LMFDB - Embedded newform 3726.2.a.q.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:	$1$
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	$-0.580429\pi$
$5$	−1.11802	−0.499992	−0.249996	+	0.968247i	$0.580429\pi$
$6$	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	$-0.536038\pi$
$11$	−0.749396	−0.225951	−0.112976	+	0.993598i	$0.536038\pi$
$12$	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	$-0.599293\pi$
$17$	−2.53078	−0.613805	−0.306903	+	0.951741i	$0.599293\pi$
$18$	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	$-0.638905\pi$
$29$	−4.55223	−0.845329	−0.422664	+	0.906286i	$0.638905\pi$
$30$	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	$-0.324679\pi$
$41$	6.70224	1.04671	0.523357	+	0.852113i	$0.324679\pi$
$42$	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	$-0.685273\pi$
$47$	−7.53766	−1.09948	−0.549740	+	0.835336i	$0.685273\pi$
$48$	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	$-0.383643\pi$
$53$	5.20468	0.714918	0.357459	+	0.933929i	$0.383643\pi$
$54$	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	$-0.429986\pi$
$59$	3.35185	0.436373	0.218187	+	0.975907i	$0.429986\pi$
$60$	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	$-0.688299\pi$
$71$	−9.39777	−1.11531	−0.557655	+	0.830073i	$0.688299\pi$
$72$	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	$-0.680865\pi$
$83$	−9.80499	−1.07624	−0.538119	+	0.842869i	$0.680865\pi$
$84$	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	$-0.331954\pi$
$89$	9.50470	1.00750	0.503748	+	0.863851i	$0.331954\pi$
$90$	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	$-0.595064\pi$
$101$	−5.91398	−0.588463	−0.294231	+	0.955734i	$0.595064\pi$
$102$	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	$-0.689245\pi$
$107$	−11.5879	−1.12024	−0.560121	+	0.828411i	$0.689245\pi$
$108$	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	$-0.795290\pi$
$113$	−17.0132	−1.60046	−0.800232	+	0.599691i	$0.795290\pi$
$114$	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	$-0.846272\pi$
$131$	−20.2730	−1.77126	−0.885629	+	0.464394i	$0.846272\pi$
$132$	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	$-0.667366\pi$
$137$	−11.7492	−1.00380	−0.501902	+	0.864924i	$0.667366\pi$
$138$	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	$-0.653632\pi$
$149$	−11.3308	−0.928253	−0.464127	+	0.885769i	$0.653632\pi$
$150$	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	$-0.272737\pi$
$167$	16.9247	1.30967	0.654837	+	0.755770i	$0.272737\pi$
$168$	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	$-0.431233\pi$
$173$	5.63895	0.428721	0.214361	+	0.976755i	$0.431233\pi$
$174$	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	$-0.835129\pi$
$179$	−23.2484	−1.73767	−0.868833	+	0.495105i	$0.835129\pi$
$180$	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	$-0.472223\pi$
$191$	2.40899	0.174309	0.0871543	+	0.996195i	$0.472223\pi$
$192$	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	$-0.417547\pi$
$197$	7.19038	0.512293	0.256147	+	0.966638i	$0.417547\pi$
$198$	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	$-0.453973\pi$
$227$	4.34200	0.288189	0.144094	+	0.989564i	$0.453973\pi$
$228$	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	$-0.464133\pi$
$233$	3.43272	0.224885	0.112442	+	0.993658i	$0.464133\pi$
$234$	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	$-0.443730\pi$
$239$	5.43744	0.351719	0.175859	+	0.984415i	$0.443730\pi$
$240$	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	$-0.350191\pi$
$251$	14.3682	0.906912	0.453456	+	0.891279i	$0.350191\pi$
$252$	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	$-0.558572\pi$
$257$	−5.86656	−0.365946	−0.182973	+	0.983118i	$0.558572\pi$
$258$	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	$-0.387877\pi$
$263$	11.1901	0.690009	0.345004	+	0.938601i	$0.387877\pi$
$264$	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	$-0.440678\pi$
$269$	6.07796	0.370580	0.185290	+	0.982684i	$0.440678\pi$
