LMFDB - Embedded newform 3726.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3726,2,Mod(1,3726)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3726, 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("3726.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3726 = 2 \cdot 3^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3726.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:	$29.7522597931$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.8472.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 27x - 33 $ x^3 - x^2 - 27*x - 33
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$6.20673$ of defining polynomial
Character	$\chi$	$=$	3726.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +5.20673 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +5.20673 q^{7} -1.00000 q^{8} -1.00000 q^{10} +4.55504 q^{11} +1.65170 q^{13} -5.20673 q^{14} +1.00000 q^{16} +1.65170 q^{17} +4.00000 q^{19} +1.00000 q^{20} -4.55504 q^{22} +1.00000 q^{23} -4.00000 q^{25} -1.65170 q^{26} +5.20673 q^{28} -0.348303 q^{29} +8.55504 q^{31} -1.00000 q^{32} -1.65170 q^{34} +5.20673 q^{35} +5.00000 q^{37} -4.00000 q^{38} -1.00000 q^{40} -9.85843 q^{41} +7.20673 q^{43} +4.55504 q^{44} -1.00000 q^{46} +3.20673 q^{47} +20.1101 q^{49} +4.00000 q^{50} +1.65170 q^{52} -13.8584 q^{53} +4.55504 q^{55} -5.20673 q^{56} +0.348303 q^{58} +9.20673 q^{59} -11.4135 q^{61} -8.55504 q^{62} +1.00000 q^{64} +1.65170 q^{65} +1.34830 q^{67} +1.65170 q^{68} -5.20673 q^{70} -5.34830 q^{71} -16.6651 q^{73} -5.00000 q^{74} +4.00000 q^{76} +23.7169 q^{77} -9.20673 q^{79} +1.00000 q^{80} +9.85843 q^{82} -0.555036 q^{83} +1.65170 q^{85} -7.20673 q^{86} -4.55504 q^{88} -3.65170 q^{89} +8.59995 q^{91} +1.00000 q^{92} -3.20673 q^{94} +4.00000 q^{95} -6.41347 q^{97} -20.1101 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 2 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 2 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 2 q^{7} - 3 q^{8} - 3 q^{10} + q^{11} + 2 q^{14} + 3 q^{16} + 12 q^{19} + 3 q^{20} - q^{22} + 3 q^{23} - 12 q^{25} - 2 q^{28} - 6 q^{29} + 13 q^{31} - 3 q^{32} - 2 q^{35} + 15 q^{37} - 12 q^{38} - 3 q^{40} - 7 q^{41} + 4 q^{43} + q^{44} - 3 q^{46} - 8 q^{47} + 35 q^{49} + 12 q^{50} - 19 q^{53} + q^{55} + 2 q^{56} + 6 q^{58} + 10 q^{59} + q^{61} - 13 q^{62} + 3 q^{64} + 9 q^{67} + 2 q^{70} - 21 q^{71} - 12 q^{73} - 15 q^{74} + 12 q^{76} + 26 q^{77} - 10 q^{79} + 3 q^{80} + 7 q^{82} + 11 q^{83} - 4 q^{86} - q^{88} - 6 q^{89} + 28 q^{91} + 3 q^{92} + 8 q^{94} + 12 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 2 * q^7 - 3 * q^8 - 3 * q^10 + q^11 + 2 * q^14 + 3 * q^16 + 12 * q^19 + 3 * q^20 - q^22 + 3 * q^23 - 12 * q^25 - 2 * q^28 - 6 * q^29 + 13 * q^31 - 3 * q^32 - 2 * q^35 + 15 * q^37 - 12 * q^38 - 3 * q^40 - 7 * q^41 + 4 * q^43 + q^44 - 3 * q^46 - 8 * q^47 + 35 * q^49 + 12 * q^50 - 19 * q^53 + q^55 + 2 * q^56 + 6 * q^58 + 10 * q^59 + q^61 - 13 * q^62 + 3 * q^64 + 9 * q^67 + 2 * q^70 - 21 * q^71 - 12 * q^73 - 15 * q^74 + 12 * q^76 + 26 * q^77 - 10 * q^79 + 3 * q^80 + 7 * q^82 + 11 * q^83 - 4 * q^86 - q^88 - 6 * q^89 + 28 * q^91 + 3 * q^92 + 8 * q^94 + 12 * q^95 + 16 * q^97 - 35 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	5.20673	1.96796	0.983980	−	0.178278i	$-0.0570527\pi$
$7$	5.20673	1.96796	0.983980	+	0.178278i	$0.0570527\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	4.55504	1.37340	0.686698	−	0.726943i	$-0.259060\pi$
$11$	4.55504	1.37340	0.686698	+	0.726943i	$0.259060\pi$
$12$	0	0
$13$	1.65170	0.458098	0.229049	−	0.973415i	$-0.426438\pi$
$13$	1.65170	0.458098	0.229049	+	0.973415i	$0.426438\pi$
$14$	−5.20673	−1.39156
$15$	0	0
$16$	1.00000	0.250000
$17$	1.65170	0.400595	0.200298	−	0.979735i	$-0.435809\pi$
$17$	1.65170	0.400595	0.200298	+	0.979735i	$0.435809\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−4.55504	−0.971137
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−1.65170	−0.323924
$27$	0	0
$28$	5.20673	0.983980
$29$	−0.348303	−0.0646783	−0.0323391	−	0.999477i	$-0.510296\pi$
$29$	−0.348303	−0.0646783	−0.0323391	+	0.999477i	$0.510296\pi$
$30$	0	0
$31$	8.55504	1.53653	0.768265	−	0.640132i	$-0.221120\pi$
$31$	8.55504	1.53653	0.768265	+	0.640132i	$0.221120\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.65170	−0.283264
$35$	5.20673	0.880099
$36$	0	0
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−9.85843	−1.53963	−0.769814	−	0.638268i	$-0.779651\pi$
$41$	−9.85843	−1.53963	−0.769814	+	0.638268i	$0.779651\pi$
$42$	0	0
$43$	7.20673	1.09902	0.549508	−	0.835488i	$-0.314815\pi$
$43$	7.20673	1.09902	0.549508	+	0.835488i	$0.314815\pi$
$44$	4.55504	0.686698
$45$	0	0
$46$	−1.00000	−0.147442
$47$	3.20673	0.467750	0.233875	−	0.972267i	$-0.424859\pi$
$47$	3.20673	0.467750	0.233875	+	0.972267i	$0.424859\pi$
$48$	0	0
$49$	20.1101	2.87287
$50$	4.00000	0.565685
$51$	0	0
$52$	1.65170	0.229049
$53$	−13.8584	−1.90360	−0.951801	−	0.306717i	$-0.900770\pi$
$53$	−13.8584	−1.90360	−0.951801	+	0.306717i	$0.900770\pi$
$54$	0	0
$55$	4.55504	0.614201
$56$	−5.20673	−0.695779
$57$	0	0
$58$	0.348303	0.0457344
$59$	9.20673	1.19861	0.599307	−	0.800519i	$-0.295443\pi$
