LMFDB - Embedded newform 3726.2.a.n.1.2

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			1.2
Root			$-1.39477$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.39477 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.39477 q^{7} +1.00000 q^{8} -1.00000 q^{10} +6.13254 q^{11} +4.73778 q^{13} -2.39477 q^{14} +1.00000 q^{16} -4.73778 q^{17} +4.00000 q^{19} -1.00000 q^{20} +6.13254 q^{22} -1.00000 q^{23} -4.00000 q^{25} +4.73778 q^{26} -2.39477 q^{28} -2.73778 q^{29} -2.13254 q^{31} +1.00000 q^{32} -4.73778 q^{34} +2.39477 q^{35} +5.00000 q^{37} +4.00000 q^{38} -1.00000 q^{40} +5.34301 q^{41} -0.394768 q^{43} +6.13254 q^{44} -1.00000 q^{46} +4.39477 q^{47} -1.26509 q^{49} -4.00000 q^{50} +4.73778 q^{52} +9.34301 q^{53} -6.13254 q^{55} -2.39477 q^{56} -2.73778 q^{58} -1.60523 q^{59} +3.78954 q^{61} -2.13254 q^{62} +1.00000 q^{64} -4.73778 q^{65} -1.73778 q^{67} -4.73778 q^{68} +2.39477 q^{70} +2.26222 q^{71} +15.3976 q^{73} +5.00000 q^{74} +4.00000 q^{76} -14.6860 q^{77} -1.60523 q^{79} -1.00000 q^{80} +5.34301 q^{82} -10.1325 q^{83} +4.73778 q^{85} -0.394768 q^{86} +6.13254 q^{88} +6.73778 q^{89} -11.3459 q^{91} -1.00000 q^{92} +4.39477 q^{94} -4.00000 q^{95} +8.78954 q^{97} -1.26509 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−2.39477	−0.905137	−0.452569	−	0.891730i	$-0.649492\pi$
$7$	−2.39477	−0.905137	−0.452569	+	0.891730i	$0.649492\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	6.13254	1.84903	0.924516	−	0.381144i	$-0.124470\pi$
$11$	6.13254	1.84903	0.924516	+	0.381144i	$0.124470\pi$
$12$	0	0
$13$	4.73778	1.31402	0.657011	−	0.753881i	$-0.271820\pi$
$13$	4.73778	1.31402	0.657011	+	0.753881i	$0.271820\pi$
$14$	−2.39477	−0.640029
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.73778	−1.14908	−0.574540	−	0.818477i	$-0.694819\pi$
$17$	−4.73778	−1.14908	−0.574540	+	0.818477i	$0.694819\pi$
$18$	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
$22$	6.13254	1.30746
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.00000	−0.800000
$26$	4.73778	0.929154
$27$	0	0
$28$	−2.39477	−0.452569
$29$	−2.73778	−0.508392	−0.254196	−	0.967153i	$-0.581811\pi$
$29$	−2.73778	−0.508392	−0.254196	+	0.967153i	$0.581811\pi$
$30$	0	0
$31$	−2.13254	−0.383016	−0.191508	−	0.981491i	$-0.561338\pi$
$31$	−2.13254	−0.383016	−0.191508	+	0.981491i	$0.561338\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.73778	−0.812522
$35$	2.39477	0.404790
$36$	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
$41$	5.34301	0.834438	0.417219	−	0.908806i	$-0.363005\pi$
$41$	5.34301	0.834438	0.417219	+	0.908806i	$0.363005\pi$
$42$	0	0
$43$	−0.394768	−0.0602015	−0.0301007	−	0.999547i	$-0.509583\pi$
$43$	−0.394768	−0.0602015	−0.0301007	+	0.999547i	$0.509583\pi$
$44$	6.13254	0.924516
$45$	0	0
$46$	−1.00000	−0.147442
$47$	4.39477	0.641043	0.320521	−	0.947241i	$-0.396142\pi$
$47$	4.39477	0.641043	0.320521	+	0.947241i	$0.396142\pi$
$48$	0	0
$49$	−1.26509	−0.180727
$50$	−4.00000	−0.565685
$51$	0	0
$52$	4.73778	0.657011
$53$	9.34301	1.28336	0.641680	−	0.766972i	$-0.278238\pi$
$53$	9.34301	1.28336	0.641680	+	0.766972i	$0.278238\pi$
$54$	0	0
$55$	−6.13254	−0.826912
$56$	−2.39477	−0.320014
$57$	0	0
$58$	−2.73778	−0.359488
$59$	−1.60523	−0.208983	−0.104492	−	0.994526i	$-0.533322\pi$
$59$	−1.60523	−0.208983	−0.104492	+	0.994526i	$0.533322\pi$
$60$	0	0
$61$	3.78954	0.485200	0.242600	−	0.970126i	$-0.422000\pi$
$61$	3.78954	0.485200	0.242600	+	0.970126i	$0.422000\pi$
$62$	−2.13254	−0.270833
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.73778	−0.587649
$66$	0	0
$67$	−1.73778	−0.212303	−0.106152	−	0.994350i	$-0.533853\pi$
$67$	−1.73778	−0.212303	−0.106152	+	0.994350i	$0.533853\pi$
$68$	−4.73778	−0.574540
$69$	0	0
$70$	2.39477	0.286229
$71$	2.26222	0.268477	0.134238	−	0.990949i	$-0.457141\pi$
$71$	2.26222	0.268477	0.134238	+	0.990949i	$0.457141\pi$
$72$	0	0
$73$	15.3976	1.80216	0.901078	−	0.433656i	$-0.142777\pi$
$73$	15.3976	1.80216	0.901078	+	0.433656i	$0.142777\pi$
$74$	5.00000	0.581238
$75$	0	0
$76$	4.00000	0.458831
$77$	−14.6860	−1.67363
$78$	0	0
$79$	−1.60523	−0.180603	−0.0903014	−	0.995914i	$-0.528783\pi$
$79$	−1.60523	−0.180603	−0.0903014	+	0.995914i	$0.528783\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	5.34301	0.590037
$83$	−10.1325	−1.11219	−0.556096	−	0.831118i	$-0.687701\pi$
$83$	−10.1325	−1.11219	−0.556096	+	0.831118i	$0.687701\pi$
$84$	0	0
$85$	4.73778	0.513884
$86$	−0.394768	−0.0425689
$87$	0	0
$88$	6.13254	0.653731
$89$	6.73778	0.714203	0.357101	−	0.934066i	$-0.383765\pi$
$89$	6.73778	0.714203	0.357101	+	0.934066i	$0.383765\pi$
$90$	0	0
$91$	−11.3459	−1.18937
$92$	−1.00000	−0.104257
$93$	0	0
$94$	4.39477	0.453286
$95$	−4.00000	−0.410391
$96$	0	0
