LMFDB - Embedded newform 740.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(1,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("740.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 740 = 2^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	740.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:	$5.90892974957$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	740.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{3} +1.00000 q^{5} +1.26795 q^{7} +4.46410 q^{9} -5.46410 q^{11} +4.00000 q^{13} -2.73205 q^{15} -4.00000 q^{17} +6.19615 q^{19} -3.46410 q^{21} -8.92820 q^{23} +1.00000 q^{25} -4.00000 q^{27} +7.46410 q^{29} +10.1962 q^{31} +14.9282 q^{33} +1.26795 q^{35} +1.00000 q^{37} -10.9282 q^{39} +4.92820 q^{41} +6.00000 q^{43} +4.46410 q^{45} +1.26795 q^{47} -5.39230 q^{49} +10.9282 q^{51} +12.9282 q^{53} -5.46410 q^{55} -16.9282 q^{57} +3.66025 q^{59} +14.0000 q^{61} +5.66025 q^{63} +4.00000 q^{65} -5.26795 q^{67} +24.3923 q^{69} +8.00000 q^{71} -10.3923 q^{73} -2.73205 q^{75} -6.92820 q^{77} +6.19615 q^{79} -2.46410 q^{81} +1.66025 q^{83} -4.00000 q^{85} -20.3923 q^{87} -2.00000 q^{89} +5.07180 q^{91} -27.8564 q^{93} +6.19615 q^{95} -14.0000 q^{97} -24.3923 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} - 4 q^{11} + 8 q^{13} - 2 q^{15} - 8 q^{17} + 2 q^{19} - 4 q^{23} + 2 q^{25} - 8 q^{27} + 8 q^{29} + 10 q^{31} + 16 q^{33} + 6 q^{35} + 2 q^{37} - 8 q^{39} - 4 q^{41}+ \cdots - 28 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 - 4 * q^11 + 8 * q^13 - 2 * q^15 - 8 * q^17 + 2 * q^19 - 4 * q^23 + 2 * q^25 - 8 * q^27 + 8 * q^29 + 10 * q^31 + 16 * q^33 + 6 * q^35 + 2 * q^37 - 8 * q^39 - 4 * q^41 + 12 * q^43 + 2 * q^45 + 6 * q^47 + 10 * q^49 + 8 * q^51 + 12 * q^53 - 4 * q^55 - 20 * q^57 - 10 * q^59 + 28 * q^61 - 6 * q^63 + 8 * q^65 - 14 * q^67 + 28 * q^69 + 16 * q^71 - 2 * q^75 + 2 * q^79 + 2 * q^81 - 14 * q^83 - 8 * q^85 - 20 * q^87 - 4 * q^89 + 24 * q^91 - 28 * q^93 + 2 * q^95 - 28 * q^97 - 28 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−2.73205	−1.57735	−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.26795	0.479240	0.239620	−	0.970867i	$-0.422977\pi$
$7$	1.26795	0.479240	0.239620	+	0.970867i	$0.422977\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	−5.46410	−1.64749	−0.823744	−	0.566961i	$-0.808119\pi$
$11$	−5.46410	−1.64749	−0.823744	+	0.566961i	$0.808119\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−2.73205	−0.705412
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	6.19615	1.42149	0.710747	−	0.703447i	$-0.248357\pi$
$19$	6.19615	1.42149	0.710747	+	0.703447i	$0.248357\pi$
$20$	0	0
$21$	−3.46410	−0.755929
$22$	0	0
$23$	−8.92820	−1.86166	−0.930830	−	0.365454i	$-0.880914\pi$
$23$	−8.92820	−1.86166	−0.930830	+	0.365454i	$0.880914\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	7.46410	1.38605	0.693024	−	0.720914i	$-0.256278\pi$
$29$	7.46410	1.38605	0.693024	+	0.720914i	$0.256278\pi$
$30$	0	0
$31$	10.1962	1.83128	0.915642	−	0.401996i	$-0.131683\pi$
$31$	10.1962	1.83128	0.915642	+	0.401996i	$0.131683\pi$
$32$	0	0
$33$	14.9282	2.59867
$34$	0	0
$35$	1.26795	0.214323
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−10.9282	−1.74991
$40$	0	0
$41$	4.92820	0.769656	0.384828	−	0.922988i	$-0.374261\pi$
$41$	4.92820	0.769656	0.384828	+	0.922988i	$0.374261\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	4.46410	0.665469
$46$	0	0
$47$	1.26795	0.184949	0.0924747	−	0.995715i	$-0.470522\pi$
$47$	1.26795	0.184949	0.0924747	+	0.995715i	$0.470522\pi$
$48$	0	0