$59$	9.20673	1.19861	0.599307	+	0.800519i	$0.295443\pi$
$60$	0	0
$61$	−11.4135	−1.46134	−0.730672	−	0.682728i	$-0.760793\pi$
$61$	−11.4135	−1.46134	−0.730672	+	0.682728i	$0.760793\pi$
$62$	−8.55504	−1.08649
$63$	0	0
$64$	1.00000	0.125000
$65$	1.65170	0.204868
$66$	0	0
$67$	1.34830	0.164721	0.0823607	−	0.996603i	$-0.473754\pi$
$67$	1.34830	0.164721	0.0823607	+	0.996603i	$0.473754\pi$
$68$	1.65170	0.200298
$69$	0	0
$70$	−5.20673	−0.622324
$71$	−5.34830	−0.634727	−0.317363	−	0.948304i	$-0.602797\pi$
$71$	−5.34830	−0.634727	−0.317363	+	0.948304i	$0.602797\pi$
$72$	0	0
$73$	−16.6651	−1.95050	−0.975252	−	0.221097i	$-0.929036\pi$
$73$	−16.6651	−1.95050	−0.975252	+	0.221097i	$0.929036\pi$
$74$	−5.00000	−0.581238
$75$	0	0
$76$	4.00000	0.458831
$77$	23.7169	2.70279
$78$	0	0
$79$	−9.20673	−1.03584	−0.517919	−	0.855430i	$-0.673293\pi$
$79$	−9.20673	−1.03584	−0.517919	+	0.855430i	$0.673293\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	9.85843	1.08868
$83$	−0.555036	−0.0609232	−0.0304616	−	0.999536i	$-0.509698\pi$
$83$	−0.555036	−0.0609232	−0.0304616	+	0.999536i	$0.509698\pi$
$84$	0	0
$85$	1.65170	0.179152
$86$	−7.20673	−0.777122
$87$	0	0
$88$	−4.55504	−0.485569
$89$	−3.65170	−0.387079	−0.193540	−	0.981092i	$-0.561997\pi$
$89$	−3.65170	−0.387079	−0.193540	+	0.981092i	$0.561997\pi$
$90$	0	0
$91$	8.59995	0.901519
$92$	1.00000	0.104257
$93$	0	0
$94$	−3.20673	−0.330749
$95$	4.00000	0.410391
$96$	0	0
$97$	−6.41347	−0.651189	−0.325594	−	0.945510i	$-0.605564\pi$
$97$	−6.41347	−0.651189	−0.325594	+	0.945510i	$0.605564\pi$
$98$	−20.1101	−2.03142
$99$	0	0
$100$	−4.00000	−0.400000
$101$	−10.4135	−1.03618	−0.518089	−	0.855327i	$-0.673357\pi$
$101$	−10.4135	−1.03618	−0.518089	+	0.855327i	$0.673357\pi$
$102$	0	0
$103$	−2.41347	−0.237806	−0.118903	−	0.992906i	$-0.537938\pi$
$103$	−2.41347	−0.237806	−0.118903	+	0.992906i	$0.537938\pi$
$104$	−1.65170	−0.161962
$105$	0	0
$106$	13.8584	1.34605
$107$	−7.20673	−0.696701	−0.348351	−	0.937364i	$-0.613258\pi$
$107$	−7.20673	−0.696701	−0.348351	+	0.937364i	$0.613258\pi$
$108$	0	0
$109$	6.44496	0.617316	0.308658	−	0.951173i	$-0.400120\pi$
$109$	6.44496	0.617316	0.308658	+	0.951173i	$0.400120\pi$
$110$	−4.55504	−0.434306
$111$	0	0
$112$	5.20673	0.491990
$113$	11.4584	1.07791	0.538957	−	0.842333i	$-0.318819\pi$
$113$	11.4584	1.07791	0.538957	+	0.842333i	$0.318819\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	−0.348303	−0.0323391
$117$	0	0
$118$	−9.20673	−0.847549
$119$	8.59995	0.788356
$120$	0	0
$121$	9.74836	0.886214
$122$	11.4135	1.03333
$123$	0	0
$124$	8.55504	0.768265
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−17.6202	−1.56354	−0.781770	−	0.623567i	$-0.785683\pi$
$127$	−17.6202	−1.56354	−0.781770	+	0.623567i	$0.785683\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.65170	−0.144863
$131$	−14.3168	−1.25087	−0.625433	−	0.780278i	$-0.715077\pi$
$131$	−14.3168	−1.25087	−0.625433	+	0.780278i	$0.715077\pi$
$132$	0	0
$133$	20.8269	1.80592
$134$	−1.34830	−0.116476
$135$	0	0
$136$	−1.65170	−0.141632
$137$	−19.8718	−1.69777	−0.848883	−	0.528581i	$-0.822724\pi$
$137$	−19.8718	−1.69777	−0.848883	+	0.528581i	$0.822724\pi$
$138$	0	0
$139$	0.0966605	0.00819863	0.00409932	−	0.999992i	$-0.498695\pi$
$139$	0.0966605	0.00819863	0.00409932	+	0.999992i	$0.498695\pi$
$140$	5.20673	0.440049
$141$	0	0
$142$	5.34830	0.448820
$143$	7.52354	0.629150
$144$	0	0
$145$	−0.348303	−0.0289250
$146$	16.6651	1.37921
$147$	0	0
$148$	5.00000	0.410997
$149$	−2.25164	−0.184462	−0.0922309	−	0.995738i	$-0.529400\pi$
$149$	−2.25164	−0.184462	−0.0922309	+	0.995738i	$0.529400\pi$
$150$	0	0
$151$	−1.34830	−0.109723	−0.0548617	−	0.998494i	$-0.517472\pi$
$151$	−1.34830	−0.109723	−0.0548617	+	0.998494i	$0.517472\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	−23.7169	−1.91116
$155$	8.55504	0.687157
$156$	0	0
$157$	1.74836	0.139534	0.0697671	−	0.997563i	$-0.477774\pi$
$157$	1.74836	0.139534	0.0697671	+	0.997563i	$0.477774\pi$
$158$	9.20673	0.732448
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	5.20673	0.410348
$162$	0	0
$163$	14.5101	1.13652	0.568260	−	0.822849i	$-0.307617\pi$
$163$	14.5101	1.13652	0.568260	+	0.822849i	$0.307617\pi$
$164$	−9.85843	−0.769814
$165$	0	0
$166$	0.555036	0.0430792
$167$	3.76177	0.291094	0.145547	−	0.989351i	$-0.453506\pi$
$167$	3.76177	0.291094	0.145547	+	0.989351i	$0.453506\pi$
$168$	0	0
$169$	−10.2719	−0.790146
$170$	−1.65170	−0.126679
$171$	0	0
$172$	7.20673	0.549508
$173$	−10.0652	−0.765240	−0.382620	−	0.923906i	$-0.624978\pi$
$173$	−10.0652	−0.765240	−0.382620	+	0.923906i	$0.624978\pi$
$174$	0	0
$175$	−20.8269	−1.57437
$176$	4.55504	0.343349
$177$	0	0
$178$	3.65170	0.273706
$179$	−23.7169	−1.77268	−0.886341	−	0.463034i	$-0.846761\pi$
$179$	−23.7169	−1.77268	−0.886341	+	0.463034i	$0.846761\pi$
$180$	0	0
$181$	7.25164	0.539010	0.269505	−	0.962999i	$-0.413140\pi$
$181$	7.25164	0.539010	0.269505	+	0.962999i	$0.413140\pi$
$182$	−8.59995	−0.637470
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	5.00000	0.367607
$186$	0	0
$187$	7.52354	0.550176
$188$	3.20673	0.233875
$189$	0	0
$190$	−4.00000	−0.290191
$191$	5.20673	0.376746	0.188373	−	0.982098i	$-0.439679\pi$