$97$	8.78954	0.892442	0.446221	−	0.894923i	$-0.352770\pi$
$97$	8.78954	0.892442	0.446221	+	0.894923i	$0.352770\pi$
$98$	−1.26509	−0.127793
$99$	0	0
$100$	−4.00000	−0.400000
$101$	−4.78954	−0.476577	−0.238288	−	0.971194i	$-0.576586\pi$
$101$	−4.78954	−0.476577	−0.238288	+	0.971194i	$0.576586\pi$
$102$	0	0
$103$	12.7895	1.26019	0.630095	−	0.776518i	$-0.283016\pi$
$103$	12.7895	1.26019	0.630095	+	0.776518i	$0.283016\pi$
$104$	4.73778	0.464577
$105$	0	0
$106$	9.34301	0.907473
$107$	−0.394768	−0.0381636	−0.0190818	−	0.999818i	$-0.506074\pi$
$107$	−0.394768	−0.0381636	−0.0190818	+	0.999818i	$0.506074\pi$
$108$	0	0
$109$	17.1325	1.64100	0.820500	−	0.571646i	$-0.193695\pi$
$109$	17.1325	1.64100	0.820500	+	0.571646i	$0.193695\pi$
$110$	−6.13254	−0.584715
$111$	0	0
$112$	−2.39477	−0.226284
$113$	13.0029	1.22321	0.611603	−	0.791165i	$-0.290525\pi$
$113$	13.0029	1.22321	0.611603	+	0.791165i	$0.290525\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	−2.73778	−0.254196
$117$	0	0
$118$	−1.60523	−0.147774
$119$	11.3459	1.04007
$120$	0	0
$121$	26.6081	2.41892
$122$	3.78954	0.343088
$123$	0	0
$124$	−2.13254	−0.191508
$125$	9.00000	0.804984
$126$	0	0
$127$	5.18430	0.460032	0.230016	−	0.973187i	$-0.426122\pi$
$127$	5.18430	0.460032	0.230016	+	0.973187i	$0.426122\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−4.73778	−0.415530
$131$	−14.6599	−1.28084	−0.640419	−	0.768026i	$-0.721239\pi$
$131$	−14.6599	−1.28084	−0.640419	+	0.768026i	$0.721239\pi$
$132$	0	0
$133$	−9.57907	−0.830611
$134$	−1.73778	−0.150121
$135$	0	0
$136$	−4.73778	−0.406261
$137$	−19.7924	−1.69098	−0.845489	−	0.533992i	$-0.820691\pi$
$137$	−19.7924	−1.69098	−0.845489	+	0.533992i	$0.820691\pi$
$138$	0	0
$139$	13.8703	1.17647	0.588233	−	0.808692i	$-0.299824\pi$
$139$	13.8703	1.17647	0.588233	+	0.808692i	$0.299824\pi$
$140$	2.39477	0.202395
$141$	0	0
$142$	2.26222	0.189842
$143$	29.0546	2.42967
$144$	0	0
$145$	2.73778	0.227360
$146$	15.3976	1.27432
$147$	0	0
$148$	5.00000	0.410997
$149$	−14.6081	−1.19674	−0.598371	−	0.801219i	$-0.704185\pi$
$149$	−14.6081	−1.19674	−0.598371	+	0.801219i	$0.704185\pi$
$150$	0	0
$151$	1.73778	0.141418	0.0707091	−	0.997497i	$-0.477474\pi$
$151$	1.73778	0.141418	0.0707091	+	0.997497i	$0.477474\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	−14.6860	−1.18343
$155$	2.13254	0.171290
$156$	0	0
$157$	18.6081	1.48509	0.742544	−	0.669797i	$-0.233619\pi$
$157$	18.6081	1.48509	0.742544	+	0.669797i	$0.233619\pi$
$158$	−1.60523	−0.127705
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	2.39477	0.188734
$162$	0	0
$163$	13.0808	1.02457	0.512283	−	0.858817i	$-0.328800\pi$
$163$	13.0808	1.02457	0.512283	+	0.858817i	$0.328800\pi$
$164$	5.34301	0.417219
$165$	0	0
$166$	−10.1325	−0.786438
$167$	14.5273	1.12416	0.562079	−	0.827084i	$-0.310002\pi$
$167$	14.5273	1.12416	0.562079	+	0.827084i	$0.310002\pi$
$168$	0	0
$169$	9.44653	0.726656
$170$	4.73778	0.363371
$171$	0	0
$172$	−0.394768	−0.0301007
$173$	−2.05176	−0.155992	−0.0779962	−	0.996954i	$-0.524852\pi$
$173$	−2.05176	−0.155992	−0.0779962	+	0.996954i	$0.524852\pi$
$174$	0	0
$175$	9.57907	0.724110
$176$	6.13254	0.462258
$177$	0	0
$178$	6.73778	0.505018
$179$	14.6860	1.09768	0.548842	−	0.835926i	$-0.315069\pi$
$179$	14.6860	1.09768	0.548842	+	0.835926i	$0.315069\pi$
$180$	0	0
$181$	−9.60810	−0.714164	−0.357082	−	0.934073i	$-0.616228\pi$
$181$	−9.60810	−0.714164	−0.357082	+	0.934073i	$0.616228\pi$
$182$	−11.3459	−0.841012
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−5.00000	−0.367607
$186$	0	0
$187$	−29.0546	−2.12468
$188$	4.39477	0.320521
$189$	0	0
$190$	−4.00000	−0.290191
$191$	2.39477	0.173279	0.0866397	−	0.996240i	$-0.472387\pi$
$191$	2.39477	0.173279	0.0866397	+	0.996240i	$0.472387\pi$
$192$	0	0
$193$	−23.2651	−1.67466	−0.837329	−	0.546700i	$-0.815884\pi$
$193$	−23.2651	−1.67466	−0.837329	+	0.546700i	$0.815884\pi$
$194$	8.78954	0.631052
$195$	0	0
$196$	−1.26509	−0.0903634
$197$	−25.7924	−1.83763	−0.918816	−	0.394686i	$-0.870853\pi$
$197$	−25.7924	−1.83763	−0.918816	+	0.394686i	$0.870853\pi$
$198$	0	0
$199$	6.68602	0.473959	0.236980	−	0.971515i	$-0.423843\pi$
$199$	6.68602	0.473959	0.236980	+	0.971515i	$0.423843\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	−4.78954	−0.336991
$203$	6.55634	0.460165
$204$	0	0
$205$	−5.34301	−0.373172
$206$	12.7895	0.891089
$207$	0	0
$208$	4.73778	0.328506
$209$	24.5302	1.69679
$210$	0	0
$211$	−24.5563	−1.69053	−0.845264	−	0.534349i	$-0.820557\pi$
$211$	−24.5563	−1.69053	−0.845264	+	0.534349i	$0.820557\pi$
$212$	9.34301	0.641680
$213$	0	0
$214$	−0.394768	−0.0269858
$215$	0.394768	0.0269229
$216$	0	0
$217$	5.10695	0.346682
$218$	17.1325	1.16036
$219$	0	0
$220$	−6.13254	−0.413456
$221$	−22.4465	−1.50992
$222$	0	0