$49$	−5.39230	−0.770329
$50$	0	0
$51$	10.9282	1.53025
$52$	0	0
$53$	12.9282	1.77583	0.887913	−	0.460012i	$-0.152155\pi$
$53$	12.9282	1.77583	0.887913	+	0.460012i	$0.152155\pi$
$54$	0	0
$55$	−5.46410	−0.736779
$56$	0	0
$57$	−16.9282	−2.24220
$58$	0	0
$59$	3.66025	0.476524	0.238262	−	0.971201i	$-0.423422\pi$
$59$	3.66025	0.476524	0.238262	+	0.971201i	$0.423422\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	5.66025	0.713125
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	−5.26795	−0.643582	−0.321791	−	0.946811i	$-0.604285\pi$
$67$	−5.26795	−0.643582	−0.321791	+	0.946811i	$0.604285\pi$
$68$	0	0
$69$	24.3923	2.93649
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−10.3923	−1.21633	−0.608164	−	0.793812i	$-0.708094\pi$
$73$	−10.3923	−1.21633	−0.608164	+	0.793812i	$0.708094\pi$
$74$	0	0
$75$	−2.73205	−0.315470
$76$	0	0
$77$	−6.92820	−0.789542
$78$	0	0
$79$	6.19615	0.697122	0.348561	−	0.937286i	$-0.386670\pi$
$79$	6.19615	0.697122	0.348561	+	0.937286i	$0.386670\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	1.66025	0.182237	0.0911183	−	0.995840i	$-0.470956\pi$
$83$	1.66025	0.182237	0.0911183	+	0.995840i	$0.470956\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	−20.3923	−2.18628
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	5.07180	0.531669
$92$	0	0
$93$	−27.8564	−2.88857
$94$	0	0
$95$	6.19615	0.635712
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	−24.3923	−2.45152
$100$	0	0
$101$	2.53590	0.252331	0.126166	−	0.992009i	$-0.459733\pi$
$101$	2.53590	0.252331	0.126166	+	0.992009i	$0.459733\pi$
$102$	0	0
$103$	6.39230	0.629853	0.314926	−	0.949116i	$-0.398020\pi$
$103$	6.39230	0.629853	0.314926	+	0.949116i	$0.398020\pi$
$104$	0	0
$105$	−3.46410	−0.338062
$106$	0	0
$107$	−0.196152	−0.0189628	−0.00948139	−	0.999955i	$-0.503018\pi$
$107$	−0.196152	−0.0189628	−0.00948139	+	0.999955i	$0.503018\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−2.73205	−0.259315
$112$	0	0
$113$	17.8564	1.67979	0.839895	−	0.542749i	$-0.182617\pi$
$113$	17.8564	1.67979	0.839895	+	0.542749i	$0.182617\pi$
$114$	0	0
$115$	−8.92820	−0.832559
$116$	0	0
$117$	17.8564	1.65083
$118$	0	0
$119$	−5.07180	−0.464931
$120$	0	0
$121$	18.8564	1.71422
$122$	0	0
$123$	−13.4641	−1.21402
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.33975	−0.207619	−0.103809	−	0.994597i	$-0.533103\pi$
$127$	−2.33975	−0.207619	−0.103809	+	0.994597i	$0.533103\pi$
$128$	0	0
$129$	−16.3923	−1.44326
$130$	0	0
$131$	−16.7321	−1.46189	−0.730943	−	0.682438i	$-0.760920\pi$
$131$	−16.7321	−1.46189	−0.730943	+	0.682438i	$0.760920\pi$
$132$	0	0
$133$	7.85641	0.681237
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	−21.3205	−1.82153	−0.910767	−	0.412921i	$-0.864509\pi$
$137$	−21.3205	−1.82153	−0.910767	+	0.412921i	$0.864509\pi$
$138$	0	0
$139$	−18.9282	−1.60547	−0.802735	−	0.596336i	$-0.796622\pi$
$139$	−18.9282	−1.60547	−0.802735	+	0.596336i	$0.796622\pi$
$140$	0	0
$141$	−3.46410	−0.291730
$142$	0	0
$143$	−21.8564	−1.82772
$144$	0	0
$145$	7.46410	0.619860
$146$	0	0
$147$	14.7321	1.21508
$148$	0	0
$149$	9.46410	0.775329	0.387665	−	0.921800i	$-0.373282\pi$
$149$	9.46410	0.775329	0.387665	+	0.921800i	$0.373282\pi$
$150$	0	0
$151$	−12.3923	−1.00847	−0.504236	−	0.863566i	$-0.668226\pi$
$151$	−12.3923	−1.00847	−0.504236	+	0.863566i	$0.668226\pi$
$152$	0	0
$153$	−17.8564	−1.44360
$154$	0	0
$155$	10.1962	0.818975
$156$	0	0
$157$	7.46410	0.595700	0.297850	−	0.954613i	$-0.403730\pi$
$157$	7.46410	0.595700	0.297850	+	0.954613i	$0.403730\pi$
$158$	0	0
$159$	−35.3205	−2.80110
$160$	0	0