$191$	5.20673	0.376746	0.188373	+	0.982098i	$0.439679\pi$
$192$	0	0
$193$	−1.88993	−0.136040	−0.0680200	−	0.997684i	$-0.521668\pi$
$193$	−1.88993	−0.136040	−0.0680200	+	0.997684i	$0.521668\pi$
$194$	6.41347	0.460460
$195$	0	0
$196$	20.1101	1.43643
$197$	−13.8718	−0.988328	−0.494164	−	0.869369i	$-0.664526\pi$
$197$	−13.8718	−0.988328	−0.494164	+	0.869369i	$0.664526\pi$
$198$	0	0
$199$	15.7169	1.11414	0.557069	−	0.830466i	$-0.311926\pi$
$199$	15.7169	1.11414	0.557069	+	0.830466i	$0.311926\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	10.4135	0.732689
$203$	−1.81352	−0.127284
$204$	0	0
$205$	−9.85843	−0.688543
$206$	2.41347	0.168154
$207$	0	0
$208$	1.65170	0.114525
$209$	18.2201	1.26031
$210$	0	0
$211$	−19.8135	−1.36402	−0.682010	−	0.731343i	$-0.738894\pi$
$211$	−19.8135	−1.36402	−0.682010	+	0.731343i	$0.738894\pi$
$212$	−13.8584	−0.951801
$213$	0	0
$214$	7.20673	0.492642
$215$	7.20673	0.491495
$216$	0	0
$217$	44.5438	3.02383
$218$	−6.44496	−0.436508
$219$	0	0
$220$	4.55504	0.307100
$221$	2.72810	0.183512
$222$	0	0
$223$	−15.7618	−1.05549	−0.527743	−	0.849404i	$-0.676962\pi$
$223$	−15.7618	−1.05549	−0.527743	+	0.849404i	$0.676962\pi$
$224$	−5.20673	−0.347890
$225$	0	0
$226$	−11.4584	−0.762200
$227$	−0.599946	−0.0398198	−0.0199099	−	0.999802i	$-0.506338\pi$
$227$	−0.599946	−0.0398198	−0.0199099	+	0.999802i	$0.506338\pi$
$228$	0	0
$229$	13.5550	0.895742	0.447871	−	0.894098i	$-0.352182\pi$
$229$	13.5550	0.895742	0.447871	+	0.894098i	$0.352182\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	0.348303	0.0228672
$233$	10.2516	0.671607	0.335804	−	0.941932i	$-0.390992\pi$
$233$	10.2516	0.671607	0.335804	+	0.941932i	$0.390992\pi$
$234$	0	0
$235$	3.20673	0.209184
$236$	9.20673	0.599307
$237$	0	0
$238$	−8.59995	−0.557452
$239$	28.2719	1.82876	0.914379	−	0.404860i	$-0.132680\pi$
$239$	28.2719	1.82876	0.914379	+	0.404860i	$0.132680\pi$
$240$	0	0
$241$	−9.45838	−0.609268	−0.304634	−	0.952470i	$-0.598534\pi$
$241$	−9.45838	−0.609268	−0.304634	+	0.952470i	$0.598534\pi$
$242$	−9.74836	−0.626648
$243$	0	0
$244$	−11.4135	−0.730672
$245$	20.1101	1.28479
$246$	0	0
$247$	6.60679	0.420380
$248$	−8.55504	−0.543245
$249$	0	0
$250$	9.00000	0.569210
$251$	16.0786	1.01487	0.507435	−	0.861690i	$-0.330594\pi$
$251$	16.0786	1.01487	0.507435	+	0.861690i	$0.330594\pi$
$252$	0	0
$253$	4.55504	0.286373
$254$	17.6202	1.10559
$255$	0	0
$256$	1.00000	0.0625000
$257$	−17.8269	−1.11201	−0.556007	−	0.831178i	$-0.687667\pi$
$257$	−17.8269	−1.11201	−0.556007	+	0.831178i	$0.687667\pi$
$258$	0	0
$259$	26.0337	1.61765
$260$	1.65170	0.102434
$261$	0	0
$262$	14.3168	0.884495
$263$	30.0337	1.85196	0.925978	−	0.377578i	$-0.123243\pi$
$263$	30.0337	1.85196	0.925978	+	0.377578i	$0.123243\pi$
$264$	0	0
$265$	−13.8584	−0.851317
$266$	−20.8269	−1.27698
$267$	0	0
$268$	1.34830	0.0823607
$269$	−8.15498	−0.497218	−0.248609	−	0.968604i	$-0.579973\pi$
$269$	−8.15498	−0.497218	−0.248609	+	0.968604i	$0.579973\pi$
$270$	0	0
$271$	−6.96850	−0.423306	−0.211653	−	0.977345i	$-0.567885\pi$
$271$	−6.96850	−0.423306	−0.211653	+	0.977345i	$0.567885\pi$
$272$	1.65170	0.100149
$273$	0	0
$274$	19.8718	1.20050
$275$	−18.2201	−1.09872
$276$	0	0
$277$	−31.9370	−1.91891	−0.959454	−	0.281864i	$-0.909047\pi$
$277$	−31.9370	−1.91891	−0.959454	+	0.281864i	$0.909047\pi$
$278$	−0.0966605	−0.00579731
$279$	0	0
$280$	−5.20673	−0.311162
$281$	4.54162	0.270931	0.135465	−	0.990782i	$-0.456747\pi$
$281$	4.54162	0.270931	0.135465	+	0.990782i	$0.456747\pi$
$282$	0	0
$283$	2.45838	0.146135	0.0730676	−	0.997327i	$-0.476721\pi$
$283$	2.45838	0.146135	0.0730676	+	0.997327i	$0.476721\pi$
$284$	−5.34830	−0.317363
$285$	0	0
$286$	−7.52354	−0.444876
$287$	−51.3302	−3.02993
$288$	0	0
$289$	−14.2719	−0.839523
$290$	0.348303	0.0204531
$291$	0	0
$292$	−16.6651	−0.975252
$293$	24.1101	1.40853	0.704263	−	0.709939i	$-0.251278\pi$
$293$	24.1101	1.40853	0.704263	+	0.709939i	$0.251278\pi$
$294$	0	0
$295$	9.20673	0.536037
$296$	−5.00000	−0.290619
$297$	0	0
$298$	2.25164	0.130434
$299$	1.65170	0.0955201
$300$	0	0
$301$	37.5235	2.16282
$302$	1.34830	0.0775861
$303$	0	0
$304$	4.00000	0.229416
$305$	−11.4135	−0.653533
$306$	0	0
$307$	21.0134	1.19930	0.599649	−	0.800263i	$-0.295307\pi$
$307$	21.0134	1.19930	0.599649	+	0.800263i	$0.295307\pi$
$308$	23.7169	1.35139
$309$	0	0
$310$	−8.55504	−0.485893
$311$	−5.66511	−0.321239	−0.160619	−	0.987016i	$-0.551349\pi$
$311$	−5.66511	−0.321239	−0.160619	+	0.987016i	$0.551349\pi$
$312$	0	0
$313$	23.8718	1.34932	0.674658	−	0.738130i	$-0.264291\pi$
$313$	23.8718	1.34932	0.674658	+	0.738130i	$0.264291\pi$
$314$	−1.74836	−0.0986655
$315$	0	0
$316$	−9.20673	−0.517919
$317$	−7.17524	−0.403001	−0.201501	−	0.979488i	$-0.564582\pi$
$317$	−7.17524	−0.403001	−0.201501	+	0.979488i	$0.564582\pi$
$318$	0	0
$319$	−1.58653	−0.0888288
$320$	1.00000	0.0559017
$321$	0	0
$322$	−5.20673	−0.290160
$323$	6.60679	0.367612
$324$	0	0
$325$	−6.60679	−0.366479
$326$	−14.5101	−0.803642
$327$	0	0
$328$	9.85843	0.544341
$329$	16.6966	0.920514
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−0.555036	−0.0304616
$333$	0	0
$334$	−3.76177	−0.205835
$335$	1.34830	0.0736657