$223$	2.52731	0.169241	0.0846207	−	0.996413i	$-0.473032\pi$
$223$	2.52731	0.169241	0.0846207	+	0.996413i	$0.473032\pi$
$224$	−2.39477	−0.160007
$225$	0	0
$226$	13.0029	0.864938
$227$	−19.3459	−1.28403	−0.642015	−	0.766692i	$-0.721901\pi$
$227$	−19.3459	−1.28403	−0.642015	+	0.766692i	$0.721901\pi$
$228$	0	0
$229$	2.86746	0.189487	0.0947434	−	0.995502i	$-0.469797\pi$
$229$	2.86746	0.189487	0.0947434	+	0.995502i	$0.469797\pi$
$230$	1.00000	0.0659380
$231$	0	0
$232$	−2.73778	−0.179744
$233$	6.60810	0.432911	0.216455	−	0.976293i	$-0.430550\pi$
$233$	6.60810	0.432911	0.216455	+	0.976293i	$0.430550\pi$
$234$	0	0
$235$	−4.39477	−0.286683
$236$	−1.60523	−0.104492
$237$	0	0
$238$	11.3459	0.735444
$239$	−8.55347	−0.553278	−0.276639	−	0.960974i	$-0.589221\pi$
$239$	−8.55347	−0.553278	−0.276639	+	0.960974i	$0.589221\pi$
$240$	0	0
$241$	15.0029	0.966419	0.483210	−	0.875505i	$-0.339471\pi$
$241$	15.0029	0.966419	0.483210	+	0.875505i	$0.339471\pi$
$242$	26.6081	1.71043
$243$	0	0
$244$	3.78954	0.242600
$245$	1.26509	0.0808235
$246$	0	0
$247$	18.9511	1.20583
$248$	−2.13254	−0.135417
$249$	0	0
$250$	9.00000	0.569210
$251$	31.1872	1.96852	0.984258	−	0.176736i	$-0.0565539\pi$
$251$	31.1872	1.96852	0.984258	+	0.176736i	$0.0565539\pi$
$252$	0	0
$253$	−6.13254	−0.385550
$254$	5.18430	0.325292
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.5791	−0.784661	−0.392330	−	0.919824i	$-0.628331\pi$
$257$	−12.5791	−0.784661	−0.392330	+	0.919824i	$0.628331\pi$
$258$	0	0
$259$	−11.9738	−0.744018
$260$	−4.73778	−0.293824
$261$	0	0
$262$	−14.6599	−0.905689
$263$	7.97384	0.491688	0.245844	−	0.969309i	$-0.420935\pi$
$263$	7.97384	0.491688	0.245844	+	0.969309i	$0.420935\pi$
$264$	0	0
$265$	−9.34301	−0.573936
$266$	−9.57907	−0.587330
$267$	0	0
$268$	−1.73778	−0.106152
$269$	−22.4784	−1.37053	−0.685267	−	0.728292i	$-0.740314\pi$
$269$	−22.4784	−1.37053	−0.685267	+	0.728292i	$0.740314\pi$
$270$	0	0
$271$	18.9221	1.14943	0.574717	−	0.818352i	$-0.305112\pi$
$271$	18.9221	1.14943	0.574717	+	0.818352i	$0.305112\pi$
$272$	−4.73778	−0.287270
$273$	0	0
$274$	−19.7924	−1.19570
$275$	−24.5302	−1.47923
$276$	0	0
$277$	19.8442	1.19232	0.596160	−	0.802866i	$-0.296692\pi$
$277$	19.8442	1.19232	0.596160	+	0.802866i	$0.296692\pi$
$278$	13.8703	0.831887
$279$	0	0
$280$	2.39477	0.143115
$281$	−29.0029	−1.73017	−0.865083	−	0.501629i	$-0.832734\pi$
$281$	−29.0029	−1.73017	−0.865083	+	0.501629i	$0.832734\pi$
$282$	0	0
$283$	−22.0029	−1.30793	−0.653967	−	0.756523i	$-0.726897\pi$
$283$	−22.0029	−1.30793	−0.653967	+	0.756523i	$0.726897\pi$
$284$	2.26222	0.134238
$285$	0	0
$286$	29.0546	1.71804
$287$	−12.7953	−0.755281
$288$	0	0
$289$	5.44653	0.320384
$290$	2.73778	0.160768
$291$	0	0
$292$	15.3976	0.901078
$293$	−2.73491	−0.159775	−0.0798876	−	0.996804i	$-0.525456\pi$
$293$	−2.73491	−0.159775	−0.0798876	+	0.996804i	$0.525456\pi$
$294$	0	0
$295$	1.60523	0.0934602
$296$	5.00000	0.290619
$297$	0	0
$298$	−14.6081	−0.846224
$299$	−4.73778	−0.273993
$300$	0	0
$301$	0.945377	0.0544906
$302$	1.73778	0.0999978
$303$	0	0
$304$	4.00000	0.229416
$305$	−3.78954	−0.216988
$306$	0	0
$307$	−14.1354	−0.806750	−0.403375	−	0.915035i	$-0.632163\pi$
$307$	−14.1354	−0.806750	−0.403375	+	0.915035i	$0.632163\pi$
$308$	−14.6860	−0.836814
$309$	0	0
$310$	2.13254	0.121120
$311$	−26.3976	−1.49687	−0.748436	−	0.663207i	$-0.769195\pi$
$311$	−26.3976	−1.49687	−0.748436	+	0.663207i	$0.769195\pi$
$312$	0	0
$313$	−15.7924	−0.892639	−0.446320	−	0.894874i	$-0.647266\pi$
$313$	−15.7924	−0.892639	−0.446320	+	0.894874i	$0.647266\pi$
$314$	18.6081	1.05012
$315$	0	0
$316$	−1.60523	−0.0903014
$317$	−26.3168	−1.47810	−0.739051	−	0.673650i	$-0.764726\pi$
$317$	−26.3168	−1.47810	−0.739051	+	0.673650i	$0.764726\pi$
$318$	0	0
$319$	−16.7895	−0.940034
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	2.39477	0.133455
$323$	−18.9511	−1.05447
$324$	0	0
$325$	−18.9511	−1.05122
$326$	13.0808	0.724478
$327$	0	0
$328$	5.34301	0.295018
$329$	−10.5244	−0.580232
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−10.1325	−0.556096
$333$	0	0
$334$	14.5273	0.794899
$335$	1.73778	0.0949449
$336$	0	0
$337$	−6.26509	−0.341281	−0.170641	−	0.985333i	$-0.554584\pi$
$337$	−6.26509	−0.341281	−0.170641	+	0.985333i	$0.554584\pi$
$338$	9.44653	0.513823
$339$	0	0
$340$	4.73778	0.256942
$341$	−13.0779	−0.708209
$342$	0	0
$343$	19.7930	1.06872
$344$	−0.394768	−0.0212844
$345$	0	0
$346$	−2.05176	−0.110303
$347$	−17.0808	−0.916945	−0.458472	−	0.888709i	$-0.651603\pi$
$347$	−17.0808	−0.916945	−0.458472	+	0.888709i	$0.651603\pi$
$348$	0	0
$349$	−5.73491	−0.306983	−0.153491	−	0.988150i	$-0.549052\pi$
$349$	−5.73491	−0.306983	−0.153491	+	0.988150i	$0.549052\pi$
$350$	9.57907	0.512023
$351$	0	0
$352$	6.13254	0.326866
