734	−0.191867	+	0.981421i	$0.561454\pi$
$618$	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	$-0.162439\pi$
$641$	44.1844	1.74518	0.872590	+	0.488454i	$0.162439\pi$
$642$	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	$-0.272787\pi$
$647$	33.3071	1.30944	0.654719	+	0.755872i	$0.272787\pi$
$648$	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	$-0.443232\pi$
$653$	9.06643	0.354797	0.177398	+	0.984139i	$0.443232\pi$
$654$	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	$-0.282930\pi$
$659$	32.3611	1.26061	0.630305	+	0.776348i	$0.282930\pi$
$660$	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	$-0.236430\pi$
$677$	38.3315	1.47320	0.736600	+	0.676329i	$0.236430\pi$
$678$	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	$-0.540505\pi$
$683$	−6.63318	−0.253812	−0.126906	+	0.991915i	$0.540505\pi$
$684$	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	$-0.555996\pi$
$701$	−9.26736	−0.350023	−0.175012	+	0.984566i	$0.555996\pi$
$702$	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	$-0.583192\pi$
$719$	−13.8570	−0.516777	−0.258389	+	0.966041i	$0.583192\pi$
$720$	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	$-0.249563\pi$
$743$	38.6015	1.41615	0.708076	+	0.706136i	$0.249563\pi$
$744$	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	$-0.332261\pi$
$761$	27.7470	1.00583	0.502913	+	0.864337i	$0.332261\pi$
$762$	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	$-0.231308\pi$
$773$	41.5591	1.49478	0.747388	+	0.664388i	$0.231308\pi$
$774$	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	$-0.496399\pi$
$797$	0.638766	0.0226263	0.0113131	+	0.999936i	$0.496399\pi$
$798$	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	$-0.151551\pi$
$809$	50.5592	1.77757	0.888783	+	0.458327i	$0.151551\pi$
$810$	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	$-0.773851\pi$
$821$	−43.4413	−1.51611	−0.758056	+	0.652189i	$0.773851\pi$
$822$	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	$-0.670246\pi$
$827$	−29.3159	−1.01941	−0.509707	+	0.860348i	$0.670246\pi$
$828$	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	$-0.275281\pi$
$839$	37.5843	1.29755	0.648777	+	0.760979i	$0.275281\pi$
$840$	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	$-0.344152\pi$
$857$	27.5347	0.940566	0.470283	+	0.882516i	$0.344152\pi$
$858$	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	$-0.473530\pi$
$863$	4.88014	0.166122	0.0830610	+	0.996544i	$0.473530\pi$
$864$	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	$-0.205897\pi$
$881$	47.3713	1.59598	0.797990	+	0.602670i	$0.205897\pi$
$882$	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	$-0.625958\pi$
$887$	−22.9600	−0.770922	−0.385461	+	0.922724i	$0.625958\pi$
$888$	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	$-0.713200\pi$
$911$	−37.4762	−1.24164	−0.620820	+	0.783953i	$0.713200\pi$
$912$	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	$-0.771656\pi$
$929$	−45.9351	−1.50708	−0.753540	+	0.657402i	$0.771656\pi$
$930$	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	$-0.202057\pi$
$941$	49.4003	1.61040	0.805202	+	0.593000i	$0.202057\pi$
$942$	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	$-0.671930\pi$
$947$	−31.6505	−1.02850	−0.514252	+	0.857639i	$0.671930\pi$
$948$	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	$-0.399758\pi$
$953$	19.1237	0.619477	0.309739	+	0.950822i	$0.399758\pi$
$954$	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	$-0.505338\pi$
$971$	−1.04511	−0.0335391	−0.0167695	+	0.999859i	$0.505338\pi$
$972$	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	$-0.402309\pi$
$977$	18.8862	0.604222	0.302111	+	0.953273i	$0.402309\pi$
$978$	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	$-0.488378\pi$
$983$	2.28891	0.0730050	0.0365025	+	0.999334i	$0.488378\pi$
$984$	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.bb.1.1		5
3.2	odd	2		3969.2.a.ba.1.5		5
7.3	odd	6		567.2.e.e.163.5		10
7.5	odd	6		567.2.e.e.487.5		10
7.6	odd	2		3969.2.a.bc.1.1		5
9.2	odd	6		441.2.f.f.148.1		10
9.4	even	3		1323.2.f.f.883.5		10
9.5	odd	6		441.2.f.f.295.1		10
9.7	even	3		1323.2.f.f.442.5		10
21.5	even	6		567.2.e.f.487.1		10
21.17	even	6		567.2.e.f.163.1		10
21.20	even	2		3969.2.a.z.1.5		5
63.2	odd	6		441.2.g.f.67.1		10
63.4	even	3		1323.2.g.f.667.5		10
63.5	even	6		63.2.h.b.25.5	yes	10
63.11	odd	6		441.2.h.f.373.5		10
63.13	odd	6		1323.2.f.e.883.5		10
63.16	even	3		1323.2.g.f.361.5		10
63.20	even	6		441.2.f.e.148.1		10
63.23	odd	6		441.2.h.f.214.5		10
63.25	even	3		1323.2.h.f.226.1		10
63.31	odd	6		189.2.g.b.100.5		10
63.32	odd	6		441.2.g.f.79.1		10
63.34	odd	6		1323.2.f.e.442.5		10
63.38	even	6		63.2.h.b.58.5	yes	10
63.40	odd	6		189.2.h.b.46.1		10
63.41	even	6		441.2.f.e.295.1		10
63.47	even	6		63.2.g.b.4.1	✓	10
63.52	odd	6		189.2.h.b.37.1		10
63.58	even	3		1323.2.h.f.802.1		10
63.59	even	6		63.2.g.b.16.1	yes	10
63.61	odd	6		189.2.g.b.172.5		10
252.31	even	6		3024.2.t.i.289.3		10
252.47	odd	6		1008.2.t.i.193.2		10
252.59	odd	6		1008.2.t.i.961.2		10
252.103	even	6		3024.2.q.i.2881.3		10
252.115	even	6		3024.2.q.i.2305.3		10
252.131	odd	6		1008.2.q.i.529.5		10
252.187	even	6		3024.2.t.i.1873.3		10
252.227	odd	6		1008.2.q.i.625.5		10

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

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