$336$	0	0
$337$	15.1101	0.823098	0.411549	−	0.911388i	$-0.364988\pi$
$337$	15.1101	0.823098	0.411549	+	0.911388i	$0.364988\pi$
$338$	10.2719	0.558718
$339$	0	0
$340$	1.65170	0.0895758
$341$	38.9685	2.11026
$342$	0	0
$343$	68.2607	3.68573
$344$	−7.20673	−0.388561
$345$	0	0
$346$	10.0652	0.541107
$347$	18.5101	0.993676	0.496838	−	0.867843i	$-0.334494\pi$
$347$	18.5101	0.993676	0.496838	+	0.867843i	$0.334494\pi$
$348$	0	0
$349$	−27.1101	−1.45117	−0.725584	−	0.688133i	$-0.758431\pi$
$349$	−27.1101	−1.45117	−0.725584	+	0.688133i	$0.758431\pi$
$350$	20.8269	1.11325
$351$	0	0
$352$	−4.55504	−0.242784
$353$	25.5753	1.36124	0.680618	−	0.732639i	$-0.261711\pi$
$353$	25.5753	1.36124	0.680618	+	0.732639i	$0.261711\pi$
$354$	0	0
$355$	−5.34830	−0.283858
$356$	−3.65170	−0.193540
$357$	0	0
$358$	23.7169	1.25348
$359$	10.6966	0.564545	0.282273	−	0.959334i	$-0.408912\pi$
$359$	10.6966	0.564545	0.282273	+	0.959334i	$0.408912\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−7.25164	−0.381138
$363$	0	0
$364$	8.59995	0.450760
$365$	−16.6651	−0.872292
$366$	0	0
$367$	−13.3034	−0.694432	−0.347216	−	0.937785i	$-0.612873\pi$
$367$	−13.3034	−0.694432	−0.347216	+	0.937785i	$0.612873\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−5.00000	−0.259938
$371$	−72.1572	−3.74621
$372$	0	0
$373$	9.25164	0.479032	0.239516	−	0.970892i	$-0.423011\pi$
$373$	9.25164	0.479032	0.239516	+	0.970892i	$0.423011\pi$
$374$	−7.52354	−0.389033
$375$	0	0
$376$	−3.20673	−0.165375
$377$	−0.575291	−0.0296290
$378$	0	0
$379$	16.2382	0.834102	0.417051	−	0.908883i	$-0.363064\pi$
$379$	16.2382	0.834102	0.417051	+	0.908883i	$0.363064\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	−5.20673	−0.266400
$383$	−5.58653	−0.285459	−0.142729	−	0.989762i	$-0.545588\pi$
$383$	−5.58653	−0.285459	−0.142729	+	0.989762i	$0.545588\pi$
$384$	0	0
$385$	23.7169	1.20872
$386$	1.88993	0.0961948
$387$	0	0
$388$	−6.41347	−0.325594
$389$	9.85843	0.499842	0.249921	−	0.968266i	$-0.419595\pi$
$389$	9.85843	0.499842	0.249921	+	0.968266i	$0.419595\pi$
$390$	0	0
$391$	1.65170	0.0835299
$392$	−20.1101	−1.01571
$393$	0	0
$394$	13.8718	0.698853
$395$	−9.20673	−0.463241
$396$	0	0
$397$	−33.3686	−1.67472	−0.837360	−	0.546652i	$-0.815902\pi$
$397$	−33.3686	−1.67472	−0.837360	+	0.546652i	$0.815902\pi$
$398$	−15.7169	−0.787815
$399$	0	0
$400$	−4.00000	−0.200000
$401$	6.15498	0.307365	0.153683	−	0.988120i	$-0.450887\pi$
$401$	6.15498	0.307365	0.153683	+	0.988120i	$0.450887\pi$
$402$	0	0
$403$	14.1303	0.703882
$404$	−10.4135	−0.518089
$405$	0	0
$406$	1.81352	0.0900035
$407$	22.7752	1.12892
$408$	0	0
$409$	−18.7169	−0.925489	−0.462745	−	0.886492i	$-0.653135\pi$
$409$	−18.7169	−0.925489	−0.462745	+	0.886492i	$0.653135\pi$
$410$	9.85843	0.486873
$411$	0	0
$412$	−2.41347	−0.118903
$413$	47.9370	2.35883
$414$	0	0
$415$	−0.555036	−0.0272457
$416$	−1.65170	−0.0809811
$417$	0	0
$418$	−18.2201	−0.891176
$419$	28.6336	1.39884	0.699422	−	0.714709i	$-0.253441\pi$
$419$	28.6336	1.39884	0.699422	+	0.714709i	$0.253441\pi$
$420$	0	0
$421$	12.1101	0.590209	0.295104	−	0.955465i	$-0.404646\pi$
$421$	12.1101	0.590209	0.295104	+	0.955465i	$0.404646\pi$
$422$	19.8135	0.964507
$423$	0	0
$424$	13.8584	0.673025
$425$	−6.60679	−0.320476
$426$	0	0
$427$	−59.4269	−2.87587
$428$	−7.20673	−0.348351
$429$	0	0
$430$	−7.20673	−0.347540
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−24.0652	−1.15650	−0.578249	−	0.815860i	$-0.696264\pi$
$433$	−24.0652	−1.15650	−0.578249	+	0.815860i	$0.696264\pi$
$434$	−44.5438	−2.13817
$435$	0	0
$436$	6.44496	0.308658
$437$	4.00000	0.191346
$438$	0	0
$439$	−8.87184	−0.423430	−0.211715	−	0.977331i	$-0.567905\pi$
$439$	−8.87184	−0.423430	−0.211715	+	0.977331i	$0.567905\pi$
$440$	−4.55504	−0.217153
$441$	0	0
$442$	−2.72810	−0.129763
$443$	4.37980	0.208091	0.104045	−	0.994573i	$-0.466821\pi$
$443$	4.37980	0.208091	0.104045	+	0.994573i	$0.466821\pi$
$444$	0	0
$445$	−3.65170	−0.173107
$446$	15.7618	0.746341
$447$	0	0
$448$	5.20673	0.245995
$449$	−24.0786	−1.13634	−0.568169	−	0.822912i	$-0.692348\pi$
$449$	−24.0786	−1.13634	−0.568169	+	0.822912i	$0.692348\pi$
$450$	0	0
$451$	−44.9055	−2.11452
$452$	11.4584	0.538957
$453$	0	0
$454$	0.599946	0.0281568
$455$	8.59995	0.403172
$456$	0	0
$457$	9.87184	0.461785	0.230893	−	0.972979i	$-0.425835\pi$
$457$	9.87184	0.461785	0.230893	+	0.972979i	$0.425835\pi$
$458$	−13.5550	−0.633385
$459$	0	0
$460$	1.00000	0.0466252
$461$	33.3302	1.55234	0.776172	−	0.630522i	$-0.217159\pi$
$461$	33.3302	1.55234	0.776172	+	0.630522i	$0.217159\pi$
$462$	0	0
$463$	−5.85843	−0.272264	−0.136132	−	0.990691i	$-0.543467\pi$
$463$	−5.85843	−0.272264	−0.136132	+	0.990691i	$0.543467\pi$
$464$	−0.348303	−0.0161696
$465$	0	0
$466$	−10.2516	−0.474898
$467$	−27.4786	−1.27156	−0.635780	−	0.771871i	$-0.719321\pi$
$467$	−27.4786	−1.27156	−0.635780	+	0.771871i	$0.719321\pi$
$468$	0	0
$469$	7.02025	0.324165
$470$	−3.20673	−0.147916
$471$	0	0
$472$	−9.20673	−0.423774
$473$	32.8269	1.50938
$474$	0	0
$475$	−16.0000	−0.734130
$476$	8.59995	0.394178
$477$	0	0
$478$	−28.2719	−1.29313
$479$	12.7303	0.581661	0.290831	−	0.956775i	$-0.406068\pi$
$479$	12.7303	0.581661	0.290831	+	0.956775i	$0.406068\pi$