$353$	−12.0290	−0.640240	−0.320120	−	0.947377i	$-0.603723\pi$
$353$	−12.0290	−0.640240	−0.320120	+	0.947377i	$0.603723\pi$
$354$	0	0
$355$	−2.26222	−0.120066
$356$	6.73778	0.357101
$357$	0	0
$358$	14.6860	0.776180
$359$	−4.52445	−0.238791	−0.119396	−	0.992847i	$-0.538096\pi$
$359$	−4.52445	−0.238791	−0.119396	+	0.992847i	$0.538096\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−9.60810	−0.504990
$363$	0	0
$364$	−11.3459	−0.594685
$365$	−15.3976	−0.805949
$366$	0	0
$367$	−19.4756	−1.01662	−0.508308	−	0.861175i	$-0.669729\pi$
$367$	−19.4756	−1.01662	−0.508308	+	0.861175i	$0.669729\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−5.00000	−0.259938
$371$	−22.3743	−1.16162
$372$	0	0
$373$	−7.60810	−0.393932	−0.196966	−	0.980410i	$-0.563109\pi$
$373$	−7.60810	−0.393932	−0.196966	+	0.980410i	$0.563109\pi$
$374$	−29.0546	−1.50238
$375$	0	0
$376$	4.39477	0.226643
$377$	−12.9710	−0.668039
$378$	0	0
$379$	34.5273	1.77355	0.886774	−	0.462202i	$-0.152941\pi$
$379$	34.5273	1.77355	0.886774	+	0.462202i	$0.152941\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	2.39477	0.122527
$383$	20.7895	1.06230	0.531148	−	0.847279i	$-0.321761\pi$
$383$	20.7895	1.06230	0.531148	+	0.847279i	$0.321761\pi$
$384$	0	0
$385$	14.6860	0.748469
$386$	−23.2651	−1.18416
$387$	0	0
$388$	8.78954	0.446221
$389$	−5.34301	−0.270901	−0.135451	−	0.990784i	$-0.543248\pi$
$389$	−5.34301	−0.270901	−0.135451	+	0.990784i	$0.543248\pi$
$390$	0	0
$391$	4.73778	0.239600
$392$	−1.26509	−0.0638966
$393$	0	0
$394$	−25.7924	−1.29940
$395$	1.60523	0.0807680
$396$	0	0
$397$	−27.4238	−1.37636	−0.688180	−	0.725540i	$-0.741590\pi$
$397$	−27.4238	−1.37636	−0.688180	+	0.725540i	$0.741590\pi$
$398$	6.68602	0.335140
$399$	0	0
$400$	−4.00000	−0.200000
$401$	24.4784	1.22239	0.611197	−	0.791479i	$-0.290688\pi$
$401$	24.4784	1.22239	0.611197	+	0.791479i	$0.290688\pi$
$402$	0	0
$403$	−10.1035	−0.503292
$404$	−4.78954	−0.238288
$405$	0	0
$406$	6.55634	0.325386
$407$	30.6627	1.51989
$408$	0	0
$409$	−9.68602	−0.478943	−0.239471	−	0.970903i	$-0.576974\pi$
$409$	−9.68602	−0.478943	−0.239471	+	0.970903i	$0.576974\pi$
$410$	−5.34301	−0.263872
$411$	0	0
$412$	12.7895	0.630095
$413$	3.84416	0.189159
$414$	0	0
$415$	10.1325	0.497387
$416$	4.73778	0.232289
$417$	0	0
$418$	24.5302	1.19981
$419$	29.3197	1.43236	0.716181	−	0.697915i	$-0.245889\pi$
$419$	29.3197	1.43236	0.716181	+	0.697915i	$0.245889\pi$
$420$	0	0
$421$	−9.26509	−0.451553	−0.225776	−	0.974179i	$-0.572492\pi$
$421$	−9.26509	−0.451553	−0.225776	+	0.974179i	$0.572492\pi$
$422$	−24.5563	−1.19538
$423$	0	0
$424$	9.34301	0.453737
$425$	18.9511	0.919264
$426$	0	0
$427$	−9.07506	−0.439173
$428$	−0.394768	−0.0190818
$429$	0	0
$430$	0.394768	0.0190374
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−11.9482	−0.574196	−0.287098	−	0.957901i	$-0.592691\pi$
$433$	−11.9482	−0.574196	−0.287098	+	0.957901i	$0.592691\pi$
$434$	5.10695	0.245141
$435$	0	0
$436$	17.1325	0.820500
$437$	−4.00000	−0.191346
$438$	0	0
$439$	30.7924	1.46964	0.734821	−	0.678262i	$-0.237266\pi$
$439$	30.7924	1.46964	0.734821	+	0.678262i	$0.237266\pi$
$440$	−6.13254	−0.292358
$441$	0	0
$442$	−22.4465	−1.06767
$443$	−27.1843	−1.29157	−0.645783	−	0.763521i	$-0.723469\pi$
$443$	−27.1843	−1.29157	−0.645783	+	0.763521i	$0.723469\pi$
$444$	0	0
$445$	−6.73778	−0.319401
$446$	2.52731	0.119672
$447$	0	0
$448$	−2.39477	−0.113142
$449$	−23.1872	−1.09427	−0.547135	−	0.837044i	$-0.684282\pi$
$449$	−23.1872	−1.09427	−0.547135	+	0.837044i	$0.684282\pi$
$450$	0	0
$451$	32.7662	1.54290
$452$	13.0029	0.611603
$453$	0	0
$454$	−19.3459	−0.907947
$455$	11.3459	0.531903
$456$	0	0
$457$	−29.7924	−1.39363	−0.696815	−	0.717251i	$-0.745400\pi$
$457$	−29.7924	−1.39363	−0.696815	+	0.717251i	$0.745400\pi$
$458$	2.86746	0.133987
$459$	0	0
$460$	1.00000	0.0466252
$461$	30.7953	1.43428	0.717139	−	0.696930i	$-0.245451\pi$
$461$	30.7953	1.43428	0.717139	+	0.696930i	$0.245451\pi$
$462$	0	0
$463$	−1.34301	−0.0624149	−0.0312075	−	0.999513i	$-0.509935\pi$
$463$	−1.34301	−0.0624149	−0.0312075	+	0.999513i	$0.509935\pi$
$464$	−2.73778	−0.127098
$465$	0	0
$466$	6.60810	0.306114
$467$	0.158706	0.00734405	0.00367202	−	0.999993i	$-0.498831\pi$
$467$	0.158706	0.00734405	0.00367202	+	0.999993i	$0.498831\pi$
$468$	0	0
$469$	4.16157	0.192163
$470$	−4.39477	−0.202716
$471$	0	0
$472$	−1.60523	−0.0738868
$473$	−2.42093	−0.111314
$474$	0	0
$475$	−16.0000	−0.734130
$476$	11.3459	0.520037
$477$	0	0
$478$	−8.55347	−0.391227
$479$	31.4494	1.43696	0.718480	−	0.695548i	$-0.244838\pi$
$479$	31.4494	1.43696	0.718480	+	0.695548i	$0.244838\pi$
$480$	0	0
$481$	23.6889	1.08012
$482$	15.0029	0.683362
$483$	0	0
$484$	26.6081	1.20946
$485$	−8.78954	−0.399112
$486$	0	0
$487$	25.7378	1.16629	0.583145	−	0.812368i	$-0.301822\pi$
