LMFDB - Embedded newform 3726.2.a.v.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:	$5$
Coefficient field:	5.5.1030257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 11x^{3} - 4x^{2} + 7x - 1 $ x^5 - x^4 - 11x^3 - 4x^2 + 7*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 414)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.25060$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.11802 q^{5} -1.85241 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.11802 q^{5} -1.85241 q^{7} +1.00000 q^{8} +1.11802 q^{10} +0.749396 q^{11} +4.65202 q^{13} -1.85241 q^{14} +1.00000 q^{16} +2.53078 q^{17} +2.66702 q^{19} +1.11802 q^{20} +0.749396 q^{22} -1.00000 q^{23} -3.75004 q^{25} +4.65202 q^{26} -1.85241 q^{28} +4.55223 q^{29} +1.05280 q^{31} +1.00000 q^{32} +2.53078 q^{34} -2.07102 q^{35} +0.587232 q^{37} +2.66702 q^{38} +1.11802 q^{40} -6.70224 q^{41} +1.70159 q^{43} +0.749396 q^{44} -1.00000 q^{46} +7.53766 q^{47} -3.56858 q^{49} -3.75004 q^{50} +4.65202 q^{52} -5.20468 q^{53} +0.837836 q^{55} -1.85241 q^{56} +4.55223 q^{58} -3.35185 q^{59} -0.967200 q^{61} +1.05280 q^{62} +1.00000 q^{64} +5.20103 q^{65} +7.55723 q^{67} +2.53078 q^{68} -2.07102 q^{70} +9.39777 q^{71} +3.04657 q^{73} +0.587232 q^{74} +2.66702 q^{76} -1.38819 q^{77} +10.2916 q^{79} +1.11802 q^{80} -6.70224 q^{82} +9.80499 q^{83} +2.82945 q^{85} +1.70159 q^{86} +0.749396 q^{88} -9.50470 q^{89} -8.61745 q^{91} -1.00000 q^{92} +7.53766 q^{94} +2.98177 q^{95} +9.98672 q^{97} -3.56858 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} + 5 q^{5} + 3 q^{7} + 5 q^{8}+O(q^{10}) $ 5 * q + 5 * q^2 + 5 * q^4 + 5 * q^5 + 3 * q^7 + 5 * q^8	$ 5 q + 5 q^{2} + 5 q^{4} + 5 q^{5} + 3 q^{7} + 5 q^{8} + 5 q^{10} + 11 q^{11} + 3 q^{14} + 5 q^{16} + 11 q^{17} + 9 q^{19} + 5 q^{20} + 11 q^{22} - 5 q^{23} + 12 q^{25} + 3 q^{28} - 2 q^{29} + 4 q^{31} + 5 q^{32} + 11 q^{34} - 7 q^{35} + 4 q^{37} + 9 q^{38} + 5 q^{40} + 8 q^{41} + 5 q^{43} + 11 q^{44} - 5 q^{46} + 19 q^{47} + 4 q^{49} + 12 q^{50} - 3 q^{53} - 2 q^{55} + 3 q^{56} - 2 q^{58} + 19 q^{59} - 13 q^{61} + 4 q^{62} + 5 q^{64} - q^{65} + q^{67} + 11 q^{68} - 7 q^{70} + 27 q^{71} - 7 q^{73} + 4 q^{74} + 9 q^{76} + 19 q^{77} + 3 q^{79} + 5 q^{80} + 8 q^{82} + 9 q^{83} - 5 q^{85} + 5 q^{86} + 11 q^{88} + 9 q^{89} - 19 q^{91} - 5 q^{92} + 19 q^{94} + 17 q^{95} - 20 q^{97} + 4 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 + 5 * q^5 + 3 * q^7 + 5 * q^8 + 5 * q^10 + 11 * q^11 + 3 * q^14 + 5 * q^16 + 11 * q^17 + 9 * q^19 + 5 * q^20 + 11 * q^22 - 5 * q^23 + 12 * q^25 + 3 * q^28 - 2 * q^29 + 4 * q^31 + 5 * q^32 + 11 * q^34 - 7 * q^35 + 4 * q^37 + 9 * q^38 + 5 * q^40 + 8 * q^41 + 5 * q^43 + 11 * q^44 - 5 * q^46 + 19 * q^47 + 4 * q^49 + 12 * q^50 - 3 * q^53 - 2 * q^55 + 3 * q^56 - 2 * q^58 + 19 * q^59 - 13 * q^61 + 4 * q^62 + 5 * q^64 - q^65 + q^67 + 11 * q^68 - 7 * q^70 + 27 * q^71 - 7 * q^73 + 4 * q^74 + 9 * q^76 + 19 * q^77 + 3 * q^79 + 5 * q^80 + 8 * q^82 + 9 * q^83 - 5 * q^85 + 5 * q^86 + 11 * q^88 + 9 * q^89 - 19 * q^91 - 5 * q^92 + 19 * q^94 + 17 * q^95 - 20 * q^97 + 4 * 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
$5$	1.11802	0.499992	0.249996	−	0.968247i	$-0.419571\pi$
$5$	1.11802	0.499992	0.249996	+	0.968247i	$0.419571\pi$
$6$	0	0
$7$	−1.85241	−0.700145	−0.350073	−	0.936723i	$-0.613843\pi$
$7$	−1.85241	−0.700145	−0.350073	+	0.936723i	$0.613843\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.11802	0.353547
$11$	0.749396	0.225951	0.112976	−	0.993598i	$-0.463962\pi$
$11$	0.749396	0.225951	0.112976	+	0.993598i	$0.463962\pi$
$12$	0	0
$13$	4.65202	1.29024	0.645120	−	0.764082i	$-0.276808\pi$
$13$	4.65202	1.29024	0.645120	+	0.764082i	$0.276808\pi$
$14$	−1.85241	−0.495077
$15$	0	0
$16$	1.00000	0.250000
$17$	2.53078	0.613805	0.306903	−	0.951741i	$-0.400707\pi$
$17$	2.53078	0.613805	0.306903	+	0.951741i	$0.400707\pi$
$18$	0	0
$19$	2.66702	0.611857	0.305929	−	0.952054i	$-0.401033\pi$
$19$	2.66702	0.611857	0.305929	+	0.952054i	$0.401033\pi$
$20$	1.11802	0.249996
$21$	0	0
$22$	0.749396	0.159772
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.75004	−0.750008
$26$	4.65202	0.912337
$27$	0	0
$28$	−1.85241	−0.350073
$29$	4.55223	0.845329	0.422664	−	0.906286i	$-0.361095\pi$
$29$	4.55223	0.845329	0.422664	+	0.906286i	$0.361095\pi$
$30$	0	0
$31$	1.05280	0.189088	0.0945439	−	0.995521i	$-0.469861\pi$
$31$	1.05280	0.189088	0.0945439	+	0.995521i	$0.469861\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.53078	0.434026
$35$	−2.07102	−0.350067
$36$	0	0
$37$	0.587232	0.0965403	0.0482702	−	0.998834i	$-0.484629\pi$
$37$	0.587232	0.0965403	0.0482702	+	0.998834i	$0.484629\pi$
$38$	2.66702	0.432649
$39$	0	0
$40$	1.11802	0.176774
$41$	−6.70224	−1.04671	−0.523357	−	0.852113i	$-0.675321\pi$
$41$	−6.70224	−1.04671	−0.523357	+	0.852113i	$0.675321\pi$
$42$	0	0
$43$	1.70159	0.259491	0.129745	−	0.991547i	$-0.458584\pi$
$43$	1.70159	0.259491	0.129745	+	0.991547i	$0.458584\pi$
$44$	0.749396	0.112976
$45$	0	0
$46$	−1.00000	−0.147442
$47$	7.53766	1.09948	0.549740	−	0.835336i	$-0.314727\pi$
$47$	7.53766	1.09948	0.549740	+	0.835336i	$0.314727\pi$
$48$	0	0
$49$	−3.56858	−0.509797
$50$	−3.75004	−0.530336
$51$	0	0
$52$	4.65202	0.645120
$53$	−5.20468	−0.714918	−0.357459	−	0.933929i	$-0.616357\pi$