$480$	0	0
$481$	8.25848	0.376554
$482$	9.45838	0.430817
$483$	0	0
$484$	9.74836	0.443107
$485$	−6.41347	−0.291221
$486$	0	0
$487$	22.6517	1.02645	0.513223	−	0.858255i	$-0.328451\pi$
$487$	22.6517	1.02645	0.513223	+	0.858255i	$0.328451\pi$
$488$	11.4135	0.516663
$489$	0	0
$490$	−20.1101	−0.908480
$491$	6.41347	0.289436	0.144718	−	0.989473i	$-0.453773\pi$
$491$	6.41347	0.289436	0.144718	+	0.989473i	$0.453773\pi$
$492$	0	0
$493$	−0.575291	−0.0259098
$494$	−6.60679	−0.297253
$495$	0	0
$496$	8.55504	0.384132
$497$	−27.8472	−1.24912
$498$	0	0
$499$	−5.39321	−0.241433	−0.120717	−	0.992687i	$-0.538519\pi$
$499$	−5.39321	−0.241433	−0.120717	+	0.992687i	$0.538519\pi$
$500$	−9.00000	−0.402492
$501$	0	0
$502$	−16.0786	−0.717622
$503$	19.7169	0.879131	0.439566	−	0.898210i	$-0.355132\pi$
$503$	19.7169	0.879131	0.439566	+	0.898210i	$0.355132\pi$
$504$	0	0
$505$	−10.4135	−0.463393
$506$	−4.55504	−0.202496
$507$	0	0
$508$	−17.6202	−0.781770
$509$	−3.39321	−0.150401	−0.0752007	−	0.997168i	$-0.523960\pi$
$509$	−3.39321	−0.150401	−0.0752007	+	0.997168i	$0.523960\pi$
$510$	0	0
$511$	−86.7708	−3.83851
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	17.8269	0.786312
$515$	−2.41347	−0.106350
$516$	0	0
$517$	14.6068	0.642406
$518$	−26.0337	−1.14385
$519$	0	0
$520$	−1.65170	−0.0724317
$521$	21.7169	0.951433	0.475716	−	0.879599i	$-0.342189\pi$
$521$	21.7169	0.951433	0.475716	+	0.879599i	$0.342189\pi$
$522$	0	0
$523$	−25.3820	−1.10988	−0.554938	−	0.831892i	$-0.687258\pi$
$523$	−25.3820	−1.10988	−0.554938	+	0.831892i	$0.687258\pi$
$524$	−14.3168	−0.625433
$525$	0	0
$526$	−30.0337	−1.30953
$527$	14.1303	0.615527
$528$	0	0
$529$	1.00000	0.0434783
$530$	13.8584	0.601972
$531$	0	0
$532$	20.8269	0.902962
$533$	−16.2831	−0.705301
$534$	0	0
$535$	−7.20673	−0.311574
$536$	−1.34830	−0.0582378
$537$	0	0
$538$	8.15498	0.351586
$539$	91.6021	3.94558
$540$	0	0
$541$	−16.4786	−0.708472	−0.354236	−	0.935156i	$-0.615259\pi$
$541$	−16.4786	−0.708472	−0.354236	+	0.935156i	$0.615259\pi$
$542$	6.96850	0.299323
$543$	0	0
$544$	−1.65170	−0.0708159
$545$	6.44496	0.276072
$546$	0	0
$547$	25.8403	1.10485	0.552427	−	0.833561i	$-0.313702\pi$
$547$	25.8403	1.10485	0.552427	+	0.833561i	$0.313702\pi$
$548$	−19.8718	−0.848883
$549$	0	0
$550$	18.2201	0.776910
$551$	−1.39321	−0.0593528
$552$	0	0
$553$	−47.9370	−2.03849
$554$	31.9370	1.35687
$555$	0	0
$556$	0.0966605	0.00409932
$557$	23.6336	1.00139	0.500694	−	0.865624i	$-0.333078\pi$
$557$	23.6336	1.00139	0.500694	+	0.865624i	$0.333078\pi$
$558$	0	0
$559$	11.9033	0.503458
$560$	5.20673	0.220025
$561$	0	0
$562$	−4.54162	−0.191577
$563$	−4.19332	−0.176727	−0.0883637	−	0.996088i	$-0.528164\pi$
$563$	−4.19332	−0.176727	−0.0883637	+	0.996088i	$0.528164\pi$
$564$	0	0
$565$	11.4584	0.482058
$566$	−2.45838	−0.103333
$567$	0	0
$568$	5.34830	0.224410
$569$	−20.9551	−0.878483	−0.439242	−	0.898369i	$-0.644753\pi$
$569$	−20.9551	−0.878483	−0.439242	+	0.898369i	$0.644753\pi$
$570$	0	0
$571$	12.7484	0.533502	0.266751	−	0.963765i	$-0.414050\pi$
$571$	12.7484	0.533502	0.266751	+	0.963765i	$0.414050\pi$
$572$	7.52354	0.314575
$573$	0	0
$574$	51.3302	2.14248
$575$	−4.00000	−0.166812
$576$	0	0
$577$	1.69661	0.0706306	0.0353153	−	0.999376i	$-0.488756\pi$
$577$	1.69661	0.0706306	0.0353153	+	0.999376i	$0.488756\pi$
$578$	14.2719	0.593633
$579$	0	0
$580$	−0.348303	−0.0144625
$581$	−2.88993	−0.119894
$582$	0	0
$583$	−63.1257	−2.61440
$584$	16.6651	0.689607
$585$	0	0
$586$	−24.1101	−0.995978
$587$	−0.730273	−0.0301416	−0.0150708	−	0.999886i	$-0.504797\pi$
$587$	−0.730273	−0.0301416	−0.0150708	+	0.999886i	$0.504797\pi$
$588$	0	0
$589$	34.2201	1.41002
$590$	−9.20673	−0.379035
$591$	0	0
$592$	5.00000	0.205499
$593$	26.1101	1.07221	0.536106	−	0.844151i	$-0.319895\pi$
$593$	26.1101	1.07221	0.536106	+	0.844151i	$0.319895\pi$
$594$	0	0
$595$	8.59995	0.352563
$596$	−2.25164	−0.0922309
$597$	0	0
$598$	−1.65170	−0.0675429
$599$	48.3056	1.97371	0.986856	−	0.161603i	$-0.0516663\pi$
$599$	48.3056	1.97371	0.986856	+	0.161603i	$0.0516663\pi$
$600$	0	0
$601$	34.0203	1.38772	0.693858	−	0.720112i	$-0.255910\pi$
$601$	34.0203	1.38772	0.693858	+	0.720112i	$0.255910\pi$
$602$	−37.5235	−1.52935
$603$	0	0
$604$	−1.34830	−0.0548617
$605$	9.74836	0.396327
$606$	0	0
$607$	21.1886	0.860021	0.430010	−	0.902824i	$-0.358510\pi$
$607$	21.1886	0.860021	0.430010	+	0.902824i	$0.358510\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	11.4135	0.462118
$611$	5.29655	0.214276
$612$	0	0
$613$	−22.8899	−0.924515	−0.462258	−	0.886746i	$-0.652961\pi$
$613$	−22.8899	−0.924515	−0.462258	+	0.886746i	$0.652961\pi$
$614$	−21.0134	−0.848032
$615$	0	0
$616$	−23.7169	−0.955580
$617$	−35.6785	−1.43636	−0.718182	−	0.695856i	$-0.755025\pi$
$617$	−35.6785	−1.43636	−0.718182	+	0.695856i	$0.755025\pi$
$618$	0	0
$619$	−22.3686	−0.899068	−0.449534	−	0.893263i	$-0.648410\pi$
$619$	−22.3686	−0.899068	−0.449534	+	0.893263i	$0.648410\pi$
$620$	8.55504	0.343579
$621$	0	0
$622$	5.66511	0.227150
$623$	−19.0134	−0.761756
$624$	0	0
$625$	11.0000	0.440000
$626$	−23.8718	−0.954111
$627$	0	0
$628$	1.74836	0.0697671
$629$	8.25848	0.329287
$630$	0	0