$487$	25.7378	1.16629	0.583145	+	0.812368i	$0.301822\pi$
$488$	3.78954	0.171544
$489$	0	0
$490$	1.26509	0.0571509
$491$	8.78954	0.396666	0.198333	−	0.980135i	$-0.436447\pi$
$491$	8.78954	0.396666	0.198333	+	0.980135i	$0.436447\pi$
$492$	0	0
$493$	12.9710	0.584183
$494$	18.9511	0.852651
$495$	0	0
$496$	−2.13254	−0.0957541
$497$	−5.41750	−0.243008
$498$	0	0
$499$	6.95111	0.311174	0.155587	−	0.987822i	$-0.450273\pi$
$499$	6.95111	0.311174	0.155587	+	0.987822i	$0.450273\pi$
$500$	9.00000	0.402492
$501$	0	0
$502$	31.1872	1.39195
$503$	−10.6860	−0.476466	−0.238233	−	0.971208i	$-0.576568\pi$
$503$	−10.6860	−0.476466	−0.238233	+	0.971208i	$0.576568\pi$
$504$	0	0
$505$	4.78954	0.213132
$506$	−6.13254	−0.272625
$507$	0	0
$508$	5.18430	0.230016
$509$	−8.95111	−0.396751	−0.198375	−	0.980126i	$-0.563567\pi$
$509$	−8.95111	−0.396751	−0.198375	+	0.980126i	$0.563567\pi$
$510$	0	0
$511$	−36.8738	−1.63120
$512$	1.00000	0.0441942
$513$	0	0
$514$	−12.5791	−0.554839
$515$	−12.7895	−0.563574
$516$	0	0
$517$	26.9511	1.18531
$518$	−11.9738	−0.526100
$519$	0	0
$520$	−4.73778	−0.207765
$521$	−12.6860	−0.555785	−0.277892	−	0.960612i	$-0.589636\pi$
$521$	−12.6860	−0.555785	−0.277892	+	0.960612i	$0.589636\pi$
$522$	0	0
$523$	15.7116	0.687021	0.343510	−	0.939149i	$-0.388384\pi$
$523$	15.7116	0.687021	0.343510	+	0.939149i	$0.388384\pi$
$524$	−14.6599	−0.640419
$525$	0	0
$526$	7.97384	0.347676
$527$	10.1035	0.440116
$528$	0	0
$529$	1.00000	0.0434783
$530$	−9.34301	−0.405834
$531$	0	0
$532$	−9.57907	−0.415305
$533$	25.3140	1.09647
$534$	0	0
$535$	0.394768	0.0170673
$536$	−1.73778	−0.0750605
$537$	0	0
$538$	−22.4784	−0.969113
$539$	−7.75821	−0.334170
$540$	0	0
$541$	10.8413	0.466104	0.233052	−	0.972464i	$-0.425129\pi$
$541$	10.8413	0.466104	0.233052	+	0.972464i	$0.425129\pi$
$542$	18.9221	0.812773
$543$	0	0
$544$	−4.73778	−0.203130
$545$	−17.1325	−0.733878
$546$	0	0
$547$	−39.7145	−1.69807	−0.849034	−	0.528338i	$-0.822815\pi$
$547$	−39.7145	−1.69807	−0.849034	+	0.528338i	$0.822815\pi$
$548$	−19.7924	−0.845489
$549$	0	0
$550$	−24.5302	−1.04597
$551$	−10.9511	−0.466533
$552$	0	0
$553$	3.84416	0.163470
$554$	19.8442	0.843098
$555$	0	0
$556$	13.8703	0.588233
$557$	34.3197	1.45417	0.727086	−	0.686546i	$-0.240874\pi$
$557$	34.3197	1.45417	0.727086	+	0.686546i	$0.240874\pi$
$558$	0	0
$559$	−1.87032	−0.0791061
$560$	2.39477	0.101197
$561$	0	0
$562$	−29.0029	−1.22341
$563$	31.7406	1.33771	0.668854	−	0.743394i	$-0.266785\pi$
$563$	31.7406	1.33771	0.668854	+	0.743394i	$0.266785\pi$
$564$	0	0
$565$	−13.0029	−0.547035
$566$	−22.0029	−0.924849
$567$	0	0
$568$	2.26222	0.0949208
$569$	30.2133	1.26661	0.633304	−	0.773903i	$-0.281698\pi$
$569$	30.2133	1.26661	0.633304	+	0.773903i	$0.281698\pi$
$570$	0	0
$571$	29.6081	1.23906	0.619530	−	0.784973i	$-0.287323\pi$
$571$	29.6081	1.23906	0.619530	+	0.784973i	$0.287323\pi$
$572$	29.0546	1.21483
$573$	0	0
$574$	−12.7953	−0.534064
$575$	4.00000	0.166812
$576$	0	0
$577$	−4.47555	−0.186320	−0.0931599	−	0.995651i	$-0.529697\pi$
$577$	−4.47555	−0.186320	−0.0931599	+	0.995651i	$0.529697\pi$
$578$	5.44653	0.226546
$579$	0	0
$580$	2.73778	0.113680
$581$	24.2651	1.00669
$582$	0	0
$583$	57.2964	2.37298
$584$	15.3976	0.637159
$585$	0	0
$586$	−2.73491	−0.112978
$587$	−43.4494	−1.79335	−0.896674	−	0.442691i	$-0.854024\pi$
$587$	−43.4494	−1.79335	−0.896674	+	0.442691i	$0.854024\pi$
$588$	0	0
$589$	−8.53018	−0.351480
$590$	1.60523	0.0660864
$591$	0	0
$592$	5.00000	0.205499
$593$	−4.73491	−0.194440	−0.0972198	−	0.995263i	$-0.530995\pi$
$593$	−4.73491	−0.194440	−0.0972198	+	0.995263i	$0.530995\pi$
$594$	0	0
$595$	−11.3459	−0.465135
$596$	−14.6081	−0.598371
$597$	0	0
$598$	−4.73778	−0.193742
$599$	9.42036	0.384906	0.192453	−	0.981306i	$-0.438356\pi$
$599$	9.42036	0.384906	0.192453	+	0.981306i	$0.438356\pi$
$600$	0	0
$601$	31.1616	1.27111	0.635553	−	0.772057i	$-0.280772\pi$
$601$	31.1616	1.27111	0.635553	+	0.772057i	$0.280772\pi$
$602$	0.945377	0.0385307
$603$	0	0
$604$	1.73778	0.0707091
$605$	−26.6081	−1.08177
$606$	0	0
$607$	−47.4523	−1.92603	−0.963014	−	0.269452i	$-0.913157\pi$
$607$	−47.4523	−1.92603	−0.963014	+	0.269452i	$0.913157\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	−3.78954	−0.153434
$611$	20.8214	0.842345
$612$	0	0
$613$	−44.2651	−1.78785	−0.893925	−	0.448216i	$-0.852059\pi$
$613$	−44.2651	−1.78785	−0.893925	+	0.448216i	$0.852059\pi$
$614$	−14.1354	−0.570459
$615$	0	0
$616$	−14.6860	−0.591717
$617$	−31.5330	−1.26947	−0.634736	−	0.772729i	$-0.718891\pi$
$617$	−31.5330	−1.26947	−0.634736	+	0.772729i	$0.718891\pi$
$618$	0	0
$619$	−16.4238	−0.660128	−0.330064	−	0.943959i	$-0.607070\pi$
$619$	−16.4238	−0.660128	−0.330064	+	0.943959i	$0.607070\pi$
$620$	2.13254	0.0856450
$621$	0	0