$53$	−5.20468	−0.714918	−0.357459	+	0.933929i	$0.616357\pi$
$54$	0	0
$55$	0.837836	0.112974
$56$	−1.85241	−0.247539
$57$	0	0
$58$	4.55223	0.597738
$59$	−3.35185	−0.436373	−0.218187	−	0.975907i	$-0.570014\pi$
$59$	−3.35185	−0.436373	−0.218187	+	0.975907i	$0.570014\pi$
$60$	0	0
$61$	−0.967200	−0.123837	−0.0619186	−	0.998081i	$-0.519722\pi$
$61$	−0.967200	−0.123837	−0.0619186	+	0.998081i	$0.519722\pi$
$62$	1.05280	0.133705
$63$	0	0
$64$	1.00000	0.125000
$65$	5.20103	0.645109
$66$	0	0
$67$	7.55723	0.923262	0.461631	−	0.887072i	$-0.347264\pi$
$67$	7.55723	0.923262	0.461631	+	0.887072i	$0.347264\pi$
$68$	2.53078	0.306903
$69$	0	0
$70$	−2.07102	−0.247534
$71$	9.39777	1.11531	0.557655	−	0.830073i	$-0.311701\pi$
$71$	9.39777	1.11531	0.557655	+	0.830073i	$0.311701\pi$
$72$	0	0
$73$	3.04657	0.356574	0.178287	−	0.983979i	$-0.442945\pi$
$73$	3.04657	0.356574	0.178287	+	0.983979i	$0.442945\pi$
$74$	0.587232	0.0682643
$75$	0	0
$76$	2.66702	0.305929
$77$	−1.38819	−0.158199
$78$	0	0
$79$	10.2916	1.15790	0.578949	−	0.815364i	$-0.303463\pi$
$79$	10.2916	1.15790	0.578949	+	0.815364i	$0.303463\pi$
$80$	1.11802	0.124998
$81$	0	0
$82$	−6.70224	−0.740139
$83$	9.80499	1.07624	0.538119	−	0.842869i	$-0.319135\pi$
$83$	9.80499	1.07624	0.538119	+	0.842869i	$0.319135\pi$
$84$	0	0
$85$	2.82945	0.306897
$86$	1.70159	0.183488
$87$	0	0
$88$	0.749396	0.0798859
$89$	−9.50470	−1.00750	−0.503748	−	0.863851i	$-0.668046\pi$
$89$	−9.50470	−1.00750	−0.503748	+	0.863851i	$0.668046\pi$
$90$	0	0
$91$	−8.61745	−0.903355
$92$	−1.00000	−0.104257
$93$	0	0
$94$	7.53766	0.777450
$95$	2.98177	0.305924
$96$	0	0
$97$	9.98672	1.01400	0.506999	−	0.861947i	$-0.330755\pi$
$97$	9.98672	1.01400	0.506999	+	0.861947i	$0.330755\pi$
$98$	−3.56858	−0.360481
$99$	0	0
$100$	−3.75004	−0.375004
$101$	5.91398	0.588463	0.294231	−	0.955734i	$-0.404936\pi$
$101$	5.91398	0.588463	0.294231	+	0.955734i	$0.404936\pi$
$102$	0	0
$103$	6.68659	0.658850	0.329425	−	0.944182i	$-0.393145\pi$
$103$	6.68659	0.658850	0.329425	+	0.944182i	$0.393145\pi$
$104$	4.65202	0.456168
$105$	0	0
$106$	−5.20468	−0.505524
$107$	11.5879	1.12024	0.560121	−	0.828411i	$-0.310755\pi$
$107$	11.5879	1.12024	0.560121	+	0.828411i	$0.310755\pi$
$108$	0	0
$109$	−19.3290	−1.85138	−0.925691	−	0.378281i	$-0.876515\pi$
$109$	−19.3290	−1.85138	−0.925691	+	0.378281i	$0.876515\pi$
$110$	0.837836	0.0798845
$111$	0	0
$112$	−1.85241	−0.175036
$113$	17.0132	1.60046	0.800232	−	0.599691i	$-0.204710\pi$
$113$	17.0132	1.60046	0.800232	+	0.599691i	$0.204710\pi$
$114$	0	0
$115$	−1.11802	−0.104255
$116$	4.55223	0.422664
$117$	0	0
$118$	−3.35185	−0.308563
$119$	−4.68805	−0.429753
$120$	0	0
$121$	−10.4384	−0.948946
$122$	−0.967200	−0.0875662
$123$	0	0
$124$	1.05280	0.0945439
$125$	−9.78268	−0.874989
$126$	0	0
$127$	6.17781	0.548192	0.274096	−	0.961702i	$-0.411621\pi$
$127$	6.17781	0.548192	0.274096	+	0.961702i	$0.411621\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	5.20103	0.456161
$131$	20.2730	1.77126	0.885629	−	0.464394i	$-0.153728\pi$
$131$	20.2730	1.77126	0.885629	+	0.464394i	$0.153728\pi$
$132$	0	0
$133$	−4.94042	−0.428389
$134$	7.55723	0.652845
$135$	0	0
$136$	2.53078	0.217013
$137$	11.7492	1.00380	0.501902	−	0.864924i	$-0.332634\pi$
$137$	11.7492	1.00380	0.501902	+	0.864924i	$0.332634\pi$
$138$	0	0
$139$	−2.11828	−0.179670	−0.0898351	−	0.995957i	$-0.528634\pi$
$139$	−2.11828	−0.179670	−0.0898351	+	0.995957i	$0.528634\pi$
$140$	−2.07102	−0.175033
$141$	0	0
$142$	9.39777	0.788643
$143$	3.48621	0.291531
$144$	0	0
$145$	5.08947	0.422657
$146$	3.04657	0.252136
$147$	0	0
$148$	0.587232	0.0482702
$149$	11.3308	0.928253	0.464127	−	0.885769i	$-0.346368\pi$
$149$	11.3308	0.928253	0.464127	+	0.885769i	$0.346368\pi$
$150$	0	0
$151$	−10.1234	−0.823830	−0.411915	−	0.911222i	$-0.635140\pi$
$151$	−10.1234	−0.823830	−0.411915	+	0.911222i	$0.635140\pi$
$152$	2.66702	0.216324
$153$	0	0
$154$	−1.38819	−0.111863
$155$	1.17704	0.0945423
$156$	0	0
$157$	−23.6834	−1.89014	−0.945071	−	0.326865i	$-0.894008\pi$
$157$	−23.6834	−1.89014	−0.945071	+	0.326865i	$0.894008\pi$
$158$	10.2916	0.818758
$159$	0	0
$160$	1.11802	0.0883869
$161$	1.85241	0.145990
$162$	0	0
$163$	14.7718	1.15701	0.578507	−	0.815678i	$-0.303636\pi$
$163$	14.7718	1.15701	0.578507	+	0.815678i	$0.303636\pi$
$164$	−6.70224	−0.523357
$165$	0	0
$166$	9.80499	0.761015
$167$	−16.9247	−1.30967	−0.654837	−	0.755770i	$-0.727263\pi$
$167$	−16.9247	−1.30967	−0.654837	+	0.755770i	$0.727263\pi$
$168$	0	0
$169$	8.64132	0.664717
$170$	2.82945	0.217009
$171$	0	0
$172$	1.70159	0.129745
$173$	−5.63895	−0.428721	−0.214361	−	0.976755i	$-0.568767\pi$
$173$	−5.63895	−0.428721	−0.214361	+	0.976755i	$0.568767\pi$
$174$	0	0
$175$	6.94661	0.525115
$176$	0.749396	0.0564878
$177$	0	0
$178$	−9.50470	−0.712407
$179$	23.2484	1.73767	0.868833	−	0.495105i	$-0.164871\pi$
$179$	23.2484	1.73767	0.868833	+	0.495105i	$0.164871\pi$
$180$	0	0
$181$	−8.03806	−0.597464	−0.298732	−	0.954337i	$-0.596564\pi$
$181$	−8.03806	−0.597464	−0.298732	+	0.954337i	$0.596564\pi$
$182$	−8.61745	−0.638768
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	0.656534	0.0482693
$186$	0	0
$187$	1.89656	0.138690
$188$	7.53766	0.549740
$189$	0	0
$190$	2.98177	0.216321
$191$	−2.40899	−0.174309	−0.0871543	−	0.996195i	$-0.527777\pi$