$161$	−11.3205	−0.892181
$162$	0	0
$163$	15.4641	1.21124	0.605621	−	0.795753i	$-0.292925\pi$
$163$	15.4641	1.21124	0.605621	+	0.795753i	$0.292925\pi$
$164$	0	0
$165$	14.9282	1.16216
$166$	0	0
$167$	6.39230	0.494651	0.247326	−	0.968932i	$-0.420448\pi$
$167$	6.39230	0.494651	0.247326	+	0.968932i	$0.420448\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	27.6603	2.11523
$172$	0	0
$173$	−4.53590	−0.344858	−0.172429	−	0.985022i	$-0.555162\pi$
$173$	−4.53590	−0.344858	−0.172429	+	0.985022i	$0.555162\pi$
$174$	0	0
$175$	1.26795	0.0958479
$176$	0	0
$177$	−10.0000	−0.751646
$178$	0	0
$179$	4.73205	0.353690	0.176845	−	0.984239i	$-0.443411\pi$
$179$	4.73205	0.353690	0.176845	+	0.984239i	$0.443411\pi$
$180$	0	0
$181$	−15.3205	−1.13876	−0.569382	−	0.822073i	$-0.692818\pi$
$181$	−15.3205	−1.13876	−0.569382	+	0.822073i	$0.692818\pi$
$182$	0	0
$183$	−38.2487	−2.82743
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	21.8564	1.59830
$188$	0	0
$189$	−5.07180	−0.368919
$190$	0	0
$191$	14.5885	1.05558	0.527792	−	0.849374i	$-0.323020\pi$
$191$	14.5885	1.05558	0.527792	+	0.849374i	$0.323020\pi$
$192$	0	0
$193$	8.92820	0.642666	0.321333	−	0.946966i	$-0.395869\pi$
$193$	8.92820	0.642666	0.321333	+	0.946966i	$0.395869\pi$
$194$	0	0
$195$	−10.9282	−0.782585
$196$	0	0
$197$	9.32051	0.664059	0.332029	−	0.943269i	$-0.392267\pi$
$197$	9.32051	0.664059	0.332029	+	0.943269i	$0.392267\pi$
$198$	0	0
$199$	13.1244	0.930361	0.465180	−	0.885216i	$-0.345989\pi$
$199$	13.1244	0.930361	0.465180	+	0.885216i	$0.345989\pi$
$200$	0	0
$201$	14.3923	1.01515
$202$	0	0
$203$	9.46410	0.664250
$204$	0	0
$205$	4.92820	0.344201
$206$	0	0
$207$	−39.8564	−2.77021
$208$	0	0
$209$	−33.8564	−2.34190
$210$	0	0
$211$	−10.9282	−0.752329	−0.376164	−	0.926553i	$-0.622757\pi$
$211$	−10.9282	−0.752329	−0.376164	+	0.926553i	$0.622757\pi$
$212$	0	0
$213$	−21.8564	−1.49758
$214$	0	0
$215$	6.00000	0.409197
$216$	0	0
$217$	12.9282	0.877624
$218$	0	0
$219$	28.3923	1.91857
$220$	0	0
$221$	−16.0000	−1.07628
$222$	0	0
$223$	8.19615	0.548855	0.274427	−	0.961608i	$-0.411512\pi$
$223$	8.19615	0.548855	0.274427	+	0.961608i	$0.411512\pi$
$224$	0	0
$225$	4.46410	0.297607
$226$	0	0
$227$	−0.535898	−0.0355688	−0.0177844	−	0.999842i	$-0.505661\pi$
$227$	−0.535898	−0.0355688	−0.0177844	+	0.999842i	$0.505661\pi$
$228$	0	0
$229$	7.07180	0.467317	0.233659	−	0.972319i	$-0.424930\pi$
$229$	7.07180	0.467317	0.233659	+	0.972319i	$0.424930\pi$
$230$	0	0
$231$	18.9282	1.24538
$232$	0	0
$233$	−4.53590	−0.297157	−0.148578	−	0.988901i	$-0.547470\pi$
$233$	−4.53590	−0.297157	−0.148578	+	0.988901i	$0.547470\pi$
$234$	0	0
$235$	1.26795	0.0827119
$236$	0	0
$237$	−16.9282	−1.09960
$238$	0	0
$239$	28.4449	1.83995	0.919973	−	0.391983i	$-0.128211\pi$
$239$	28.4449	1.83995	0.919973	+	0.391983i	$0.128211\pi$
$240$	0	0
$241$	−16.5359	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$241$	−16.5359	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$242$	0	0
$243$	18.7321	1.20166
$244$	0	0
$245$	−5.39230	−0.344502
$246$	0	0
$247$	24.7846	1.57701
$248$	0	0
$249$	−4.53590	−0.287451
$250$	0	0
$251$	−16.7321	−1.05612	−0.528059	−	0.849208i	$-0.677080\pi$
$251$	−16.7321	−1.05612	−0.528059	+	0.849208i	$0.677080\pi$
$252$	0	0
$253$	48.7846	3.06706
$254$	0	0
$255$	10.9282	0.684351
$256$	0	0
$257$	1.07180	0.0668568	0.0334284	−	0.999441i	$-0.489357\pi$
$257$	1.07180	0.0668568	0.0334284	+	0.999441i	$0.489357\pi$
$258$	0	0
$259$	1.26795	0.0787865
$260$	0	0
$261$	33.3205	2.06249
$262$	0	0
$263$	−7.12436	−0.439307	−0.219653	−	0.975578i	$-0.570493\pi$