$631$	−18.7933	−0.748148	−0.374074	−	0.927399i	$-0.622039\pi$
$631$	−18.7933	−0.748148	−0.374074	+	0.927399i	$0.622039\pi$
$632$	9.20673	0.366224
$633$	0	0
$634$	7.17524	0.284965
$635$	−17.6202	−0.699236
$636$	0	0
$637$	33.2157	1.31606
$638$	1.58653	0.0628114
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	31.7820	1.25531	0.627657	−	0.778490i	$-0.284014\pi$
$641$	31.7820	1.25531	0.627657	+	0.778490i	$0.284014\pi$
$642$	0	0
$643$	25.9819	1.02463	0.512313	−	0.858799i	$-0.328789\pi$
$643$	25.9819	1.02463	0.512313	+	0.858799i	$0.328789\pi$
$644$	5.20673	0.205174
$645$	0	0
$646$	−6.60679	−0.259941
$647$	−13.3820	−0.526100	−0.263050	−	0.964782i	$-0.584728\pi$
$647$	−13.3820	−0.526100	−0.263050	+	0.964782i	$0.584728\pi$
$648$	0	0
$649$	41.9370	1.64617
$650$	6.60679	0.259140
$651$	0	0
$652$	14.5101	0.568260
$653$	−11.1101	−0.434771	−0.217385	−	0.976086i	$-0.569753\pi$
$653$	−11.1101	−0.434771	−0.217385	+	0.976086i	$0.569753\pi$
$654$	0	0
$655$	−14.3168	−0.559404
$656$	−9.85843	−0.384907
$657$	0	0
$658$	−16.6966	−0.650901
$659$	−8.79327	−0.342537	−0.171269	−	0.985224i	$-0.554787\pi$
$659$	−8.79327	−0.342537	−0.171269	+	0.985224i	$0.554787\pi$
$660$	0	0
$661$	−16.2404	−0.631678	−0.315839	−	0.948813i	$-0.602286\pi$
$661$	−16.2404	−0.631678	−0.315839	+	0.948813i	$0.602286\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	0.555036	0.0215396
$665$	20.8269	0.807634
$666$	0	0
$667$	−0.348303	−0.0134863
$668$	3.76177	0.145547
$669$	0	0
$670$	−1.34830	−0.0520895
$671$	−51.9888	−2.00700
$672$	0	0
$673$	−40.1618	−1.54812	−0.774062	−	0.633109i	$-0.781778\pi$
$673$	−40.1618	−1.54812	−0.774062	+	0.633109i	$0.781778\pi$
$674$	−15.1101	−0.582018
$675$	0	0
$676$	−10.2719	−0.395073
$677$	39.7954	1.52946	0.764731	−	0.644349i	$-0.222872\pi$
$677$	39.7954	1.52946	0.764731	+	0.644349i	$0.222872\pi$
$678$	0	0
$679$	−33.3932	−1.28151
$680$	−1.65170	−0.0633397
$681$	0	0
$682$	−38.9685	−1.49218
$683$	−20.7303	−0.793222	−0.396611	−	0.917987i	$-0.629814\pi$
$683$	−20.7303	−0.793222	−0.396611	+	0.917987i	$0.629814\pi$
$684$	0	0
$685$	−19.8718	−0.759264
$686$	−68.2607	−2.60620
$687$	0	0
$688$	7.20673	0.274754
$689$	−22.8899	−0.872037
$690$	0	0
$691$	−23.2404	−0.884107	−0.442053	−	0.896989i	$-0.645750\pi$
$691$	−23.2404	−0.884107	−0.442053	+	0.896989i	$0.645750\pi$
$692$	−10.0652	−0.382620
$693$	0	0
$694$	−18.5101	−0.702635
$695$	0.0966605	0.00366654
$696$	0	0
$697$	−16.2831	−0.616768
$698$	27.1101	1.02613
$699$	0	0
$700$	−20.8269	−0.787184
$701$	31.6336	1.19479	0.597393	−	0.801949i	$-0.296203\pi$
$701$	31.6336	1.19479	0.597393	+	0.801949i	$0.296203\pi$
$702$	0	0
$703$	20.0000	0.754314
$704$	4.55504	0.171674
$705$	0	0
$706$	−25.5753	−0.962539
$707$	−54.2201	−2.03916
$708$	0	0
$709$	28.0471	1.05333	0.526665	−	0.850073i	$-0.323442\pi$
$709$	28.0471	1.05333	0.526665	+	0.850073i	$0.323442\pi$
$710$	5.34830	0.200718
$711$	0	0
$712$	3.65170	0.136853
$713$	8.55504	0.320389
$714$	0	0
$715$	7.52354	0.281364
$716$	−23.7169	−0.886341
$717$	0	0
$718$	−10.6966	−0.399194
$719$	−38.9685	−1.45328	−0.726640	−	0.687018i	$-0.758919\pi$
$719$	−38.9685	−1.45328	−0.726640	+	0.687018i	$0.758919\pi$
$720$	0	0
$721$	−12.5663	−0.467993
$722$	3.00000	0.111648
$723$	0	0
$724$	7.25164	0.269505
$725$	1.39321	0.0517426
$726$	0	0
$727$	17.2136	0.638416	0.319208	−	0.947685i	$-0.396583\pi$
$727$	17.2136	0.638416	0.319208	+	0.947685i	$0.396583\pi$
$728$	−8.59995	−0.318735
$729$	0	0
$730$	16.6651	0.616803
$731$	11.9033	0.440261
$732$	0	0
$733$	23.7169	0.876002	0.438001	−	0.898974i	$-0.355687\pi$
$733$	23.7169	0.876002	0.438001	+	0.898974i	$0.355687\pi$
$734$	13.3034	0.491037
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	6.14157	0.226228
$738$	0	0
$739$	−36.6336	−1.34759	−0.673795	−	0.738919i	$-0.735337\pi$
$739$	−36.6336	−1.34759	−0.673795	+	0.738919i	$0.735337\pi$
$740$	5.00000	0.183804
$741$	0	0
$742$	72.1572	2.64897
$743$	−44.2538	−1.62351	−0.811757	−	0.583995i	$-0.801489\pi$
$743$	−44.2538	−1.62351	−0.811757	+	0.583995i	$0.801489\pi$
$744$	0	0
$745$	−2.25164	−0.0824938
$746$	−9.25164	−0.338727
$747$	0	0
$748$	7.52354	0.275088
$749$	−37.5235	−1.37108
$750$	0	0
$751$	8.19332	0.298979	0.149489	−	0.988763i	$-0.452237\pi$
$751$	8.19332	0.298979	0.149489	+	0.988763i	$0.452237\pi$
$752$	3.20673	0.116938
$753$	0	0
$754$	0.575291	0.0209509
$755$	−1.34830	−0.0490698
$756$	0	0
$757$	−4.74836	−0.172582	−0.0862910	−	0.996270i	$-0.527501\pi$
$757$	−4.74836	−0.172582	−0.0862910	+	0.996270i	$0.527501\pi$
$758$	−16.2382	−0.589799
$759$	0	0
$760$	−4.00000	−0.145095
$761$	−47.9055	−1.73657	−0.868287	−	0.496063i	$-0.834779\pi$
$761$	−47.9055	−1.73657	−0.868287	+	0.496063i	$0.834779\pi$
$762$	0	0
$763$	33.5572	1.21485
$764$	5.20673	0.188373
$765$	0	0
$766$	5.58653	0.201850
$767$	15.2067	0.549083
$768$	0	0
$769$	4.56845	0.164742	0.0823712	−	0.996602i	$-0.473751\pi$
$769$	4.56845	0.164742	0.0823712	+	0.996602i	$0.473751\pi$
$770$	−23.7169	−0.854696
$771$	0	0
$772$	−1.88993	−0.0680200
$773$	−44.2404	−1.59122	−0.795608	−	0.605811i	$-0.792849\pi$
$773$	−44.2404	−1.59122	−0.795608	+	0.605811i	$0.792849\pi$
$774$	0	0
$775$	−34.2201	−1.22922
$776$	6.41347	0.230230
$777$	0	0