$191$	−2.40899	−0.174309	−0.0871543	+	0.996195i	$0.527777\pi$
$192$	0	0
$193$	−13.9985	−1.00763	−0.503817	−	0.863810i	$-0.668071\pi$
$193$	−13.9985	−1.00763	−0.503817	+	0.863810i	$0.668071\pi$
$194$	9.98672	0.717005
$195$	0	0
$196$	−3.56858	−0.254898
$197$	−7.19038	−0.512293	−0.256147	−	0.966638i	$-0.582453\pi$
$197$	−7.19038	−0.512293	−0.256147	+	0.966638i	$0.582453\pi$
$198$	0	0
$199$	16.7553	1.18775	0.593874	−	0.804558i	$-0.297598\pi$
$199$	16.7553	1.18775	0.593874	+	0.804558i	$0.297598\pi$
$200$	−3.75004	−0.265168
$201$	0	0
$202$	5.91398	0.416106
$203$	−8.43260	−0.591853
$204$	0	0
$205$	−7.49321	−0.523348
$206$	6.68659	0.465877
$207$	0	0
$208$	4.65202	0.322560
$209$	1.99866	0.138250
$210$	0	0
$211$	−16.0522	−1.10508	−0.552540	−	0.833487i	$-0.686341\pi$
$211$	−16.0522	−1.10508	−0.552540	+	0.833487i	$0.686341\pi$
$212$	−5.20468	−0.357459
$213$	0	0
$214$	11.5879	0.792131
$215$	1.90241	0.129743
$216$	0	0
$217$	−1.95021	−0.132389
$218$	−19.3290	−1.30912
$219$	0	0
$220$	0.837836	0.0564869
$221$	11.7733	0.791955
$222$	0	0
$223$	−21.8540	−1.46345	−0.731725	−	0.681600i	$-0.761284\pi$
$223$	−21.8540	−1.46345	−0.731725	+	0.681600i	$0.761284\pi$
$224$	−1.85241	−0.123769
$225$	0	0
$226$	17.0132	1.13170
$227$	−4.34200	−0.288189	−0.144094	−	0.989564i	$-0.546027\pi$
$227$	−4.34200	−0.288189	−0.144094	+	0.989564i	$0.546027\pi$
$228$	0	0
$229$	−23.4555	−1.54998	−0.774992	−	0.631971i	$-0.782246\pi$
$229$	−23.4555	−1.54998	−0.774992	+	0.631971i	$0.782246\pi$
$230$	−1.11802	−0.0737197
$231$	0	0
$232$	4.55223	0.298869
$233$	−3.43272	−0.224885	−0.112442	−	0.993658i	$-0.535867\pi$
$233$	−3.43272	−0.224885	−0.112442	+	0.993658i	$0.535867\pi$
$234$	0	0
$235$	8.42722	0.549731
$236$	−3.35185	−0.218187
$237$	0	0
$238$	−4.68805	−0.303881
$239$	−5.43744	−0.351719	−0.175859	−	0.984415i	$-0.556270\pi$
$239$	−5.43744	−0.351719	−0.175859	+	0.984415i	$0.556270\pi$
$240$	0	0
$241$	18.8254	1.21265	0.606324	−	0.795218i	$-0.292643\pi$
$241$	18.8254	1.21265	0.606324	+	0.795218i	$0.292643\pi$
$242$	−10.4384	−0.671006
$243$	0	0
$244$	−0.967200	−0.0619186
$245$	−3.98972	−0.254894
$246$	0	0
$247$	12.4071	0.789442
$248$	1.05280	0.0668526
$249$	0	0
$250$	−9.78268	−0.618711
$251$	−14.3682	−0.906912	−0.453456	−	0.891279i	$-0.649809\pi$
$251$	−14.3682	−0.906912	−0.453456	+	0.891279i	$0.649809\pi$
$252$	0	0
$253$	−0.749396	−0.0471141
$254$	6.17781	0.387630
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.86656	0.365946	0.182973	−	0.983118i	$-0.441428\pi$
$257$	5.86656	0.365946	0.182973	+	0.983118i	$0.441428\pi$
$258$	0	0
$259$	−1.08779	−0.0675922
$260$	5.20103	0.322554
$261$	0	0
$262$	20.2730	1.25247
$263$	−11.1901	−0.690009	−0.345004	−	0.938601i	$-0.612123\pi$
$263$	−11.1901	−0.690009	−0.345004	+	0.938601i	$0.612123\pi$
$264$	0	0
$265$	−5.81892	−0.357453
$266$	−4.94042	−0.302917
$267$	0	0
$268$	7.55723	0.461631
$269$	−6.07796	−0.370580	−0.185290	−	0.982684i	$-0.559322\pi$
$269$	−6.07796	−0.370580	−0.185290	+	0.982684i	$0.559322\pi$
$270$	0	0
$271$	9.46281	0.574825	0.287412	−	0.957807i	$-0.407205\pi$
$271$	9.46281	0.574825	0.287412	+	0.957807i	$0.407205\pi$
$272$	2.53078	0.153451
$273$	0	0
$274$	11.7492	0.709797
$275$	−2.81027	−0.169465
$276$	0	0
$277$	−19.3044	−1.15989	−0.579945	−	0.814655i	$-0.696926\pi$
$277$	−19.3044	−1.15989	−0.579945	+	0.814655i	$0.696926\pi$
$278$	−2.11828	−0.127046
$279$	0	0
$280$	−2.07102	−0.123767
$281$	19.1872	1.14461	0.572306	−	0.820040i	$-0.306049\pi$
$281$	19.1872	1.14461	0.572306	+	0.820040i	$0.306049\pi$
$282$	0	0
$283$	−19.5688	−1.16325	−0.581623	−	0.813458i	$-0.697582\pi$
$283$	−19.5688	−1.16325	−0.581623	+	0.813458i	$0.697582\pi$
$284$	9.39777	0.557655
$285$	0	0
$286$	3.48621	0.206144
$287$	12.4153	0.732852
$288$	0	0
$289$	−10.5951	−0.623243
$290$	5.08947	0.298864
$291$	0	0
$292$	3.04657	0.178287
$293$	13.3941	0.782492	0.391246	−	0.920286i	$-0.372044\pi$
$293$	13.3941	0.782492	0.391246	+	0.920286i	$0.372044\pi$
$294$	0	0
$295$	−3.74742	−0.218183
$296$	0.587232	0.0341322
$297$	0	0
$298$	11.3308	0.656374
$299$	−4.65202	−0.269033
$300$	0	0
$301$	−3.15205	−0.181681
$302$	−10.1234	−0.582536
$303$	0	0
$304$	2.66702	0.152964
$305$	−1.08134	−0.0619176
$306$	0	0
$307$	−26.6861	−1.52306	−0.761529	−	0.648131i	$-0.775551\pi$
$307$	−26.6861	−1.52306	−0.761529	+	0.648131i	$0.775551\pi$
$308$	−1.38819	−0.0790994
$309$	0	0
$310$	1.17704	0.0668515
$311$	18.7797	1.06490	0.532450	−	0.846462i	$-0.321271\pi$
$311$	18.7797	1.06490	0.532450	+	0.846462i	$0.321271\pi$
$312$	0	0
$313$	−28.7642	−1.62585	−0.812924	−	0.582370i	$-0.802126\pi$
$313$	−28.7642	−1.62585	−0.812924	+	0.582370i	$0.802126\pi$
$314$	−23.6834	−1.33653
$315$	0	0
$316$	10.2916	0.578949
$317$	−7.57138	−0.425251	−0.212625	−	0.977134i	$-0.568201\pi$
$317$	−7.57138	−0.425251	−0.212625	+	0.977134i	$0.568201\pi$
$318$	0	0
$319$	3.41143	0.191003
$320$	1.11802	0.0624989
$321$	0	0
$322$	1.85241	0.103231
$323$	6.74966	0.375561
$324$	0	0
$325$	−17.4453	−0.967690
$326$	14.7718	0.818132
$327$	0	0
$328$	−6.70224	−0.370069
$329$	−13.9628	−0.769796
$330$	0	0
$331$	−16.4896	−0.906350	−0.453175	−	0.891422i	$-0.649709\pi$
$331$	−16.4896	−0.906350	−0.453175	+	0.891422i	$0.649709\pi$
$332$	9.80499	0.538119
$333$	0	0
$334$	−16.9247	−0.926080