$263$	−7.12436	−0.439307	−0.219653	+	0.975578i	$0.570493\pi$
$264$	0	0
$265$	12.9282	0.794173
$266$	0	0
$267$	5.46410	0.334398
$268$	0	0
$269$	−12.3923	−0.755572	−0.377786	−	0.925893i	$-0.623315\pi$
$269$	−12.3923	−0.755572	−0.377786	+	0.925893i	$0.623315\pi$
$270$	0	0
$271$	−5.85641	−0.355751	−0.177876	−	0.984053i	$-0.556922\pi$
$271$	−5.85641	−0.355751	−0.177876	+	0.984053i	$0.556922\pi$
$272$	0	0
$273$	−13.8564	−0.838628
$274$	0	0
$275$	−5.46410	−0.329498
$276$	0	0
$277$	−28.7846	−1.72950	−0.864750	−	0.502203i	$-0.832523\pi$
$277$	−28.7846	−1.72950	−0.864750	+	0.502203i	$0.832523\pi$
$278$	0	0
$279$	45.5167	2.72501
$280$	0	0
$281$	−6.39230	−0.381333	−0.190666	−	0.981655i	$-0.561065\pi$
$281$	−6.39230	−0.381333	−0.190666	+	0.981655i	$0.561065\pi$
$282$	0	0
$283$	8.53590	0.507406	0.253703	−	0.967282i	$-0.418351\pi$
$283$	8.53590	0.507406	0.253703	+	0.967282i	$0.418351\pi$
$284$	0	0
$285$	−16.9282	−1.00274
$286$	0	0
$287$	6.24871	0.368850
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	38.2487	2.24218
$292$	0	0
$293$	0.928203	0.0542262	0.0271131	−	0.999632i	$-0.491369\pi$
$293$	0.928203	0.0542262	0.0271131	+	0.999632i	$0.491369\pi$
$294$	0	0
$295$	3.66025	0.213108
$296$	0	0
$297$	21.8564	1.26824
$298$	0	0
$299$	−35.7128	−2.06533
$300$	0	0
$301$	7.60770	0.438500
$302$	0	0
$303$	−6.92820	−0.398015
$304$	0	0
$305$	14.0000	0.801638
$306$	0	0
$307$	1.26795	0.0723657	0.0361828	−	0.999345i	$-0.488480\pi$
$307$	1.26795	0.0723657	0.0361828	+	0.999345i	$0.488480\pi$
$308$	0	0
$309$	−17.4641	−0.993498
$310$	0	0
$311$	26.1962	1.48545	0.742724	−	0.669598i	$-0.233534\pi$
$311$	26.1962	1.48545	0.742724	+	0.669598i	$0.233534\pi$
$312$	0	0
$313$	−11.8564	−0.670164	−0.335082	−	0.942189i	$-0.608764\pi$
$313$	−11.8564	−0.670164	−0.335082	+	0.942189i	$0.608764\pi$
$314$	0	0
$315$	5.66025	0.318919
$316$	0	0
$317$	15.4641	0.868550	0.434275	−	0.900780i	$-0.357005\pi$
$317$	15.4641	0.868550	0.434275	+	0.900780i	$0.357005\pi$
$318$	0	0
$319$	−40.7846	−2.28350
$320$	0	0
$321$	0.535898	0.0299109
$322$	0	0
$323$	−24.7846	−1.37905
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−5.46410	−0.302166
$328$	0	0
$329$	1.60770	0.0886351
$330$	0	0
$331$	11.6603	0.640906	0.320453	−	0.947264i	$-0.396165\pi$
$331$	11.6603	0.640906	0.320453	+	0.947264i	$0.396165\pi$
$332$	0	0
$333$	4.46410	0.244631
$334$	0	0
$335$	−5.26795	−0.287819
$336$	0	0
$337$	11.4641	0.624489	0.312245	−	0.950002i	$-0.398919\pi$
$337$	11.4641	0.624489	0.312245	+	0.950002i	$0.398919\pi$
$338$	0	0
$339$	−48.7846	−2.64962
$340$	0	0
$341$	−55.7128	−3.01702
$342$	0	0
$343$	−15.7128	−0.848412
$344$	0	0
$345$	24.3923	1.31324
$346$	0	0
$347$	−26.7846	−1.43787	−0.718937	−	0.695076i	$-0.755371\pi$
$347$	−26.7846	−1.43787	−0.718937	+	0.695076i	$0.755371\pi$
$348$	0	0
$349$	−22.2487	−1.19095	−0.595473	−	0.803375i	$-0.703035\pi$
$349$	−22.2487	−1.19095	−0.595473	+	0.803375i	$0.703035\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	0	0
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	0	0
$357$	13.8564	0.733359
$358$	0	0
$359$	−5.46410	−0.288384	−0.144192	−	0.989550i	$-0.546058\pi$
$359$	−5.46410	−0.288384	−0.144192	+	0.989550i	$0.546058\pi$
$360$	0	0
$361$	19.3923	1.02065
$362$	0	0
$363$	−51.5167	−2.70392
$364$	0	0
$365$	−10.3923	−0.543958
$366$	0	0
$367$	−14.0526	−0.733538	−0.366769	−	0.930312i	$-0.619536\pi$
$367$	−14.0526	−0.733538	−0.366769	+	0.930312i	$0.619536\pi$
$368$	0	0
$369$	22.0000	1.14527
$370$	0	0
$371$	16.3923	0.851046
$372$	0	0
$373$	16.2487	0.841326	0.420663	−	0.907217i	$-0.361797\pi$