$270$	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	$-0.693951\pi$
$281$	−19.1872	−1.14461	−0.572306	+	0.820040i	$0.693951\pi$
$282$	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	$-0.627956\pi$
$293$	−13.3941	−0.782492	−0.391246	+	0.920286i	$0.627956\pi$
$294$	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	$-0.678729\pi$
$311$	−18.7797	−1.06490	−0.532450	+	0.846462i	$0.678729\pi$
$312$	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	$-0.431799\pi$
$317$	7.57138	0.425251	0.212625	+	0.977134i	$0.431799\pi$
$318$	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	$-0.335215\pi$
$347$	18.4369	0.989744	0.494872	+	0.868966i	$0.335215\pi$
$348$	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	$-0.357232\pi$
$353$	16.2944	0.867264	0.433632	+	0.901090i	$0.357232\pi$
$354$	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	$-0.410247\pi$
$359$	10.5440	0.556491	0.278245	+	0.960510i	$0.410247\pi$
$360$	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	$-0.743436\pi$
$383$	−27.1001	−1.38475	−0.692375	+	0.721538i	$0.743436\pi$
$384$	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	$-0.377922\pi$
$389$	14.7602	0.748371	0.374186	+	0.927354i	$0.377922\pi$
$390$	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	$-0.702674\pi$
$401$	−23.8121	−1.18912	−0.594561	+	0.804051i	$0.702674\pi$
$402$	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	$-0.165465\pi$
$419$	35.5312	1.73581	0.867907	+	0.496727i	$0.165465\pi$
$420$	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	$-0.641064\pi$
$431$	−17.8043	−0.857605	−0.428803	+	0.903398i	$0.641064\pi$
$432$	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	$-0.379844\pi$
$443$	15.5154	0.737159	0.368579	+	0.929596i	$0.379844\pi$
$444$	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	$-0.492273\pi$
$449$	1.02862	0.0485438	0.0242719	+	0.999705i	$0.492273\pi$
$450$	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	$-0.636824\pi$
$461$	−17.8952	−0.833463	−0.416731	+	0.909030i	$0.636824\pi$
$462$	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	$-0.304348\pi$
$467$	24.9243	1.15336	0.576681	+	0.816970i	$0.304348\pi$
$468$	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	$-0.693165\pi$
$479$	−24.9623	−1.14056	−0.570278	+	0.821452i	$0.693165\pi$
$480$	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	$-0.713033\pi$
$491$	−27.4946	−1.24082	−0.620408	+	0.784279i	$0.713033\pi$
$492$	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	$-0.566904\pi$
$503$	−9.35861	−0.417280	−0.208640	+	0.977993i	$0.566904\pi$
$504$	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	$-0.796002\pi$
$509$	−36.1685	−1.60314	−0.801571	+	0.597899i	$0.796002\pi$
$510$	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	$-0.335856\pi$
$521$	22.5114	0.986242	0.493121	+	0.869961i	$0.335856\pi$
$522$	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	$-0.478519\pi$
$557$	3.18302	0.134869	0.0674343	+	0.997724i	$0.478519\pi$
$558$	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	$-0.490025\pi$
$563$	1.48690	0.0626654	0.0313327	+	0.999509i	$0.490025\pi$
$564$	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	$-0.713885\pi$
$569$	−29.6981	−1.24501	−0.622506	+	0.782615i	$0.713885\pi$
$570$	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	$-0.290906\pi$
$587$	29.5900	1.22131	0.610656	+	0.791896i	$0.290906\pi$
$588$	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	$-0.557159\pi$
$593$	−8.69875	−0.357215	−0.178607	+	0.983920i	$0.557159\pi$
$594$	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	$-0.281118\pi$
$599$	31.0686	1.26943	0.634714	+	0.772747i	$0.281118\pi$
$600$	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	$-0.359132\pi$
$617$	21.2748	0.856491	0.428246	+	0.903662i	$0.359132\pi$
$618$	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	$-0.385193\pi$
$641$	17.8698	0.705814	0.352907	+	0.935658i	$0.385193\pi$
$642$	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	$-0.528016\pi$