$622$	−26.3976	−1.05845
$623$	−16.1354	−0.646452
$624$	0	0
$625$	11.0000	0.440000
$626$	−15.7924	−0.631191
$627$	0	0
$628$	18.6081	0.742544
$629$	−23.6889	−0.944538
$630$	0	0
$631$	−26.3948	−1.05076	−0.525380	−	0.850868i	$-0.676077\pi$
$631$	−26.3948	−1.05076	−0.525380	+	0.850868i	$0.676077\pi$
$632$	−1.60523	−0.0638527
$633$	0	0
$634$	−26.3168	−1.04518
$635$	−5.18430	−0.205733
$636$	0	0
$637$	−5.99371	−0.237479
$638$	−16.7895	−0.664704
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−10.6343	−0.420028	−0.210014	−	0.977698i	$-0.567351\pi$
$641$	−10.6343	−0.420028	−0.210014	+	0.977698i	$0.567351\pi$
$642$	0	0
$643$	−35.0575	−1.38253	−0.691266	−	0.722600i	$-0.742947\pi$
$643$	−35.0575	−1.38253	−0.691266	+	0.722600i	$0.742947\pi$
$644$	2.39477	0.0943671
$645$	0	0
$646$	−18.9511	−0.745621
$647$	−27.7116	−1.08946	−0.544728	−	0.838613i	$-0.683367\pi$
$647$	−27.7116	−1.08946	−0.544728	+	0.838613i	$0.683367\pi$
$648$	0	0
$649$	−9.84416	−0.386417
$650$	−18.9511	−0.743324
$651$	0	0
$652$	13.0808	0.512283
$653$	−10.2651	−0.401704	−0.200852	−	0.979622i	$-0.564371\pi$
$653$	−10.2651	−0.401704	−0.200852	+	0.979622i	$0.564371\pi$
$654$	0	0
$655$	14.6599	0.572808
$656$	5.34301	0.208609
$657$	0	0
$658$	−10.5244	−0.410286
$659$	16.3948	0.638649	0.319325	−	0.947645i	$-0.396544\pi$
$659$	16.3948	0.638649	0.319325	+	0.947645i	$0.396544\pi$
$660$	0	0
$661$	29.3686	1.14231	0.571153	−	0.820843i	$-0.306496\pi$
$661$	29.3686	1.14231	0.571153	+	0.820843i	$0.306496\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	−10.1325	−0.393219
$665$	9.57907	0.371460
$666$	0	0
$667$	2.73778	0.106007
$668$	14.5273	0.562079
$669$	0	0
$670$	1.73778	0.0671362
$671$	23.2395	0.897151
$672$	0	0
$673$	−41.8186	−1.61199	−0.805994	−	0.591924i	$-0.798368\pi$
$673$	−41.8186	−1.61199	−0.805994	+	0.591924i	$0.798368\pi$
$674$	−6.26509	−0.241322
$675$	0	0
$676$	9.44653	0.363328
$677$	16.5011	0.634191	0.317095	−	0.948394i	$-0.397292\pi$
$677$	16.5011	0.634191	0.317095	+	0.948394i	$0.397292\pi$
$678$	0	0
$679$	−21.0489	−0.807782
$680$	4.73778	0.181685
$681$	0	0
$682$	−13.0779	−0.500779
$683$	−23.4494	−0.897266	−0.448633	−	0.893716i	$-0.648089\pi$
$683$	−23.4494	−0.897266	−0.448633	+	0.893716i	$0.648089\pi$
$684$	0	0
$685$	19.7924	0.756229
$686$	19.7930	0.755699
$687$	0	0
$688$	−0.394768	−0.0150504
$689$	44.2651	1.68637
$690$	0	0
$691$	22.3686	0.850942	0.425471	−	0.904972i	$-0.360108\pi$
$691$	22.3686	0.850942	0.425471	+	0.904972i	$0.360108\pi$
$692$	−2.05176	−0.0779962
$693$	0	0
$694$	−17.0808	−0.648378
$695$	−13.8703	−0.526131
$696$	0	0
$697$	−25.3140	−0.958835
$698$	−5.73491	−0.217070
$699$	0	0
$700$	9.57907	0.362055
$701$	26.3197	0.994082	0.497041	−	0.867727i	$-0.334420\pi$
$701$	26.3197	0.994082	0.497041	+	0.867727i	$0.334420\pi$
$702$	0	0
$703$	20.0000	0.754314
$704$	6.13254	0.231129
$705$	0	0
$706$	−12.0290	−0.452718
$707$	11.4698	0.431367
$708$	0	0
$709$	−45.1092	−1.69411	−0.847057	−	0.531503i	$-0.821628\pi$
$709$	−45.1092	−1.69411	−0.847057	+	0.531503i	$0.821628\pi$
$710$	−2.26222	−0.0848997
$711$	0	0
$712$	6.73778	0.252509
$713$	2.13254	0.0798644
$714$	0	0
$715$	−29.0546	−1.08658
$716$	14.6860	0.548842
$717$	0	0
$718$	−4.52445	−0.168851
$719$	13.0779	0.487724	0.243862	−	0.969810i	$-0.421586\pi$
$719$	13.0779	0.487724	0.243862	+	0.969810i	$0.421586\pi$
$720$	0	0
$721$	−30.6280	−1.14064
$722$	−3.00000	−0.111648
$723$	0	0
$724$	−9.60810	−0.357082
$725$	10.9511	0.406714
$726$	0	0
$727$	41.9022	1.55407	0.777034	−	0.629459i	$-0.216724\pi$
$727$	41.9022	1.55407	0.777034	+	0.629459i	$0.216724\pi$
$728$	−11.3459	−0.420506
$729$	0	0
$730$	−15.3976	−0.569892
$731$	1.87032	0.0691763
$732$	0	0
$733$	14.6860	0.542440	0.271220	−	0.962517i	$-0.412573\pi$
$733$	14.6860	0.542440	0.271220	+	0.962517i	$0.412573\pi$
$734$	−19.4756	−0.718856
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−10.6570	−0.392555
$738$	0	0
$739$	21.3197	0.784258	0.392129	−	0.919910i	$-0.371739\pi$
$739$	21.3197	0.784258	0.392129	+	0.919910i	$0.371739\pi$
$740$	−5.00000	−0.183804
$741$	0	0
$742$	−22.3743	−0.821388
$743$	−36.5040	−1.33920	−0.669601	−	0.742721i	$-0.733535\pi$
$743$	−36.5040	−1.33920	−0.669601	+	0.742721i	$0.733535\pi$
$744$	0	0
$745$	14.6081	0.535199
$746$	−7.60810	−0.278552
$747$	0	0
$748$	−29.0546	−1.06234
$749$	0.945377	0.0345433
$750$	0	0
$751$	35.7406	1.30419	0.652097	−	0.758135i	$-0.273889\pi$
$751$	35.7406	1.30419	0.652097	+	0.758135i	$0.273889\pi$
$752$	4.39477	0.160261
$753$	0	0
$754$	−12.9710	−0.472375
$755$	−1.73778	−0.0632442
$756$	0	0
$757$	−21.6081	−0.785360	−0.392680	−	0.919675i	$-0.628452\pi$
$757$	−21.6081	−0.785360	−0.392680	+	0.919675i	$0.628452\pi$
$758$	34.5273	1.25409
$759$	0	0
$760$	−4.00000	−0.145095