$373$	16.2487	0.841326	0.420663	+	0.907217i	$0.361797\pi$
$374$	0	0
$375$	−2.73205	−0.141082
$376$	0	0
$377$	29.8564	1.53768
$378$	0	0
$379$	−0.392305	−0.0201513	−0.0100757	−	0.999949i	$-0.503207\pi$
$379$	−0.392305	−0.0201513	−0.0100757	+	0.999949i	$0.503207\pi$
$380$	0	0
$381$	6.39230	0.327488
$382$	0	0
$383$	11.8564	0.605834	0.302917	−	0.953017i	$-0.402039\pi$
$383$	11.8564	0.605834	0.302917	+	0.953017i	$0.402039\pi$
$384$	0	0
$385$	−6.92820	−0.353094
$386$	0	0
$387$	26.7846	1.36154
$388$	0	0
$389$	20.9282	1.06110	0.530551	−	0.847653i	$-0.321985\pi$
$389$	20.9282	1.06110	0.530551	+	0.847653i	$0.321985\pi$
$390$	0	0
$391$	35.7128	1.80607
$392$	0	0
$393$	45.7128	2.30591
$394$	0	0
$395$	6.19615	0.311762
$396$	0	0
$397$	−25.7128	−1.29049	−0.645245	−	0.763976i	$-0.723245\pi$
$397$	−25.7128	−1.29049	−0.645245	+	0.763976i	$0.723245\pi$
$398$	0	0
$399$	−21.4641	−1.07455
$400$	0	0
$401$	−3.85641	−0.192580	−0.0962899	−	0.995353i	$-0.530698\pi$
$401$	−3.85641	−0.192580	−0.0962899	+	0.995353i	$0.530698\pi$
$402$	0	0
$403$	40.7846	2.03163
$404$	0	0
$405$	−2.46410	−0.122442
$406$	0	0
$407$	−5.46410	−0.270845
$408$	0	0
$409$	0.535898	0.0264985	0.0132492	−	0.999912i	$-0.495783\pi$
$409$	0.535898	0.0264985	0.0132492	+	0.999912i	$0.495783\pi$
$410$	0	0
$411$	58.2487	2.87320
$412$	0	0
$413$	4.64102	0.228369
$414$	0	0
$415$	1.66025	0.0814987
$416$	0	0
$417$	51.7128	2.53239
$418$	0	0
$419$	−6.53590	−0.319300	−0.159650	−	0.987174i	$-0.551036\pi$
$419$	−6.53590	−0.319300	−0.159650	+	0.987174i	$0.551036\pi$
$420$	0	0
$421$	−22.7846	−1.11045	−0.555227	−	0.831699i	$-0.687369\pi$
$421$	−22.7846	−1.11045	−0.555227	+	0.831699i	$0.687369\pi$
$422$	0	0
$423$	5.66025	0.275211
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	17.7513	0.859045
$428$	0	0
$429$	59.7128	2.88296
$430$	0	0
$431$	−25.5167	−1.22909	−0.614547	−	0.788880i	$-0.710661\pi$
$431$	−25.5167	−1.22909	−0.614547	+	0.788880i	$0.710661\pi$
$432$	0	0
$433$	−9.32051	−0.447915	−0.223958	−	0.974599i	$-0.571898\pi$
$433$	−9.32051	−0.447915	−0.223958	+	0.974599i	$0.571898\pi$
$434$	0	0
$435$	−20.3923	−0.977736
$436$	0	0
$437$	−55.3205	−2.64634
$438$	0	0
$439$	−2.87564	−0.137247	−0.0686235	−	0.997643i	$-0.521861\pi$
$439$	−2.87564	−0.137247	−0.0686235	+	0.997643i	$0.521861\pi$
$440$	0	0
$441$	−24.0718	−1.14628
$442$	0	0
$443$	3.80385	0.180726	0.0903631	−	0.995909i	$-0.471197\pi$
$443$	3.80385	0.180726	0.0903631	+	0.995909i	$0.471197\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	0	0
$447$	−25.8564	−1.22297
$448$	0	0
$449$	8.53590	0.402834	0.201417	−	0.979506i	$-0.435445\pi$
$449$	8.53590	0.402834	0.201417	+	0.979506i	$0.435445\pi$
$450$	0	0
$451$	−26.9282	−1.26800
$452$	0	0
$453$	33.8564	1.59071
$454$	0	0
$455$	5.07180	0.237769
$456$	0	0
$457$	−11.0718	−0.517917	−0.258958	−	0.965888i	$-0.583379\pi$
$457$	−11.0718	−0.517917	−0.258958	+	0.965888i	$0.583379\pi$
$458$	0	0
$459$	16.0000	0.746816
$460$	0	0
$461$	32.5359	1.51535	0.757674	−	0.652633i	$-0.226336\pi$
$461$	32.5359	1.51535	0.757674	+	0.652633i	$0.226336\pi$
$462$	0	0
$463$	24.9282	1.15851	0.579256	−	0.815146i	$-0.303343\pi$
$463$	24.9282	1.15851	0.579256	+	0.815146i	$0.303343\pi$
$464$	0	0
$465$	−27.8564	−1.29181
$466$	0	0
$467$	34.3923	1.59149	0.795743	−	0.605634i	$-0.207081\pi$
$467$	34.3923	1.59149	0.795743	+	0.605634i	$0.207081\pi$
$468$	0	0
$469$	−6.67949	−0.308430
$470$	0	0
$471$	−20.3923	−0.939628
$472$	0	0
$473$	−32.7846	−1.50744
$474$	0	0
$475$	6.19615	0.284299
$476$	0	0
$477$	57.7128	2.64249
$478$	0	0
$479$	2.87564	0.131392	0.0656958	−	0.997840i	$-0.479073\pi$