$335$	8.44910	0.461623
$336$	0	0
$337$	−0.235115	−0.0128075	−0.00640377	−	0.999979i	$-0.502038\pi$
$337$	−0.235115	−0.0128075	−0.00640377	+	0.999979i	$0.502038\pi$
$338$	8.64132	0.470026
$339$	0	0
$340$	2.82945	0.153449
$341$	0.788961	0.0427246
$342$	0	0
$343$	19.5773	1.05708
$344$	1.70159	0.0917438
$345$	0	0
$346$	−5.63895	−0.303152
$347$	−18.4369	−0.989744	−0.494872	−	0.868966i	$-0.664785\pi$
$347$	−18.4369	−0.989744	−0.494872	+	0.868966i	$0.664785\pi$
$348$	0	0
$349$	33.9492	1.81726	0.908631	−	0.417601i	$-0.137129\pi$
$349$	33.9492	1.81726	0.908631	+	0.417601i	$0.137129\pi$
$350$	6.94661	0.371312
$351$	0	0
$352$	0.749396	0.0399429
$353$	−16.2944	−0.867264	−0.433632	−	0.901090i	$-0.642768\pi$
$353$	−16.2944	−0.867264	−0.433632	+	0.901090i	$0.642768\pi$
$354$	0	0
$355$	10.5068	0.557645
$356$	−9.50470	−0.503748
$357$	0	0
$358$	23.2484	1.22872
$359$	−10.5440	−0.556491	−0.278245	−	0.960510i	$-0.589753\pi$
$359$	−10.5440	−0.556491	−0.278245	+	0.960510i	$0.589753\pi$
$360$	0	0
$361$	−11.8870	−0.625631
$362$	−8.03806	−0.422471
$363$	0	0
$364$	−8.61745	−0.451677
$365$	3.40611	0.178284
$366$	0	0
$367$	−10.6220	−0.554465	−0.277232	−	0.960803i	$-0.589417\pi$
$367$	−10.6220	−0.554465	−0.277232	+	0.960803i	$0.589417\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	0.656534	0.0341316
$371$	9.64121	0.500547
$372$	0	0
$373$	29.0826	1.50584	0.752919	−	0.658113i	$-0.228645\pi$
$373$	29.0826	1.50584	0.752919	+	0.658113i	$0.228645\pi$
$374$	1.89656	0.0980687
$375$	0	0
$376$	7.53766	0.388725
$377$	21.1771	1.09068
$378$	0	0
$379$	0.209854	0.0107795	0.00538973	−	0.999985i	$-0.498284\pi$
$379$	0.209854	0.0107795	0.00538973	+	0.999985i	$0.498284\pi$
$380$	2.98177	0.152962
$381$	0	0
$382$	−2.40899	−0.123255
$383$	27.1001	1.38475	0.692375	−	0.721538i	$-0.256564\pi$
$383$	27.1001	1.38475	0.692375	+	0.721538i	$0.256564\pi$
$384$	0	0
$385$	−1.55202	−0.0790980
$386$	−13.9985	−0.712505
$387$	0	0
$388$	9.98672	0.506999
$389$	−14.7602	−0.748371	−0.374186	−	0.927354i	$-0.622078\pi$
$389$	−14.7602	−0.748371	−0.374186	+	0.927354i	$0.622078\pi$
$390$	0	0
$391$	−2.53078	−0.127987
$392$	−3.56858	−0.180240
$393$	0	0
$394$	−7.19038	−0.362246
$395$	11.5062	0.578939
$396$	0	0
$397$	−21.2022	−1.06411	−0.532054	−	0.846710i	$-0.678580\pi$
$397$	−21.2022	−1.06411	−0.532054	+	0.846710i	$0.678580\pi$
$398$	16.7553	0.839865
$399$	0	0
$400$	−3.75004	−0.187502
$401$	23.8121	1.18912	0.594561	−	0.804051i	$-0.297326\pi$
$401$	23.8121	1.18912	0.594561	+	0.804051i	$0.297326\pi$
$402$	0	0
$403$	4.89763	0.243968
$404$	5.91398	0.294231
$405$	0	0
$406$	−8.43260	−0.418503
$407$	0.440069	0.0218134
$408$	0	0
$409$	−25.1569	−1.24393	−0.621965	−	0.783045i	$-0.713666\pi$
$409$	−25.1569	−1.24393	−0.621965	+	0.783045i	$0.713666\pi$
$410$	−7.49321	−0.370063
$411$	0	0
$412$	6.68659	0.329425
$413$	6.20900	0.305525
$414$	0	0
$415$	10.9621	0.538110
$416$	4.65202	0.228084
$417$	0	0
$418$	1.99866	0.0977575
$419$	−35.5312	−1.73581	−0.867907	−	0.496727i	$-0.834535\pi$
$419$	−35.5312	−1.73581	−0.867907	+	0.496727i	$0.834535\pi$
$420$	0	0
$421$	−22.6694	−1.10484	−0.552420	−	0.833566i	$-0.686296\pi$
$421$	−22.6694	−1.10484	−0.552420	+	0.833566i	$0.686296\pi$
$422$	−16.0522	−0.781409
$423$	0	0
$424$	−5.20468	−0.252762
$425$	−9.49054	−0.460359
$426$	0	0
$427$	1.79165	0.0867041
$428$	11.5879	0.560121
$429$	0	0
$430$	1.90241	0.0917423
$431$	17.8043	0.857605	0.428803	−	0.903398i	$-0.358936\pi$
$431$	17.8043	0.857605	0.428803	+	0.903398i	$0.358936\pi$
$432$	0	0
$433$	30.4445	1.46307	0.731535	−	0.681804i	$-0.238804\pi$
$433$	30.4445	1.46307	0.731535	+	0.681804i	$0.238804\pi$
$434$	−1.95021	−0.0936131
$435$	0	0
$436$	−19.3290	−0.925691
$437$	−2.66702	−0.127581
$438$	0	0
$439$	2.42249	0.115619	0.0578096	−	0.998328i	$-0.481588\pi$
$439$	2.42249	0.115619	0.0578096	+	0.998328i	$0.481588\pi$
$440$	0.837836	0.0399423
$441$	0	0
$442$	11.7733	0.559997
$443$	−15.5154	−0.737159	−0.368579	−	0.929596i	$-0.620156\pi$
$443$	−15.5154	−0.737159	−0.368579	+	0.929596i	$0.620156\pi$
$444$	0	0
$445$	−10.6264	−0.503740
$446$	−21.8540	−1.03482
$447$	0	0
$448$	−1.85241	−0.0875181
$449$	−1.02862	−0.0485438	−0.0242719	−	0.999705i	$-0.507727\pi$
$449$	−1.02862	−0.0485438	−0.0242719	+	0.999705i	$0.507727\pi$
$450$	0	0
$451$	−5.02263	−0.236506
$452$	17.0132	0.800232
$453$	0	0
$454$	−4.34200	−0.203780
$455$	−9.63444	−0.451670
$456$	0	0
$457$	−3.84837	−0.180019	−0.0900095	−	0.995941i	$-0.528690\pi$
$457$	−3.84837	−0.180019	−0.0900095	+	0.995941i	$0.528690\pi$
$458$	−23.4555	−1.09600
$459$	0	0
$460$	−1.11802	−0.0521277
$461$	17.8952	0.833463	0.416731	−	0.909030i	$-0.363176\pi$
$461$	17.8952	0.833463	0.416731	+	0.909030i	$0.363176\pi$
$462$	0	0
$463$	10.0013	0.464799	0.232400	−	0.972620i	$-0.425342\pi$
$463$	10.0013	0.464799	0.232400	+	0.972620i	$0.425342\pi$
$464$	4.55223	0.211332
$465$	0	0
$466$	−3.43272	−0.159018
$467$	−24.9243	−1.15336	−0.576681	−	0.816970i	$-0.695652\pi$
$467$	−24.9243	−1.15336	−0.576681	+	0.816970i	$0.695652\pi$
$468$	0	0
$469$	−13.9991	−0.646418
$470$	8.42722	0.388719
$471$	0	0
$472$	−3.35185	−0.154281
$473$	1.27517	0.0586323
$474$	0	0
$475$	−10.0015	−0.458898
$476$	−4.68805	−0.214876
$477$	0	0
$478$	−5.43744	−0.248703
$479$	24.9623	1.14056	0.570278	−	0.821452i	$-0.306835\pi$