$647$	−4.47180	−0.175804	−0.0879022	+	0.996129i	$0.528016\pi$
$648$	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	$-0.431584\pi$
$653$	10.9005	0.426569	0.213284	+	0.976990i	$0.431584\pi$
$654$	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	$-0.315504\pi$
$659$	28.1199	1.09540	0.547698	+	0.836676i	$0.315504\pi$
$660$	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	$-0.455905\pi$
$677$	7.18581	0.276173	0.138087	+	0.990420i	$0.455905\pi$
$678$	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	$-0.420348\pi$
$683$	12.9433	0.495263	0.247631	+	0.968854i	$0.420348\pi$
$684$	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	$-0.370266\pi$
$701$	20.9895	0.792764	0.396382	+	0.918086i	$0.370266\pi$
$702$	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	$-0.379801\pi$
$719$	19.7731	0.737414	0.368707	+	0.929546i	$0.379801\pi$
$720$	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	$-0.263077\pi$
$743$	36.9328	1.35493	0.677467	+	0.735553i	$0.263077\pi$
$744$	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	$-0.244577\pi$
$761$	39.6718	1.43810	0.719051	+	0.694958i	$0.244577\pi$
$762$	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	$-0.195311\pi$
$773$	45.4626	1.63518	0.817588	+	0.575804i	$0.195311\pi$
$774$	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	$-0.387879\pi$
$797$	19.4795	0.690001	0.345001	+	0.938602i	$0.387879\pi$
$798$	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	$-0.789493\pi$
$809$	−44.8930	−1.57836	−0.789178	+	0.614165i	$0.789493\pi$
$810$	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	$-0.754009\pi$
$821$	−41.0287	−1.43191	−0.715956	+	0.698146i	$0.754009\pi$
$822$	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	$-0.545773\pi$
$827$	−8.24227	−0.286612	−0.143306	+	0.989678i	$0.545773\pi$
$828$	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	$-0.613449\pi$
$839$	−20.2129	−0.697826	−0.348913	+	0.937155i	$0.613449\pi$
$840$	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	$-0.586387\pi$
$857$	−15.6954	−0.536145	−0.268072	+	0.963399i	$0.586387\pi$
$858$	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	$-0.371712\pi$
$863$	23.0436	0.784413	0.392207	+	0.919877i	$0.371712\pi$
$864$	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	$-0.283774\pi$
$881$	37.2946	1.25649	0.628243	+	0.778017i	$0.283774\pi$
$882$	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	$-0.236904\pi$
$887$	43.8156	1.47118	0.735592	+	0.677425i	$0.236904\pi$
$888$	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	$-0.462404\pi$
$911$	7.11321	0.235671	0.117836	+	0.993033i	$0.462404\pi$
$912$	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	$-0.551303\pi$
$929$	−9.78244	−0.320952	−0.160476	+	0.987040i	$0.551303\pi$
$930$	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	$-0.597600\pi$
$941$	−18.5181	−0.603673	−0.301836	+	0.953360i	$0.597600\pi$
$942$	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	$-0.257648\pi$
$947$	42.4621	1.37983	0.689916	+	0.723890i	$0.257648\pi$
$948$	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	$-0.417477\pi$
$953$	15.8281	0.512721	0.256361	+	0.966581i	$0.417477\pi$
$954$	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	$-0.308359\pi$
$971$	35.2953	1.13268	0.566341	+	0.824171i	$0.308359\pi$
$972$	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	$-0.883465\pi$
$977$	−58.3711	−1.86746	−0.933728	+	0.357982i	$0.883465\pi$
$978$	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	$-0.741775\pi$
$983$	−43.1792	−1.37720	−0.688602	+	0.725140i	$0.741775\pi$
$984$	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.q.1.3		5
3.2	odd	2		3726.2.a.v.1.3		5
9.2	odd	6		414.2.e.b.139.5	✓	10
9.4	even	3		1242.2.e.d.829.3		10
9.5	odd	6		414.2.e.b.277.5	yes	10
9.7	even	3		1242.2.e.d.415.3		10

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

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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more