$479$	2.87564	0.131392	0.0656958	+	0.997840i	$0.479073\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	30.9282	1.40728
$484$	0	0
$485$	−14.0000	−0.635707
$486$	0	0
$487$	15.4641	0.700745	0.350373	−	0.936610i	$-0.386055\pi$
$487$	15.4641	0.700745	0.350373	+	0.936610i	$0.386055\pi$
$488$	0	0
$489$	−42.2487	−1.91055
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	−29.8564	−1.34466
$494$	0	0
$495$	−24.3923	−1.09635
$496$	0	0
$497$	10.1436	0.455002
$498$	0	0
$499$	−19.6603	−0.880114	−0.440057	−	0.897970i	$-0.645042\pi$
$499$	−19.6603	−0.880114	−0.440057	+	0.897970i	$0.645042\pi$
$500$	0	0
$501$	−17.4641	−0.780239
$502$	0	0
$503$	−4.14359	−0.184754	−0.0923769	−	0.995724i	$-0.529446\pi$
$503$	−4.14359	−0.184754	−0.0923769	+	0.995724i	$0.529446\pi$
$504$	0	0
$505$	2.53590	0.112846
$506$	0	0
$507$	−8.19615	−0.364004
$508$	0	0
$509$	11.8564	0.525526	0.262763	−	0.964860i	$-0.415366\pi$
$509$	11.8564	0.525526	0.262763	+	0.964860i	$0.415366\pi$
$510$	0	0
$511$	−13.1769	−0.582912
$512$	0	0
$513$	−24.7846	−1.09427
$514$	0	0
$515$	6.39230	0.281679
$516$	0	0
$517$	−6.92820	−0.304702
$518$	0	0
$519$	12.3923	0.543962
$520$	0	0
$521$	−45.4641	−1.99182	−0.995909	−	0.0903593i	$-0.971198\pi$
$521$	−45.4641	−1.99182	−0.995909	+	0.0903593i	$0.971198\pi$
$522$	0	0
$523$	30.7846	1.34612	0.673058	−	0.739589i	$-0.264980\pi$
$523$	30.7846	1.34612	0.673058	+	0.739589i	$0.264980\pi$
$524$	0	0
$525$	−3.46410	−0.151186
$526$	0	0
$527$	−40.7846	−1.77661
$528$	0	0
$529$	56.7128	2.46577
$530$	0	0
$531$	16.3397	0.709085
$532$	0	0
$533$	19.7128	0.853857
$534$	0	0
$535$	−0.196152	−0.00848041
$536$	0	0
$537$	−12.9282	−0.557893
$538$	0	0
$539$	29.4641	1.26911
$540$	0	0
$541$	10.3923	0.446800	0.223400	−	0.974727i	$-0.428284\pi$
$541$	10.3923	0.446800	0.223400	+	0.974727i	$0.428284\pi$
$542$	0	0
$543$	41.8564	1.79623
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	8.92820	0.381742	0.190871	−	0.981615i	$-0.438869\pi$
$547$	8.92820	0.381742	0.190871	+	0.981615i	$0.438869\pi$
$548$	0	0
$549$	62.4974	2.66732
$550$	0	0
$551$	46.2487	1.97026
$552$	0	0
$553$	7.85641	0.334088
$554$	0	0
$555$	−2.73205	−0.115969
$556$	0	0
$557$	−1.85641	−0.0786585	−0.0393292	−	0.999226i	$-0.512522\pi$
$557$	−1.85641	−0.0786585	−0.0393292	+	0.999226i	$0.512522\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	−59.7128	−2.52108
$562$	0	0
$563$	24.9282	1.05060	0.525299	−	0.850918i	$-0.323953\pi$
$563$	24.9282	1.05060	0.525299	+	0.850918i	$0.323953\pi$
$564$	0	0
$565$	17.8564	0.751225
$566$	0	0
$567$	−3.12436	−0.131211
$568$	0	0
$569$	13.3205	0.558425	0.279212	−	0.960229i	$-0.409927\pi$
$569$	13.3205	0.558425	0.279212	+	0.960229i	$0.409927\pi$
$570$	0	0
$571$	−15.7128	−0.657561	−0.328780	−	0.944406i	$-0.606638\pi$
$571$	−15.7128	−0.657561	−0.328780	+	0.944406i	$0.606638\pi$
$572$	0	0
$573$	−39.8564	−1.66503
$574$	0	0
$575$	−8.92820	−0.372332
$576$	0	0
$577$	26.6410	1.10908	0.554540	−	0.832157i	$-0.312894\pi$
$577$	26.6410	1.10908	0.554540	+	0.832157i	$0.312894\pi$
$578$	0	0
$579$	−24.3923	−1.01371
$580$	0	0
$581$	2.10512	0.0873350
$582$	0	0
$583$	−70.6410	−2.92565
$584$	0	0
$585$	17.8564	0.738272
$586$	0	0
$587$	0.928203	0.0383110	0.0191555	−	0.999817i	$-0.493902\pi$
$587$	0.928203	0.0383110	0.0191555	+	0.999817i	$0.493902\pi$
$588$	0	0
$589$	63.1769	2.60316
$590$	0	0
$591$	−25.4641	−1.04745
$592$	0	0
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	0	0
$595$	−5.07180	−0.207923
$596$	0	0
$597$	−35.8564	−1.46751
$598$	0	0
$599$	21.4641	0.876999	0.438500	−	0.898731i	$-0.355510\pi$