$479$	24.9623	1.14056	0.570278	+	0.821452i	$0.306835\pi$
$480$	0	0
$481$	2.73182	0.124560
$482$	18.8254	0.857472
$483$	0	0
$484$	−10.4384	−0.474473
$485$	11.1653	0.506990
$486$	0	0
$487$	−25.0877	−1.13683	−0.568416	−	0.822741i	$-0.692444\pi$
$487$	−25.0877	−1.13683	−0.568416	+	0.822741i	$0.692444\pi$
$488$	−0.967200	−0.0437831
$489$	0	0
$490$	−3.98972	−0.180237
$491$	27.4946	1.24082	0.620408	−	0.784279i	$-0.286967\pi$
$491$	27.4946	1.24082	0.620408	+	0.784279i	$0.286967\pi$
$492$	0	0
$493$	11.5207	0.518867
$494$	12.4071	0.558220
$495$	0	0
$496$	1.05280	0.0472719
$497$	−17.4085	−0.780879
$498$	0	0
$499$	13.3626	0.598194	0.299097	−	0.954223i	$-0.403315\pi$
$499$	13.3626	0.598194	0.299097	+	0.954223i	$0.403315\pi$
$500$	−9.78268	−0.437495
$501$	0	0
$502$	−14.3682	−0.641284
$503$	9.35861	0.417280	0.208640	−	0.977993i	$-0.433096\pi$
$503$	9.35861	0.417280	0.208640	+	0.977993i	$0.433096\pi$
$504$	0	0
$505$	6.61192	0.294226
$506$	−0.749396	−0.0333147
$507$	0	0
$508$	6.17781	0.274096
$509$	36.1685	1.60314	0.801571	−	0.597899i	$-0.203998\pi$
$509$	36.1685	1.60314	0.801571	+	0.597899i	$0.203998\pi$
$510$	0	0
$511$	−5.64349	−0.249653
$512$	1.00000	0.0441942
$513$	0	0
$514$	5.86656	0.258763
$515$	7.47571	0.329419
$516$	0	0
$517$	5.64869	0.248429
$518$	−1.08779	−0.0477949
$519$	0	0
$520$	5.20103	0.228080
$521$	−22.5114	−0.986242	−0.493121	−	0.869961i	$-0.664144\pi$
$521$	−22.5114	−0.986242	−0.493121	+	0.869961i	$0.664144\pi$
$522$	0	0
$523$	−37.8110	−1.65336	−0.826680	−	0.562673i	$-0.809773\pi$
$523$	−37.8110	−1.65336	−0.826680	+	0.562673i	$0.809773\pi$
$524$	20.2730	0.885629
$525$	0	0
$526$	−11.1901	−0.487910
$527$	2.66440	0.116063
$528$	0	0
$529$	1.00000	0.0434783
$530$	−5.81892	−0.252758
$531$	0	0
$532$	−4.94042	−0.214194
$533$	−31.1790	−1.35051
$534$	0	0
$535$	12.9554	0.560112
$536$	7.55723	0.326423
$537$	0	0
$538$	−6.07796	−0.262039
$539$	−2.67428	−0.115189
$540$	0	0
$541$	−33.9521	−1.45971	−0.729857	−	0.683599i	$-0.760414\pi$
$541$	−33.9521	−1.45971	−0.729857	+	0.683599i	$0.760414\pi$
$542$	9.46281	0.406463
$543$	0	0
$544$	2.53078	0.108506
$545$	−21.6101	−0.925675
$546$	0	0
$547$	−2.25113	−0.0962512	−0.0481256	−	0.998841i	$-0.515325\pi$
$547$	−2.25113	−0.0962512	−0.0481256	+	0.998841i	$0.515325\pi$
$548$	11.7492	0.501902
$549$	0	0
$550$	−2.81027	−0.119830
$551$	12.1409	0.517221
$552$	0	0
$553$	−19.0643	−0.810697
$554$	−19.3044	−0.820167
$555$	0	0
$556$	−2.11828	−0.0898351
$557$	−3.18302	−0.134869	−0.0674343	−	0.997724i	$-0.521481\pi$
$557$	−3.18302	−0.134869	−0.0674343	+	0.997724i	$0.521481\pi$
$558$	0	0
$559$	7.91586	0.334805
$560$	−2.07102	−0.0875167
$561$	0	0
$562$	19.1872	0.809363
$563$	−1.48690	−0.0626654	−0.0313327	−	0.999509i	$-0.509975\pi$
$563$	−1.48690	−0.0626654	−0.0313327	+	0.999509i	$0.509975\pi$
$564$	0	0
$565$	19.0210	0.800218
$566$	−19.5688	−0.822539
$567$	0	0
$568$	9.39777	0.394322
$569$	29.6981	1.24501	0.622506	−	0.782615i	$-0.286115\pi$
$569$	29.6981	1.24501	0.622506	+	0.782615i	$0.286115\pi$
$570$	0	0
$571$	30.3796	1.27135	0.635674	−	0.771958i	$-0.280722\pi$
$571$	30.3796	1.27135	0.635674	+	0.771958i	$0.280722\pi$
$572$	3.48621	0.145766
$573$	0	0
$574$	12.4153	0.518204
$575$	3.75004	0.156388
$576$	0	0
$577$	−6.01032	−0.250213	−0.125106	−	0.992143i	$-0.539927\pi$
$577$	−6.01032	−0.250213	−0.125106	+	0.992143i	$0.539927\pi$
$578$	−10.5951	−0.440700
$579$	0	0
$580$	5.08947	0.211329
$581$	−18.1629	−0.753522
$582$	0	0
$583$	−3.90037	−0.161537
$584$	3.04657	0.126068
$585$	0	0
$586$	13.3941	0.553306
$587$	−29.5900	−1.22131	−0.610656	−	0.791896i	$-0.709094\pi$
$587$	−29.5900	−1.22131	−0.610656	+	0.791896i	$0.709094\pi$
$588$	0	0
$589$	2.80783	0.115695
$590$	−3.74742	−0.154279
$591$	0	0
$592$	0.587232	0.0241351
$593$	8.69875	0.357215	0.178607	−	0.983920i	$-0.442841\pi$
$593$	8.69875	0.357215	0.178607	+	0.983920i	$0.442841\pi$
$594$	0	0
$595$	−5.24131	−0.214873
$596$	11.3308	0.464127
$597$	0	0
$598$	−4.65202	−0.190235
$599$	−31.0686	−1.26943	−0.634714	−	0.772747i	$-0.718882\pi$
$599$	−31.0686	−1.26943	−0.634714	+	0.772747i	$0.718882\pi$
$600$	0	0
$601$	15.8350	0.645925	0.322963	−	0.946412i	$-0.395321\pi$
$601$	15.8350	0.645925	0.322963	+	0.946412i	$0.395321\pi$
$602$	−3.15205	−0.128468
$603$	0	0
$604$	−10.1234	−0.411915
$605$	−11.6703	−0.474465
$606$	0	0
$607$	6.76607	0.274626	0.137313	−	0.990528i	$-0.456153\pi$
$607$	6.76607	0.274626	0.137313	+	0.990528i	$0.456153\pi$
$608$	2.66702	0.108162
$609$	0	0
$610$	−1.08134	−0.0437824
$611$	35.0654	1.41859
$612$	0	0
$613$	29.0133	1.17184	0.585919	−	0.810370i	$-0.300734\pi$
$613$	29.0133	1.17184	0.585919	+	0.810370i	$0.300734\pi$
$614$	−26.6861	−1.07696
$615$	0	0
$616$	−1.38819	−0.0559317
$617$	−21.2748	−0.856491	−0.428246	−	0.903662i	$-0.640868\pi$
$617$	−21.2748	−0.856491	−0.428246	+	0.903662i	$0.640868\pi$
$618$	0	0
$619$	28.0858	1.12886	0.564432	−	0.825480i	$-0.309095\pi$
$619$	28.0858	1.12886	0.564432	+	0.825480i	$0.309095\pi$
$620$	1.17704	0.0472711
$621$	0	0
$622$	18.7797	0.752998
$623$	17.6066	0.705393
$624$	0	0
$625$	7.81303	0.312521
$626$	−28.7642	−1.14965
$627$	0	0
$628$	−23.6834	−0.945071
$629$	1.48616	0.0592569
$630$	0	0
$631$	−37.2627	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$631$	−37.2627	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$632$	10.2916	0.409379