$778$	−9.85843	−0.353442
$779$	−39.4337	−1.41286
$780$	0	0
$781$	−24.3617	−0.871731
$782$	−1.65170	−0.0590646
$783$	0	0
$784$	20.1101	0.718217
$785$	1.74836	0.0624016
$786$	0	0
$787$	−45.0203	−1.60480	−0.802399	−	0.596787i	$-0.796444\pi$
$787$	−45.0203	−1.60480	−0.802399	+	0.596787i	$0.796444\pi$
$788$	−13.8718	−0.494164
$789$	0	0
$790$	9.20673	0.327561
$791$	59.6607	2.12129
$792$	0	0
$793$	−18.8516	−0.669440
$794$	33.3686	1.18421
$795$	0	0
$796$	15.7169	0.557069
$797$	−6.58653	−0.233307	−0.116653	−	0.993173i	$-0.537217\pi$
$797$	−6.58653	−0.233307	−0.116653	+	0.993173i	$0.537217\pi$
$798$	0	0
$799$	5.29655	0.187379
$800$	4.00000	0.141421
$801$	0	0
$802$	−6.15498	−0.217340
$803$	−75.9102	−2.67881
$804$	0	0
$805$	5.20673	0.183513
$806$	−14.1303	−0.497720
$807$	0	0
$808$	10.4135	0.366344
$809$	35.1416	1.23551	0.617756	−	0.786370i	$-0.288042\pi$
$809$	35.1416	1.23551	0.617756	+	0.786370i	$0.288042\pi$
$810$	0	0
$811$	−18.3168	−0.643190	−0.321595	−	0.946877i	$-0.604219\pi$
$811$	−18.3168	−0.643190	−0.321595	+	0.946877i	$0.604219\pi$
$812$	−1.81352	−0.0636421
$813$	0	0
$814$	−22.7752	−0.798270
$815$	14.5101	0.508268
$816$	0	0
$817$	28.8269	1.00853
$818$	18.7169	0.654420
$819$	0	0
$820$	−9.85843	−0.344271
$821$	37.4584	1.30731	0.653653	−	0.756794i	$-0.273236\pi$
$821$	37.4584	1.30731	0.653653	+	0.756794i	$0.273236\pi$
$822$	0	0
$823$	2.84502	0.0991711	0.0495855	−	0.998770i	$-0.484210\pi$
$823$	2.84502	0.0991711	0.0495855	+	0.998770i	$0.484210\pi$
$824$	2.41347	0.0840771
$825$	0	0
$826$	−47.9370	−1.66794
$827$	−36.5887	−1.27231	−0.636157	−	0.771560i	$-0.719477\pi$
$827$	−36.5887	−1.27231	−0.636157	+	0.771560i	$0.719477\pi$
$828$	0	0
$829$	18.8269	0.653886	0.326943	−	0.945044i	$-0.393981\pi$
$829$	18.8269	0.653886	0.326943	+	0.945044i	$0.393981\pi$
$830$	0.555036	0.0192656
$831$	0	0
$832$	1.65170	0.0572623
$833$	33.2157	1.15086
$834$	0	0
$835$	3.76177	0.130181
$836$	18.2201	0.630157
$837$	0	0
$838$	−28.6336	−0.989132
$839$	10.0337	0.346401	0.173200	−	0.984887i	$-0.444589\pi$
$839$	10.0337	0.346401	0.173200	+	0.984887i	$0.444589\pi$
$840$	0	0
$841$	−28.8787	−0.995817
$842$	−12.1101	−0.417341
$843$	0	0
$844$	−19.8135	−0.682010
$845$	−10.2719	−0.353364
$846$	0	0
$847$	50.7571	1.74403
$848$	−13.8584	−0.475900
$849$	0	0
$850$	6.60679	0.226611
$851$	5.00000	0.171398
$852$	0	0
$853$	−15.2404	−0.521821	−0.260911	−	0.965363i	$-0.584023\pi$
$853$	−15.2404	−0.521821	−0.260911	+	0.965363i	$0.584023\pi$
$854$	59.4269	2.03355
$855$	0	0
$856$	7.20673	0.246321
$857$	35.0268	1.19649	0.598247	−	0.801312i	$-0.295864\pi$
$857$	35.0268	1.19649	0.598247	+	0.801312i	$0.295864\pi$
$858$	0	0
$859$	23.7169	0.809209	0.404604	−	0.914492i	$-0.367409\pi$
$859$	23.7169	0.809209	0.404604	+	0.914492i	$0.367409\pi$
$860$	7.20673	0.245748
$861$	0	0
$862$	−12.0000	−0.408722
$863$	23.9551	0.815441	0.407720	−	0.913107i	$-0.366324\pi$
$863$	23.9551	0.815441	0.407720	+	0.913107i	$0.366324\pi$
$864$	0	0
$865$	−10.0652	−0.342226
$866$	24.0652	0.817768
$867$	0	0
$868$	44.5438	1.51191
$869$	−41.9370	−1.42262
$870$	0	0
$871$	2.22699	0.0754586
$872$	−6.44496	−0.218254
$873$	0	0
$874$	−4.00000	−0.135302
$875$	−46.8606	−1.58418
$876$	0	0
$877$	15.2382	0.514558	0.257279	−	0.966337i	$-0.417174\pi$
$877$	15.2382	0.514558	0.257279	+	0.966337i	$0.417174\pi$
$878$	8.87184	0.299410
$879$	0	0
$880$	4.55504	0.153550
$881$	18.8269	0.634296	0.317148	−	0.948376i	$-0.397275\pi$
$881$	18.8269	0.634296	0.317148	+	0.948376i	$0.397275\pi$
$882$	0	0
$883$	−2.79327	−0.0940009	−0.0470005	−	0.998895i	$-0.514966\pi$
$883$	−2.79327	−0.0940009	−0.0470005	+	0.998895i	$0.514966\pi$
$884$	2.72810	0.0917560
$885$	0	0
$886$	−4.37980	−0.147142
$887$	−41.3483	−1.38834	−0.694170	−	0.719811i	$-0.744228\pi$
$887$	−41.3483	−1.38834	−0.694170	+	0.719811i	$0.744228\pi$
$888$	0	0
$889$	−91.7437	−3.07698
$890$	3.65170	0.122405
$891$	0	0
$892$	−15.7618	−0.527743
$893$	12.8269	0.429237
$894$	0	0
$895$	−23.7169	−0.792767
$896$	−5.20673	−0.173945
$897$	0	0
$898$	24.0786	0.803512
$899$	−2.97975	−0.0993801
$900$	0	0
$901$	−22.8899	−0.762574
$902$	44.9055	1.49519
$903$	0	0
$904$	−11.4584	−0.381100
$905$	7.25164	0.241053
$906$	0	0
$907$	49.3820	1.63970	0.819851	−	0.572577i	$-0.194056\pi$
$907$	49.3820	1.63970	0.819851	+	0.572577i	$0.194056\pi$
$908$	−0.599946	−0.0199099
$909$	0	0
$910$	−8.59995	−0.285085
$911$	−2.69661	−0.0893425	−0.0446713	−	0.999002i	$-0.514224\pi$
$911$	−2.69661	−0.0893425	−0.0446713	+	0.999002i	$0.514224\pi$
$912$	0	0
$913$	−2.52821	−0.0836716
$914$	−9.87184	−0.326532
$915$	0	0
$916$	13.5550	0.447871
$917$	−74.5438	−2.46165
$918$	0	0
$919$	−10.3100	−0.340094	−0.170047	−	0.985436i	$-0.554392\pi$
$919$	−10.3100	−0.340094	−0.170047	+	0.985436i	$0.554392\pi$
$920$	−1.00000	−0.0329690
$921$	0	0
$922$	−33.3302	−1.09767
$923$	−8.83378	−0.290767
$924$	0	0
$925$	−20.0000	−0.657596
$926$	5.85843	0.192520
$927$	0	0
$928$	0.348303	0.0114336
$929$	15.4135	0.505699	0.252850	−	0.967506i	$-0.418632\pi$
$929$	15.4135	0.505699	0.252850	+	0.967506i	$0.418632\pi$
$930$	0	0
$931$	80.4403	2.63632
$932$	10.2516	0.335804
$933$	0	0
$934$	27.4786	0.899128
$935$	7.52354	0.246046
$936$	0	0