$599$	21.4641	0.876999	0.438500	+	0.898731i	$0.355510\pi$
$600$	0	0
$601$	−3.32051	−0.135446	−0.0677232	−	0.997704i	$-0.521573\pi$
$601$	−3.32051	−0.135446	−0.0677232	+	0.997704i	$0.521573\pi$
$602$	0	0
$603$	−23.5167	−0.957672
$604$	0	0
$605$	18.8564	0.766622
$606$	0	0
$607$	−19.0718	−0.774100	−0.387050	−	0.922059i	$-0.626506\pi$
$607$	−19.0718	−0.774100	−0.387050	+	0.922059i	$0.626506\pi$
$608$	0	0
$609$	−25.8564	−1.04775
$610$	0	0
$611$	5.07180	0.205183
$612$	0	0
$613$	−24.6410	−0.995241	−0.497621	−	0.867395i	$-0.665793\pi$
$613$	−24.6410	−0.995241	−0.497621	+	0.867395i	$0.665793\pi$
$614$	0	0
$615$	−13.4641	−0.542925
$616$	0	0
$617$	2.78461	0.112104	0.0560521	−	0.998428i	$-0.482149\pi$
$617$	2.78461	0.112104	0.0560521	+	0.998428i	$0.482149\pi$
$618$	0	0
$619$	−22.9282	−0.921562	−0.460781	−	0.887514i	$-0.652431\pi$
$619$	−22.9282	−0.921562	−0.460781	+	0.887514i	$0.652431\pi$
$620$	0	0
$621$	35.7128	1.43311
$622$	0	0
$623$	−2.53590	−0.101599
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	92.4974	3.69399
$628$	0	0
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−2.48334	−0.0988602	−0.0494301	−	0.998778i	$-0.515741\pi$
$631$	−2.48334	−0.0988602	−0.0494301	+	0.998778i	$0.515741\pi$
$632$	0	0
$633$	29.8564	1.18669
$634$	0	0
$635$	−2.33975	−0.0928500
$636$	0	0
$637$	−21.5692	−0.854604
$638$	0	0
$639$	35.7128	1.41278
$640$	0	0
$641$	0.392305	0.0154951	0.00774755	−	0.999970i	$-0.497534\pi$
$641$	0.392305	0.0154951	0.00774755	+	0.999970i	$0.497534\pi$
$642$	0	0
$643$	−5.60770	−0.221146	−0.110573	−	0.993868i	$-0.535269\pi$
$643$	−5.60770	−0.221146	−0.110573	+	0.993868i	$0.535269\pi$
$644$	0	0
$645$	−16.3923	−0.645446
$646$	0	0
$647$	−2.00000	−0.0786281	−0.0393141	−	0.999227i	$-0.512517\pi$
$647$	−2.00000	−0.0786281	−0.0393141	+	0.999227i	$0.512517\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	−35.3205	−1.38432
$652$	0	0
$653$	−12.9282	−0.505920	−0.252960	−	0.967477i	$-0.581404\pi$
$653$	−12.9282	−0.505920	−0.252960	+	0.967477i	$0.581404\pi$
$654$	0	0
$655$	−16.7321	−0.653775
$656$	0	0
$657$	−46.3923	−1.80994
$658$	0	0
$659$	42.9282	1.67225	0.836123	−	0.548542i	$-0.184817\pi$
$659$	42.9282	1.67225	0.836123	+	0.548542i	$0.184817\pi$
$660$	0	0
$661$	20.6410	0.802842	0.401421	−	0.915894i	$-0.368516\pi$
$661$	20.6410	0.802842	0.401421	+	0.915894i	$0.368516\pi$
$662$	0	0
$663$	43.7128	1.69766
$664$	0	0
$665$	7.85641	0.304658
$666$	0	0
$667$	−66.6410	−2.58035
$668$	0	0
$669$	−22.3923	−0.865737
$670$	0	0
$671$	−76.4974	−2.95315
$672$	0	0
$673$	−41.7128	−1.60791	−0.803955	−	0.594690i	$-0.797275\pi$
$673$	−41.7128	−1.60791	−0.803955	+	0.594690i	$0.797275\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−45.0333	−1.73077	−0.865386	−	0.501107i	$-0.832926\pi$
$677$	−45.0333	−1.73077	−0.865386	+	0.501107i	$0.832926\pi$
$678$	0	0
$679$	−17.7513	−0.681232
$680$	0	0
$681$	1.46410	0.0561045
$682$	0	0
$683$	15.0718	0.576706	0.288353	−	0.957524i	$-0.406892\pi$
$683$	15.0718	0.576706	0.288353	+	0.957524i	$0.406892\pi$
$684$	0	0
$685$	−21.3205	−0.814615
$686$	0	0
$687$	−19.3205	−0.737123
$688$	0	0
$689$	51.7128	1.97010
$690$	0	0
$691$	32.3923	1.23226	0.616131	−	0.787644i	$-0.288699\pi$
$691$	32.3923	1.23226	0.616131	+	0.787644i	$0.288699\pi$
$692$	0	0
$693$	−30.9282	−1.17487
$694$	0	0
$695$	−18.9282	−0.717988
$696$	0	0
$697$	−19.7128	−0.746676
$698$	0	0
$699$	12.3923	0.468720
$700$	0	0
$701$	15.8564	0.598888	0.299444	−	0.954114i	$-0.403199\pi$
$701$	15.8564	0.598888	0.299444	+	0.954114i	$0.403199\pi$
$702$	0	0
$703$	6.19615	0.233692