$633$	0	0
$634$	−7.57138	−0.300698
$635$	6.90689	0.274091
$636$	0	0
$637$	−16.6011	−0.657760
$638$	3.41143	0.135060
$639$	0	0
$640$	1.11802	0.0441934
$641$	−17.8698	−0.705814	−0.352907	−	0.935658i	$-0.614807\pi$
$641$	−17.8698	−0.705814	−0.352907	+	0.935658i	$0.614807\pi$
$642$	0	0
$643$	39.4980	1.55765	0.778824	−	0.627242i	$-0.215816\pi$
$643$	39.4980	1.55765	0.778824	+	0.627242i	$0.215816\pi$
$644$	1.85241	0.0729952
$645$	0	0
$646$	6.74966	0.265562
$647$	4.47180	0.175804	0.0879022	−	0.996129i	$-0.471984\pi$
$647$	4.47180	0.175804	0.0879022	+	0.996129i	$0.471984\pi$
$648$	0	0
$649$	−2.51186	−0.0985992
$650$	−17.4453	−0.684260
$651$	0	0
$652$	14.7718	0.578507
$653$	−10.9005	−0.426569	−0.213284	−	0.976990i	$-0.568416\pi$
$653$	−10.9005	−0.426569	−0.213284	+	0.976990i	$0.568416\pi$
$654$	0	0
$655$	22.6655	0.885614
$656$	−6.70224	−0.261679
$657$	0	0
$658$	−13.9628	−0.544328
$659$	−28.1199	−1.09540	−0.547698	−	0.836676i	$-0.684496\pi$
$659$	−28.1199	−1.09540	−0.547698	+	0.836676i	$0.684496\pi$
$660$	0	0
$661$	−2.67126	−0.103900	−0.0519500	−	0.998650i	$-0.516544\pi$
$661$	−2.67126	−0.103900	−0.0519500	+	0.998650i	$0.516544\pi$
$662$	−16.4896	−0.640886
$663$	0	0
$664$	9.80499	0.380507
$665$	−5.52347	−0.214191
$666$	0	0
$667$	−4.55223	−0.176263
$668$	−16.9247	−0.654837
$669$	0	0
$670$	8.44910	0.326417
$671$	−0.724816	−0.0279812
$672$	0	0
$673$	−14.4698	−0.557771	−0.278886	−	0.960324i	$-0.589965\pi$
$673$	−14.4698	−0.557771	−0.278886	+	0.960324i	$0.589965\pi$
$674$	−0.235115	−0.00905630
$675$	0	0
$676$	8.64132	0.332358
$677$	−7.18581	−0.276173	−0.138087	−	0.990420i	$-0.544095\pi$
$677$	−7.18581	−0.276173	−0.138087	+	0.990420i	$0.544095\pi$
$678$	0	0
$679$	−18.4995	−0.709945
$680$	2.82945	0.108505
$681$	0	0
$682$	0.788961	0.0302109
$683$	−12.9433	−0.495263	−0.247631	−	0.968854i	$-0.579652\pi$
$683$	−12.9433	−0.495263	−0.247631	+	0.968854i	$0.579652\pi$
$684$	0	0
$685$	13.1358	0.501894
$686$	19.5773	0.747466
$687$	0	0
$688$	1.70159	0.0648727
$689$	−24.2123	−0.922416
$690$	0	0
$691$	−23.6956	−0.901424	−0.450712	−	0.892669i	$-0.648830\pi$
$691$	−23.6956	−0.901424	−0.450712	+	0.892669i	$0.648830\pi$
$692$	−5.63895	−0.214361
$693$	0	0
$694$	−18.4369	−0.699855
$695$	−2.36827	−0.0898336
$696$	0	0
$697$	−16.9619	−0.642479
$698$	33.9492	1.28500
$699$	0	0
$700$	6.94661	0.262557
$701$	−20.9895	−0.792764	−0.396382	−	0.918086i	$-0.629734\pi$
$701$	−20.9895	−0.792764	−0.396382	+	0.918086i	$0.629734\pi$
$702$	0	0
$703$	1.56616	0.0590689
$704$	0.749396	0.0282439
$705$	0	0
$706$	−16.2944	−0.613248
$707$	−10.9551	−0.412009
$708$	0	0
$709$	−10.3146	−0.387372	−0.193686	−	0.981064i	$-0.562044\pi$
$709$	−10.3146	−0.387372	−0.193686	+	0.981064i	$0.562044\pi$
$710$	10.5068	0.394315
$711$	0	0
$712$	−9.50470	−0.356204
$713$	−1.05280	−0.0394275
$714$	0	0
$715$	3.89763	0.145763
$716$	23.2484	0.868833
$717$	0	0
$718$	−10.5440	−0.393498
$719$	−19.7731	−0.737414	−0.368707	−	0.929546i	$-0.620199\pi$
$719$	−19.7731	−0.737414	−0.368707	+	0.929546i	$0.620199\pi$
$720$	0	0
$721$	−12.3863	−0.461290
$722$	−11.8870	−0.442388
$723$	0	0
$724$	−8.03806	−0.298732
$725$	−17.0711	−0.634004
$726$	0	0
$727$	−34.6836	−1.28634	−0.643172	−	0.765721i	$-0.722382\pi$
$727$	−34.6836	−1.28634	−0.643172	+	0.765721i	$0.722382\pi$
$728$	−8.61745	−0.319384
$729$	0	0
$730$	3.40611	0.126066
$731$	4.30637	0.159277
$732$	0	0
$733$	46.3944	1.71362	0.856809	−	0.515635i	$-0.172444\pi$
$733$	46.3944	1.71362	0.856809	+	0.515635i	$0.172444\pi$
$734$	−10.6220	−0.392066
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	5.66336	0.208612
$738$	0	0
$739$	−46.7860	−1.72105	−0.860526	−	0.509406i	$-0.829865\pi$
$739$	−46.7860	−1.72105	−0.860526	+	0.509406i	$0.829865\pi$
$740$	0.656534	0.0241347
$741$	0	0
$742$	9.64121	0.353940
$743$	−36.9328	−1.35493	−0.677467	−	0.735553i	$-0.736923\pi$
$743$	−36.9328	−1.35493	−0.677467	+	0.735553i	$0.736923\pi$
$744$	0	0
$745$	12.6680	0.464119
$746$	29.0826	1.06479
$747$	0	0
$748$	1.89656	0.0693450
$749$	−21.4655	−0.784332
$750$	0	0
$751$	14.5469	0.530825	0.265412	−	0.964135i	$-0.414492\pi$
$751$	14.5469	0.530825	0.265412	+	0.964135i	$0.414492\pi$
$752$	7.53766	0.274870
$753$	0	0
$754$	21.1771	0.771224
$755$	−11.3181	−0.411908
$756$	0	0
$757$	34.3329	1.24785	0.623925	−	0.781484i	$-0.285537\pi$
$757$	34.3329	1.24785	0.623925	+	0.781484i	$0.285537\pi$
$758$	0.209854	0.00762223
$759$	0	0
$760$	2.98177	0.108160
$761$	−39.6718	−1.43810	−0.719051	−	0.694958i	$-0.755423\pi$
$761$	−39.6718	−1.43810	−0.719051	+	0.694958i	$0.755423\pi$
$762$	0	0
$763$	35.8052	1.29624
$764$	−2.40899	−0.0871543
$765$	0	0
$766$	27.1001	0.979166
$767$	−15.5929	−0.563026
$768$	0	0
$769$	−30.7106	−1.10745	−0.553726	−	0.832699i	$-0.686794\pi$
$769$	−30.7106	−1.10745	−0.553726	+	0.832699i	$0.686794\pi$
$770$	−1.55202	−0.0559308
$771$	0	0
$772$	−13.9985	−0.503817
$773$	−45.4626	−1.63518	−0.817588	−	0.575804i	$-0.804689\pi$
$773$	−45.4626	−1.63518	−0.817588	+	0.575804i	$0.804689\pi$
$774$	0	0
$775$	−3.94803	−0.141817
$776$	9.98672	0.358502
$777$	0	0
$778$	−14.7602	−0.529178
$779$	−17.8750	−0.640440
$780$	0	0
$781$	7.04265	0.252006
$782$	−2.53078	−0.0905006
$783$	0	0
$784$	−3.56858	−0.127449