$761$	−29.7662	−1.07903	−0.539513	−	0.841978i	$-0.681391\pi$
$761$	−29.7662	−1.07903	−0.539513	+	0.841978i	$0.681391\pi$
$762$	0	0
$763$	−41.0285	−1.48533
$764$	2.39477	0.0866397
$765$	0	0
$766$	20.7895	0.751156
$767$	−7.60523	−0.274609
$768$	0	0
$769$	−41.2680	−1.48816	−0.744080	−	0.668090i	$-0.767112\pi$
$769$	−41.2680	−1.48816	−0.744080	+	0.668090i	$0.767112\pi$
$770$	14.6860	0.529247
$771$	0	0
$772$	−23.2651	−0.837329
$773$	−1.36861	−0.0492253	−0.0246127	−	0.999697i	$-0.507835\pi$
$773$	−1.36861	−0.0492253	−0.0246127	+	0.999697i	$0.507835\pi$
$774$	0	0
$775$	8.53018	0.306413
$776$	8.78954	0.315526
$777$	0	0
$778$	−5.34301	−0.191556
$779$	21.3720	0.765733
$780$	0	0
$781$	13.8732	0.496422
$782$	4.73778	0.169423
$783$	0	0
$784$	−1.26509	−0.0451817
$785$	−18.6081	−0.664151
$786$	0	0
$787$	−42.1616	−1.50290	−0.751449	−	0.659791i	$-0.770645\pi$
$787$	−42.1616	−1.50290	−0.751449	+	0.659791i	$0.770645\pi$
$788$	−25.7924	−0.918816
$789$	0	0
$790$	1.60523	0.0571116
$791$	−31.1388	−1.10717
$792$	0	0
$793$	17.9540	0.637564
$794$	−27.4238	−0.973234
$795$	0	0
$796$	6.68602	0.236980
$797$	21.7895	0.771825	0.385912	−	0.922535i	$-0.373887\pi$
$797$	21.7895	0.771825	0.385912	+	0.922535i	$0.373887\pi$
$798$	0	0
$799$	−20.8214	−0.736609
$800$	−4.00000	−0.141421
$801$	0	0
$802$	24.4784	0.864363
$803$	94.4267	3.33224
$804$	0	0
$805$	−2.39477	−0.0844045
$806$	−10.1035	−0.355881
$807$	0	0
$808$	−4.78954	−0.168495
$809$	−39.6570	−1.39427	−0.697133	−	0.716942i	$-0.745541\pi$
$809$	−39.6570	−1.39427	−0.697133	+	0.716942i	$0.745541\pi$
$810$	0	0
$811$	10.6599	0.374318	0.187159	−	0.982330i	$-0.440072\pi$
$811$	10.6599	0.374318	0.187159	+	0.982330i	$0.440072\pi$
$812$	6.55634	0.230082
$813$	0	0
$814$	30.6627	1.07473
$815$	−13.0808	−0.458200
$816$	0	0
$817$	−1.57907	−0.0552447
$818$	−9.68602	−0.338664
$819$	0	0
$820$	−5.34301	−0.186586
$821$	−12.9971	−0.453603	−0.226802	−	0.973941i	$-0.572827\pi$
$821$	−12.9971	−0.453603	−0.226802	+	0.973941i	$0.572827\pi$
$822$	0	0
$823$	33.4784	1.16698	0.583492	−	0.812119i	$-0.301686\pi$
$823$	33.4784	1.16698	0.583492	+	0.812119i	$0.301686\pi$
$824$	12.7895	0.445545
$825$	0	0
$826$	3.84416	0.133755
$827$	−12.1064	−0.420980	−0.210490	−	0.977596i	$-0.567506\pi$
$827$	−12.1064	−0.420980	−0.210490	+	0.977596i	$0.567506\pi$
$828$	0	0
$829$	−11.5791	−0.402158	−0.201079	−	0.979575i	$-0.564445\pi$
$829$	−11.5791	−0.402158	−0.201079	+	0.979575i	$0.564445\pi$
$830$	10.1325	0.351706
$831$	0	0
$832$	4.73778	0.164253
$833$	5.99371	0.207670
$834$	0	0
$835$	−14.5273	−0.502738
$836$	24.5302	0.848394
$837$	0	0
$838$	29.3197	1.01283
$839$	27.9738	0.965764	0.482882	−	0.875685i	$-0.339590\pi$
$839$	27.9738	0.965764	0.482882	+	0.875685i	$0.339590\pi$
$840$	0	0
$841$	−21.5046	−0.741537
$842$	−9.26509	−0.319296
$843$	0	0
$844$	−24.5563	−0.845264
$845$	−9.44653	−0.324970
$846$	0	0
$847$	−63.7202	−2.18945
$848$	9.34301	0.320840
$849$	0	0
$850$	18.9511	0.650018
$851$	−5.00000	−0.171398
$852$	0	0
$853$	30.3686	1.03980	0.519901	−	0.854227i	$-0.325969\pi$
$853$	30.3686	1.03980	0.519901	+	0.854227i	$0.325969\pi$
$854$	−9.07506	−0.310542
$855$	0	0
$856$	−0.394768	−0.0134929
$857$	35.2708	1.20483	0.602414	−	0.798184i	$-0.294206\pi$
$857$	35.2708	1.20483	0.602414	+	0.798184i	$0.294206\pi$
$858$	0	0
$859$	14.6860	0.501080	0.250540	−	0.968106i	$-0.419392\pi$
$859$	14.6860	0.501080	0.250540	+	0.968106i	$0.419392\pi$
$860$	0.394768	0.0134615
$861$	0	0
$862$	−12.0000	−0.408722
$863$	−33.2133	−1.13059	−0.565297	−	0.824887i	$-0.691239\pi$
$863$	−33.2133	−1.13059	−0.565297	+	0.824887i	$0.691239\pi$
$864$	0	0
$865$	2.05176	0.0697619
$866$	−11.9482	−0.406018
$867$	0	0
$868$	5.10695	0.173341
$869$	−9.84416	−0.333940
$870$	0	0
$871$	−8.23320	−0.278971
$872$	17.1325	0.580181
$873$	0	0
$874$	−4.00000	−0.135302
$875$	−21.5529	−0.728621
$876$	0	0
$877$	33.5273	1.13214	0.566068	−	0.824358i	$-0.308464\pi$
$877$	33.5273	1.13214	0.566068	+	0.824358i	$0.308464\pi$
$878$	30.7924	1.03919
$879$	0	0
$880$	−6.13254	−0.206728
$881$	11.5791	0.390109	0.195054	−	0.980792i	$-0.437512\pi$
$881$	11.5791	0.390109	0.195054	+	0.980792i	$0.437512\pi$
$882$	0	0
$883$	−10.3948	−0.349812	−0.174906	−	0.984585i	$-0.555962\pi$
$883$	−10.3948	−0.349812	−0.174906	+	0.984585i	$0.555962\pi$
$884$	−22.4465	−0.754958
$885$	0	0
$886$	−27.1843	−0.913275
$887$	38.2622	1.28472	0.642360	−	0.766403i	$-0.277955\pi$
$887$	38.2622	1.28472	0.642360	+	0.766403i	$0.277955\pi$
$888$	0	0
$889$	−12.4152	−0.416392
$890$	−6.73778	−0.225851
$891$	0	0
$892$	2.52731	0.0846207
$893$	17.5791	0.588261
$894$	0	0
$895$	−14.6860	−0.490899
$896$	−2.39477	−0.0800036
$897$	0	0
$898$	−23.1872	−0.773766
$899$	5.83843	0.194723
$900$	0	0
$901$	−44.2651	−1.47468
$902$	32.7662	1.09100
$903$	0	0
$904$	13.0029	0.432469