$937$	52.5055	1.71528	0.857639	−	0.514252i	$-0.171930\pi$
$937$	52.5055	1.71528	0.857639	+	0.514252i	$0.171930\pi$
$938$	−7.02025	−0.229219
$939$	0	0
$940$	3.20673	0.104592
$941$	−15.8899	−0.517997	−0.258998	−	0.965878i	$-0.583392\pi$
$941$	−15.8899	−0.517997	−0.258998	+	0.965878i	$0.583392\pi$
$942$	0	0
$943$	−9.85843	−0.321035
$944$	9.20673	0.299654
$945$	0	0
$946$	−32.8269	−1.06730
$947$	−4.82009	−0.156632	−0.0783160	−	0.996929i	$-0.524954\pi$
$947$	−4.82009	−0.156632	−0.0783160	+	0.996929i	$0.524954\pi$
$948$	0	0
$949$	−27.5257	−0.893522
$950$	16.0000	0.519109
$951$	0	0
$952$	−8.59995	−0.278726
$953$	24.3483	0.788719	0.394359	−	0.918956i	$-0.370966\pi$
$953$	24.3483	0.788719	0.394359	+	0.918956i	$0.370966\pi$
$954$	0	0
$955$	5.20673	0.168486
$956$	28.2719	0.914379
$957$	0	0
$958$	−12.7303	−0.411296
$959$	−103.467	−3.34114
$960$	0	0
$961$	42.1886	1.36092
$962$	−8.25848	−0.266264
$963$	0	0
$964$	−9.45838	−0.304634
$965$	−1.88993	−0.0608389
$966$	0	0
$967$	−40.2382	−1.29397	−0.646987	−	0.762501i	$-0.723971\pi$
$967$	−40.2382	−1.29397	−0.646987	+	0.762501i	$0.723971\pi$
$968$	−9.74836	−0.313324
$969$	0	0
$970$	6.41347	0.205924
$971$	−61.4605	−1.97236	−0.986181	−	0.165669i	$-0.947022\pi$
$971$	−61.4605	−1.97236	−0.986181	+	0.165669i	$0.947022\pi$
$972$	0	0
$973$	0.503285	0.0161346
$974$	−22.6517	−0.725807
$975$	0	0
$976$	−11.4135	−0.365336
$977$	6.41347	0.205185	0.102592	−	0.994723i	$-0.467286\pi$
$977$	6.41347	0.205185	0.102592	+	0.994723i	$0.467286\pi$
$978$	0	0
$979$	−16.6336	−0.531613
$980$	20.1101	0.642393
$981$	0	0
$982$	−6.41347	−0.204662
$983$	−51.6202	−1.64643	−0.823214	−	0.567731i	$-0.807821\pi$
$983$	−51.6202	−1.64643	−0.823214	+	0.567731i	$0.807821\pi$
$984$	0	0
$985$	−13.8718	−0.441994
$986$	0.575291	0.0183210
$987$	0	0
$988$	6.60679	0.210190
$989$	7.20673	0.229161
$990$	0	0
$991$	44.8606	1.42504	0.712522	−	0.701650i	$-0.247553\pi$
$991$	44.8606	1.42504	0.712522	+	0.701650i	$0.247553\pi$
$992$	−8.55504	−0.271623
$993$	0	0
$994$	27.8472	0.883259
$995$	15.7169	0.498258
$996$	0	0
$997$	3.26506	0.103405	0.0517027	−	0.998663i	$-0.483535\pi$
$997$	3.26506	0.103405	0.0517027	+	0.998663i	$0.483535\pi$
$998$	5.39321	0.170719
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3726.2.a.m.1.3	✓	3
3.2	odd	2		3726.2.a.n.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3726.2.a.m.1.3	✓	3	1.1	even	1	trivial
3726.2.a.n.1.3	yes	3	3.2	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:	\( N \)	\(=\)	\( 3726 = 2 \cdot 3^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3726.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +5.20673 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +5.20673 q^{7} -1.00000 q^{8} -1.00000 q^{10} +4.55504 q^{11} +1.65170 q^{13} -5.20673 q^{14} +1.00000 q^{16} +1.65170 q^{17} +4.00000 q^{19} +1.00000 q^{20} -4.55504 q^{22} +1.00000 q^{23} -4.00000 q^{25} -1.65170 q^{26} +5.20673 q^{28} -0.348303 q^{29} +8.55504 q^{31} -1.00000 q^{32} -1.65170 q^{34} +5.20673 q^{35} +5.00000 q^{37} -4.00000 q^{38} -1.00000 q^{40} -9.85843 q^{41} +7.20673 q^{43} +4.55504 q^{44} -1.00000 q^{46} +3.20673 q^{47} +20.1101 q^{49} +4.00000 q^{50} +1.65170 q^{52} -13.8584 q^{53} +4.55504 q^{55} -5.20673 q^{56} +0.348303 q^{58} +9.20673 q^{59} -11.4135 q^{61} -8.55504 q^{62} +1.00000 q^{64} +1.65170 q^{65} +1.34830 q^{67} +1.65170 q^{68} -5.20673 q^{70} -5.34830 q^{71} -16.6651 q^{73} -5.00000 q^{74} +4.00000 q^{76} +23.7169 q^{77} -9.20673 q^{79} +1.00000 q^{80} +9.85843 q^{82} -0.555036 q^{83} +1.65170 q^{85} -7.20673 q^{86} -4.55504 q^{88} -3.65170 q^{89} +8.59995 q^{91} +1.00000 q^{92} -3.20673 q^{94} +4.00000 q^{95} -6.41347 q^{97} -20.1101 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 2 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 2 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 2 q^{7} - 3 q^{8} - 3 q^{10} + q^{11} + 2 q^{14} + 3 q^{16} + 12 q^{19} + 3 q^{20} - q^{22} + 3 q^{23} - 12 q^{25} - 2 q^{28} - 6 q^{29} + 13 q^{31} - 3 q^{32} - 2 q^{35} + 15 q^{37} - 12 q^{38} - 3 q^{40} - 7 q^{41} + 4 q^{43} + q^{44} - 3 q^{46} - 8 q^{47} + 35 q^{49} + 12 q^{50} - 19 q^{53} + q^{55} + 2 q^{56} + 6 q^{58} + 10 q^{59} + q^{61} - 13 q^{62} + 3 q^{64} + 9 q^{67} + 2 q^{70} - 21 q^{71} - 12 q^{73} - 15 q^{74} + 12 q^{76} + 26 q^{77} - 10 q^{79} + 3 q^{80} + 7 q^{82} + 11 q^{83} - 4 q^{86} - q^{88} - 6 q^{89} + 28 q^{91} + 3 q^{92} + 8 q^{94} + 12 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 2 * q^7 - 3 * q^8 - 3 * q^10 + q^11 + 2 * q^14 + 3 * q^16 + 12 * q^19 + 3 * q^20 - q^22 + 3 * q^23 - 12 * q^25 - 2 * q^28 - 6 * q^29 + 13 * q^31 - 3 * q^32 - 2 * q^35 + 15 * q^37 - 12 * q^38 - 3 * q^40 - 7 * q^41 + 4 * q^43 + q^44 - 3 * q^46 - 8 * q^47 + 35 * q^49 + 12 * q^50 - 19 * q^53 + q^55 + 2 * q^56 + 6 * q^58 + 10 * q^59 + q^61 - 13 * q^62 + 3 * q^64 + 9 * q^67 + 2 * q^70 - 21 * q^71 - 12 * q^73 - 15 * q^74 + 12 * q^76 + 26 * q^77 - 10 * q^79 + 3 * q^80 + 7 * q^82 + 11 * q^83 - 4 * q^86 - q^88 - 6 * q^89 + 28 * q^91 + 3 * q^92 + 8 * q^94 + 12 * q^95 + 16 * q^97 - 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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