$785$	−26.4784	−0.945055
$786$	0	0
$787$	−10.8392	−0.386377	−0.193188	−	0.981162i	$-0.561883\pi$
$787$	−10.8392	−0.386377	−0.193188	+	0.981162i	$0.561883\pi$
$788$	−7.19038	−0.256147
$789$	0	0
$790$	11.5062	0.409372
$791$	−31.5154	−1.12056
$792$	0	0
$793$	−4.49944	−0.159780
$794$	−21.2022	−0.752438
$795$	0	0
$796$	16.7553	0.593874
$797$	−19.4795	−0.690001	−0.345001	−	0.938602i	$-0.612121\pi$
$797$	−19.4795	−0.690001	−0.345001	+	0.938602i	$0.612121\pi$
$798$	0	0
$799$	19.0762	0.674867
$800$	−3.75004	−0.132584
$801$	0	0
$802$	23.8121	0.840836
$803$	2.28308	0.0805683
$804$	0	0
$805$	2.07102	0.0729939
$806$	4.89763	0.172512
$807$	0	0
$808$	5.91398	0.208053
$809$	44.8930	1.57836	0.789178	−	0.614165i	$-0.210507\pi$
$809$	44.8930	1.57836	0.789178	+	0.614165i	$0.210507\pi$
$810$	0	0
$811$	−23.9262	−0.840161	−0.420081	−	0.907487i	$-0.637998\pi$
$811$	−23.9262	−0.840161	−0.420081	+	0.907487i	$0.637998\pi$
$812$	−8.43260	−0.295926
$813$	0	0
$814$	0.440069	0.0154244
$815$	16.5151	0.578497
$816$	0	0
$817$	4.53820	0.158771
$818$	−25.1569	−0.879592
$819$	0	0
$820$	−7.49321	−0.261674
$821$	41.0287	1.43191	0.715956	−	0.698146i	$-0.245991\pi$
$821$	41.0287	1.43191	0.715956	+	0.698146i	$0.245991\pi$
$822$	0	0
$823$	−38.1804	−1.33089	−0.665443	−	0.746449i	$-0.731757\pi$
$823$	−38.1804	−1.33089	−0.665443	+	0.746449i	$0.731757\pi$
$824$	6.68659	0.232939
$825$	0	0
$826$	6.20900	0.216039
$827$	8.24227	0.286612	0.143306	−	0.989678i	$-0.454227\pi$
$827$	8.24227	0.286612	0.143306	+	0.989678i	$0.454227\pi$
$828$	0	0
$829$	6.28685	0.218351	0.109176	−	0.994022i	$-0.465179\pi$
$829$	6.28685	0.218351	0.109176	+	0.994022i	$0.465179\pi$
$830$	10.9621	0.380501
$831$	0	0
$832$	4.65202	0.161280
$833$	−9.03130	−0.312916
$834$	0	0
$835$	−18.9221	−0.654826
$836$	1.99866	0.0691250
$837$	0	0
$838$	−35.5312	−1.22741
$839$	20.2129	0.697826	0.348913	−	0.937155i	$-0.386551\pi$
$839$	20.2129	0.697826	0.348913	+	0.937155i	$0.386551\pi$
$840$	0	0
$841$	−8.27716	−0.285419
$842$	−22.6694	−0.781241
$843$	0	0
$844$	−16.0522	−0.552540
$845$	9.66113	0.332353
$846$	0	0
$847$	19.3362	0.664400
$848$	−5.20468	−0.178730
$849$	0	0
$850$	−9.49054	−0.325523
$851$	−0.587232	−0.0201300
$852$	0	0
$853$	−42.3073	−1.44858	−0.724288	−	0.689498i	$-0.757831\pi$
$853$	−42.3073	−1.44858	−0.724288	+	0.689498i	$0.757831\pi$
$854$	1.79165	0.0613090
$855$	0	0
$856$	11.5879	0.396066
$857$	15.6954	0.536145	0.268072	−	0.963399i	$-0.413613\pi$
$857$	15.6954	0.536145	0.268072	+	0.963399i	$0.413613\pi$
$858$	0	0
$859$	43.1289	1.47154	0.735769	−	0.677233i	$-0.236821\pi$
$859$	43.1289	1.47154	0.735769	+	0.677233i	$0.236821\pi$
$860$	1.90241	0.0648716
$861$	0	0
$862$	17.8043	0.606418
$863$	−23.0436	−0.784413	−0.392207	−	0.919877i	$-0.628288\pi$
$863$	−23.0436	−0.784413	−0.392207	+	0.919877i	$0.628288\pi$
$864$	0	0
$865$	−6.30443	−0.214357
$866$	30.4445	1.03455
$867$	0	0
$868$	−1.95021	−0.0661944
$869$	7.71250	0.261629
$870$	0	0
$871$	35.1564	1.19123
$872$	−19.3290	−0.654562
$873$	0	0
$874$	−2.66702	−0.0902135
$875$	18.1215	0.612620
$876$	0	0
$877$	−7.83418	−0.264542	−0.132271	−	0.991214i	$-0.542227\pi$
$877$	−7.83418	−0.264542	−0.132271	+	0.991214i	$0.542227\pi$
$878$	2.42249	0.0817552
$879$	0	0
$880$	0.837836	0.0282434
$881$	−37.2946	−1.25649	−0.628243	−	0.778017i	$-0.716226\pi$
$881$	−37.2946	−1.25649	−0.628243	+	0.778017i	$0.716226\pi$
$882$	0	0
$883$	−1.87822	−0.0632071	−0.0316035	−	0.999500i	$-0.510061\pi$
$883$	−1.87822	−0.0632071	−0.0316035	+	0.999500i	$0.510061\pi$
$884$	11.7733	0.395978
$885$	0	0
$886$	−15.5154	−0.521250
$887$	−43.8156	−1.47118	−0.735592	−	0.677425i	$-0.763096\pi$
$887$	−43.8156	−1.47118	−0.735592	+	0.677425i	$0.763096\pi$
$888$	0	0
$889$	−11.4438	−0.383814
$890$	−10.6264	−0.356198
$891$	0	0
$892$	−21.8540	−0.731725
$893$	20.1031	0.672725
$894$	0	0
$895$	25.9920	0.868818
$896$	−1.85241	−0.0618847
$897$	0	0
$898$	−1.02862	−0.0343256
$899$	4.79257	0.159841
$900$	0	0
$901$	−13.1719	−0.438821
$902$	−5.02263	−0.167235
$903$	0	0
$904$	17.0132	0.565849
$905$	−8.98667	−0.298727
$906$	0	0
$907$	14.5435	0.482910	0.241455	−	0.970412i	$-0.422375\pi$
$907$	14.5435	0.482910	0.241455	+	0.970412i	$0.422375\pi$
$908$	−4.34200	−0.144094
$909$	0	0
$910$	−9.63444	−0.319379
$911$	−7.11321	−0.235671	−0.117836	−	0.993033i	$-0.537596\pi$
$911$	−7.11321	−0.235671	−0.117836	+	0.993033i	$0.537596\pi$
$912$	0	0
$913$	7.34782	0.243177
$914$	−3.84837	−0.127293
$915$	0	0
$916$	−23.4555	−0.774992
$917$	−37.5538	−1.24014
$918$	0	0
$919$	39.3474	1.29795	0.648975	−	0.760810i	$-0.275198\pi$
$919$	39.3474	1.29795	0.648975	+	0.760810i	$0.275198\pi$
$920$	−1.11802	−0.0368599
$921$	0	0
$922$	17.8952	0.589347
$923$	43.7186	1.43902
$924$	0	0
$925$	−2.20214	−0.0724060
$926$	10.0013	0.328663
$927$	0	0
$928$	4.55223	0.149434
$929$	9.78244	0.320952	0.160476	−	0.987040i	$-0.448697\pi$
$929$	9.78244	0.320952	0.160476	+	0.987040i	$0.448697\pi$
$930$	0	0
$931$	−9.51749	−0.311923
$932$	−3.43272	−0.112442
$933$	0	0
$934$	−24.9243	−0.815549
$935$	2.12038	0.0693439
$936$	0	0
$937$	44.7588	1.46221	0.731103	−	0.682267i	$-0.239006\pi$
$937$	44.7588	1.46221	0.731103	+	0.682267i	$0.239006\pi$
$938$	−13.9991	−0.457086
$939$	0	0
$940$	8.42722	0.274866
$941$	18.5181	0.603673	0.301836	−	0.953360i	$-0.402400\pi$