$905$	9.60810	0.319384
$906$	0	0
$907$	8.28839	0.275211	0.137606	−	0.990487i	$-0.456059\pi$
$907$	8.28839	0.275211	0.137606	+	0.990487i	$0.456059\pi$
$908$	−19.3459	−0.642015
$909$	0	0
$910$	11.3459	0.376112
$911$	−3.47555	−0.115150	−0.0575751	−	0.998341i	$-0.518337\pi$
$911$	−3.47555	−0.115150	−0.0575751	+	0.998341i	$0.518337\pi$
$912$	0	0
$913$	−62.1383	−2.05648
$914$	−29.7924	−0.985445
$915$	0	0
$916$	2.86746	0.0947434
$917$	35.1069	1.15933
$918$	0	0
$919$	50.9568	1.68091	0.840455	−	0.541881i	$-0.182288\pi$
$919$	50.9568	1.68091	0.840455	+	0.541881i	$0.182288\pi$
$920$	1.00000	0.0329690
$921$	0	0
$922$	30.7953	1.01419
$923$	10.7179	0.352784
$924$	0	0
$925$	−20.0000	−0.657596
$926$	−1.34301	−0.0441340
$927$	0	0
$928$	−2.73778	−0.0898719
$929$	−0.210465	−0.00690513	−0.00345256	−	0.999994i	$-0.501099\pi$
$929$	−0.210465	−0.00690513	−0.00345256	+	0.999994i	$0.501099\pi$
$930$	0	0
$931$	−5.06035	−0.165846
$932$	6.60810	0.216455
$933$	0	0
$934$	0.158706	0.00519302
$935$	29.0546	0.950188
$936$	0	0
$937$	−45.1121	−1.47375	−0.736874	−	0.676030i	$-0.763699\pi$
$937$	−45.1121	−1.47375	−0.736874	+	0.676030i	$0.763699\pi$
$938$	4.16157	0.135880
$939$	0	0
$940$	−4.39477	−0.143342
$941$	37.2651	1.21481	0.607404	−	0.794393i	$-0.292211\pi$
$941$	37.2651	1.21481	0.607404	+	0.794393i	$0.292211\pi$
$942$	0	0
$943$	−5.34301	−0.173992
$944$	−1.60523	−0.0522459
$945$	0	0
$946$	−2.42093	−0.0787112
$947$	−57.8761	−1.88072	−0.940359	−	0.340182i	$-0.889511\pi$
$947$	−57.8761	−1.88072	−0.940359	+	0.340182i	$0.889511\pi$
$948$	0	0
$949$	72.9505	2.36807
$950$	−16.0000	−0.519109
$951$	0	0
$952$	11.3459	0.367722
$953$	−21.2622	−0.688751	−0.344375	−	0.938832i	$-0.611909\pi$
$953$	−21.2622	−0.688751	−0.344375	+	0.938832i	$0.611909\pi$
$954$	0	0
$955$	−2.39477	−0.0774929
$956$	−8.55347	−0.276639
$957$	0	0
$958$	31.4494	1.01608
$959$	47.3982	1.53057
$960$	0	0
$961$	−26.4523	−0.853299
$962$	23.6889	0.763760
$963$	0	0
$964$	15.0029	0.483210
$965$	23.2651	0.748930
$966$	0	0
$967$	−58.5273	−1.88211	−0.941056	−	0.338252i	$-0.890165\pi$
$967$	−58.5273	−1.88211	−0.941056	+	0.338252i	$0.890165\pi$
$968$	26.6081	0.855217
$969$	0	0
$970$	−8.78954	−0.282215
$971$	−26.8988	−0.863223	−0.431611	−	0.902060i	$-0.642055\pi$
$971$	−26.8988	−0.863223	−0.431611	+	0.902060i	$0.642055\pi$
$972$	0	0
$973$	−33.2162	−1.06486
$974$	25.7378	0.824692
$975$	0	0
$976$	3.78954	0.121300
$977$	8.78954	0.281202	0.140601	−	0.990066i	$-0.455097\pi$
$977$	8.78954	0.281202	0.140601	+	0.990066i	$0.455097\pi$
$978$	0	0
$979$	41.3197	1.32058
$980$	1.26509	0.0404118
$981$	0	0
$982$	8.78954	0.280485
$983$	28.8157	0.919078	0.459539	−	0.888158i	$-0.348015\pi$
$983$	28.8157	0.919078	0.459539	+	0.888158i	$0.348015\pi$
$984$	0	0
$985$	25.7924	0.821814
$986$	12.9710	0.413080
$987$	0	0
$988$	18.9511	0.602915
$989$	0.394768	0.0125529
$990$	0	0
$991$	−23.5529	−0.748183	−0.374091	−	0.927392i	$-0.622045\pi$
$991$	−23.5529	−0.748183	−0.374091	+	0.927392i	$0.622045\pi$
$992$	−2.13254	−0.0677083
$993$	0	0
$994$	−5.41750	−0.171833
$995$	−6.68602	−0.211961
$996$	0	0
$997$	−48.7435	−1.54372	−0.771861	−	0.635791i	$-0.780674\pi$
$997$	−48.7435	−1.54372	−0.771861	+	0.635791i	$0.780674\pi$
$998$	6.95111	0.220033
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3726.2.a.n.1.2	yes	3
3.2	odd	2		3726.2.a.m.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3726.2.a.m.1.2	✓	3	3.2	odd	2
3726.2.a.n.1.2	yes	3	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.39477 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.39477 q^{7} +1.00000 q^{8} -1.00000 q^{10} +6.13254 q^{11} +4.73778 q^{13} -2.39477 q^{14} +1.00000 q^{16} -4.73778 q^{17} +4.00000 q^{19} -1.00000 q^{20} +6.13254 q^{22} -1.00000 q^{23} -4.00000 q^{25} +4.73778 q^{26} -2.39477 q^{28} -2.73778 q^{29} -2.13254 q^{31} +1.00000 q^{32} -4.73778 q^{34} +2.39477 q^{35} +5.00000 q^{37} +4.00000 q^{38} -1.00000 q^{40} +5.34301 q^{41} -0.394768 q^{43} +6.13254 q^{44} -1.00000 q^{46} +4.39477 q^{47} -1.26509 q^{49} -4.00000 q^{50} +4.73778 q^{52} +9.34301 q^{53} -6.13254 q^{55} -2.39477 q^{56} -2.73778 q^{58} -1.60523 q^{59} +3.78954 q^{61} -2.13254 q^{62} +1.00000 q^{64} -4.73778 q^{65} -1.73778 q^{67} -4.73778 q^{68} +2.39477 q^{70} +2.26222 q^{71} +15.3976 q^{73} +5.00000 q^{74} +4.00000 q^{76} -14.6860 q^{77} -1.60523 q^{79} -1.00000 q^{80} +5.34301 q^{82} -10.1325 q^{83} +4.73778 q^{85} -0.394768 q^{86} +6.13254 q^{88} +6.73778 q^{89} -11.3459 q^{91} -1.00000 q^{92} +4.39477 q^{94} -4.00000 q^{95} +8.78954 q^{97} -1.26509 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - 2 q^{7} + 3 q^{8}+O(q^{10}) \) 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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