$941$	18.5181	0.603673	0.301836	+	0.953360i	$0.402400\pi$
$942$	0	0
$943$	6.70224	0.218255
$944$	−3.35185	−0.109093
$945$	0	0
$946$	1.27517	0.0414593
$947$	−42.4621	−1.37983	−0.689916	−	0.723890i	$-0.742352\pi$
$947$	−42.4621	−1.37983	−0.689916	+	0.723890i	$0.742352\pi$
$948$	0	0
$949$	14.1727	0.460065
$950$	−10.0015	−0.324490
$951$	0	0
$952$	−4.68805	−0.151940
$953$	−15.8281	−0.512721	−0.256361	−	0.966581i	$-0.582523\pi$
$953$	−15.8281	−0.512721	−0.256361	+	0.966581i	$0.582523\pi$
$954$	0	0
$955$	−2.69329	−0.0871528
$956$	−5.43744	−0.175859
$957$	0	0
$958$	24.9623	0.806495
$959$	−21.7644	−0.702809
$960$	0	0
$961$	−29.8916	−0.964246
$962$	2.73182	0.0880773
$963$	0	0
$964$	18.8254	0.606324
$965$	−15.6505	−0.503808
$966$	0	0
$967$	−7.94934	−0.255634	−0.127817	−	0.991798i	$-0.540797\pi$
$967$	−7.94934	−0.255634	−0.127817	+	0.991798i	$0.540797\pi$
$968$	−10.4384	−0.335503
$969$	0	0
$970$	11.1653	0.358496
$971$	−35.2953	−1.13268	−0.566341	−	0.824171i	$-0.691641\pi$
$971$	−35.2953	−1.13268	−0.566341	+	0.824171i	$0.691641\pi$
$972$	0	0
$973$	3.92392	0.125795
$974$	−25.0877	−0.803862
$975$	0	0
$976$	−0.967200	−0.0309593
$977$	58.3711	1.86746	0.933728	−	0.357982i	$-0.116535\pi$
$977$	58.3711	1.86746	0.933728	+	0.357982i	$0.116535\pi$
$978$	0	0
$979$	−7.12278	−0.227645
$980$	−3.98972	−0.127447
$981$	0	0
$982$	27.4946	0.877389
$983$	43.1792	1.37720	0.688602	−	0.725140i	$-0.258225\pi$
$983$	43.1792	1.37720	0.688602	+	0.725140i	$0.258225\pi$
$984$	0	0
$985$	−8.03895	−0.256142
$986$	11.5207	0.366894
$987$	0	0
$988$	12.4071	0.394721
$989$	−1.70159	−0.0541076
$990$	0	0
$991$	54.9028	1.74405	0.872023	−	0.489465i	$-0.162808\pi$
$991$	54.9028	1.74405	0.872023	+	0.489465i	$0.162808\pi$
$992$	1.05280	0.0334263
$993$	0	0
$994$	−17.4085	−0.552165
$995$	18.7326	0.593864
$996$	0	0
$997$	16.9492	0.536786	0.268393	−	0.963309i	$-0.413507\pi$
$997$	16.9492	0.536786	0.268393	+	0.963309i	$0.413507\pi$
$998$	13.3626	0.422987
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3726.2.a.v.1.3		5
3.2	odd	2		3726.2.a.q.1.3		5
9.2	odd	6		1242.2.e.d.415.3		10
9.4	even	3		414.2.e.b.277.5	yes	10
9.5	odd	6		1242.2.e.d.829.3		10
9.7	even	3		414.2.e.b.139.5	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.2.e.b.139.5	✓	10	9.7	even	3
414.2.e.b.277.5	yes	10	9.4	even	3
1242.2.e.d.415.3		10	9.2	odd	6
1242.2.e.d.829.3		10	9.5	odd	6
3726.2.a.q.1.3		5	3.2	odd	2
3726.2.a.v.1.3		5	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.11802 q^{5} -1.85241 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.11802 q^{5} -1.85241 q^{7} +1.00000 q^{8} +1.11802 q^{10} +0.749396 q^{11} +4.65202 q^{13} -1.85241 q^{14} +1.00000 q^{16} +2.53078 q^{17} +2.66702 q^{19} +1.11802 q^{20} +0.749396 q^{22} -1.00000 q^{23} -3.75004 q^{25} +4.65202 q^{26} -1.85241 q^{28} +4.55223 q^{29} +1.05280 q^{31} +1.00000 q^{32} +2.53078 q^{34} -2.07102 q^{35} +0.587232 q^{37} +2.66702 q^{38} +1.11802 q^{40} -6.70224 q^{41} +1.70159 q^{43} +0.749396 q^{44} -1.00000 q^{46} +7.53766 q^{47} -3.56858 q^{49} -3.75004 q^{50} +4.65202 q^{52} -5.20468 q^{53} +0.837836 q^{55} -1.85241 q^{56} +4.55223 q^{58} -3.35185 q^{59} -0.967200 q^{61} +1.05280 q^{62} +1.00000 q^{64} +5.20103 q^{65} +7.55723 q^{67} +2.53078 q^{68} -2.07102 q^{70} +9.39777 q^{71} +3.04657 q^{73} +0.587232 q^{74} +2.66702 q^{76} -1.38819 q^{77} +10.2916 q^{79} +1.11802 q^{80} -6.70224 q^{82} +9.80499 q^{83} +2.82945 q^{85} +1.70159 q^{86} +0.749396 q^{88} -9.50470 q^{89} -8.61745 q^{91} -1.00000 q^{92} +7.53766 q^{94} +2.98177 q^{95} +9.98672 q^{97} -3.56858 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} + 5 q^{5} + 3 q^{7} + 5 q^{8}+O(q^{10}) \) 5 * q + 5 * q^2 + 5 * q^4 + 5 * q^5 + 3 * q^7 + 5 * q^8	\( 5 q + 5 q^{2} + 5 q^{4} + 5 q^{5} + 3 q^{7} + 5 q^{8} + 5 q^{10} + 11 q^{11} + 3 q^{14} + 5 q^{16} + 11 q^{17} + 9 q^{19} + 5 q^{20} + 11 q^{22} - 5 q^{23} + 12 q^{25} + 3 q^{28} - 2 q^{29} + 4 q^{31} + 5 q^{32} + 11 q^{34} - 7 q^{35} + 4 q^{37} + 9 q^{38} + 5 q^{40} + 8 q^{41} + 5 q^{43} + 11 q^{44} - 5 q^{46} + 19 q^{47} + 4 q^{49} + 12 q^{50} - 3 q^{53} - 2 q^{55} + 3 q^{56} - 2 q^{58} + 19 q^{59} - 13 q^{61} + 4 q^{62} + 5 q^{64} - q^{65} + q^{67} + 11 q^{68} - 7 q^{70} + 27 q^{71} - 7 q^{73} + 4 q^{74} + 9 q^{76} + 19 q^{77} + 3 q^{79} + 5 q^{80} + 8 q^{82} + 9 q^{83} - 5 q^{85} + 5 q^{86} + 11 q^{88} + 9 q^{89} - 19 q^{91} - 5 q^{92} + 19 q^{94} + 17 q^{95} - 20 q^{97} + 4 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 + 5 * q^5 + 3 * q^7 + 5 * q^8 + 5 * q^10 + 11 * q^11 + 3 * q^14 + 5 * q^16 + 11 * q^17 + 9 * q^19 + 5 * q^20 + 11 * q^22 - 5 * q^23 + 12 * q^25 + 3 * q^28 - 2 * q^29 + 4 * q^31 + 5 * q^32 + 11 * q^34 - 7 * q^35 + 4 * q^37 + 9 * q^38 + 5 * q^40 + 8 * q^41 + 5 * q^43 + 11 * q^44 - 5 * q^46 + 19 * q^47 + 4 * q^49 + 12 * q^50 - 3 * q^53 - 2 * q^55 + 3 * q^56 - 2 * q^58 + 19 * q^59 - 13 * q^61 + 4 * q^62 + 5 * q^64 - q^65 + q^67 + 11 * q^68 - 7 * q^70 + 27 * q^71 - 7 * q^73 + 4 * q^74 + 9 * q^76 + 19 * q^77 + 3 * q^79 + 5 * q^80 + 8 * q^82 + 9 * q^83 - 5 * q^85 + 5 * q^86 + 11 * q^88 + 9 * q^89 - 19 * q^91 - 5 * q^92 + 19 * q^94 + 17 * q^95 - 20 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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