$704$	0	0
$705$	−3.46410	−0.130466
$706$	0	0
$707$	3.21539	0.120927
$708$	0	0
$709$	−24.2487	−0.910679	−0.455340	−	0.890318i	$-0.650482\pi$
$709$	−24.2487	−0.910679	−0.455340	+	0.890318i	$0.650482\pi$
$710$	0	0
$711$	27.6603	1.03734
$712$	0	0
$713$	−91.0333	−3.40922
$714$	0	0
$715$	−21.8564	−0.817383
$716$	0	0
$717$	−77.7128	−2.90224
$718$	0	0
$719$	−26.2487	−0.978912	−0.489456	−	0.872028i	$-0.662805\pi$
$719$	−26.2487	−0.978912	−0.489456	+	0.872028i	$0.662805\pi$
$720$	0	0
$721$	8.10512	0.301850
$722$	0	0
$723$	45.1769	1.68015
$724$	0	0
$725$	7.46410	0.277210
$726$	0	0
$727$	6.00000	0.222528	0.111264	−	0.993791i	$-0.464510\pi$
$727$	6.00000	0.222528	0.111264	+	0.993791i	$0.464510\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	14.7321	0.543400
$736$	0	0
$737$	28.7846	1.06029
$738$	0	0
$739$	44.7846	1.64743	0.823714	−	0.567005i	$-0.191898\pi$
$739$	44.7846	1.64743	0.823714	+	0.567005i	$0.191898\pi$
$740$	0	0
$741$	−67.7128	−2.48749
$742$	0	0
$743$	35.1244	1.28859	0.644294	−	0.764778i	$-0.277151\pi$
$743$	35.1244	1.28859	0.644294	+	0.764778i	$0.277151\pi$
$744$	0	0
$745$	9.46410	0.346738
$746$	0	0
$747$	7.41154	0.271174
$748$	0	0
$749$	−0.248711	−0.00908771
$750$	0	0
$751$	−51.3205	−1.87271	−0.936356	−	0.351052i	$-0.885824\pi$
$751$	−51.3205	−1.87271	−0.936356	+	0.351052i	$0.885824\pi$
$752$	0	0
$753$	45.7128	1.66587
$754$	0	0
$755$	−12.3923	−0.451002
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	−133.282	−4.83783
$760$	0	0
$761$	13.7128	0.497089	0.248545	−	0.968620i	$-0.420048\pi$
$761$	13.7128	0.497089	0.248545	+	0.968620i	$0.420048\pi$
$762$	0	0
$763$	2.53590	0.0918057
$764$	0	0
$765$	−17.8564	−0.645600
$766$	0	0
$767$	14.6410	0.528656
$768$	0	0
$769$	12.1436	0.437909	0.218955	−	0.975735i	$-0.429735\pi$
$769$	12.1436	0.437909	0.218955	+	0.975735i	$0.429735\pi$
$770$	0	0
$771$	−2.92820	−0.105457
$772$	0	0
$773$	−34.3923	−1.23701	−0.618503	−	0.785783i	$-0.712260\pi$
$773$	−34.3923	−1.23701	−0.618503	+	0.785783i	$0.712260\pi$
$774$	0	0
$775$	10.1962	0.366257
$776$	0	0
$777$	−3.46410	−0.124274
$778$	0	0
$779$	30.5359	1.09406
$780$	0	0
$781$	−43.7128	−1.56417
$782$	0	0
$783$	−29.8564	−1.06698
$784$	0	0
$785$	7.46410	0.266405
$786$	0	0
$787$	23.8038	0.848516	0.424258	−	0.905541i	$-0.360535\pi$
$787$	23.8038	0.848516	0.424258	+	0.905541i	$0.360535\pi$
$788$	0	0
$789$	19.4641	0.692940
$790$	0	0
$791$	22.6410	0.805022
$792$	0	0
$793$	56.0000	1.98862
$794$	0	0
$795$	−35.3205	−1.25269
$796$	0	0
$797$	6.14359	0.217617	0.108809	−	0.994063i	$-0.465296\pi$
$797$	6.14359	0.217617	0.108809	+	0.994063i	$0.465296\pi$
$798$	0	0
$799$	−5.07180	−0.179427
$800$	0	0
$801$	−8.92820	−0.315463
$802$	0	0
$803$	56.7846	2.00389
$804$	0	0
$805$	−11.3205	−0.398995
$806$	0	0
$807$	33.8564	1.19180
$808$	0	0
$809$	−16.9282	−0.595164	−0.297582	−	0.954696i	$-0.596180\pi$
$809$	−16.9282	−0.595164	−0.297582	+	0.954696i	$0.596180\pi$
$810$	0	0
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	0	0
$815$	15.4641	0.541684
$816$	0	0
$817$	37.1769	1.30066
$818$	0	0
$819$	22.6410	0.791141
$820$	0	0
$821$	4.14359	0.144612	0.0723062	−	0.997382i	$-0.476964\pi$
$821$	4.14359	0.144612	0.0723062	+	0.997382i	$0.476964\pi$
$822$	0	0
$823$	19.8038	0.690319	0.345159	−	0.938544i	$-0.387825\pi$
$823$	19.8038	0.690319	0.345159	+	0.938544i	$0.387825\pi$
$824$	0	0
$825$	14.9282	0.519733
$826$	0	0
$827$	−13.6077	−0.473186	−0.236593	−	0.971609i	$-0.576031\pi$
$827$	−13.6077	−0.473186	−0.236593	+	0.971609i	$0.576031\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	78.6410	2.72803
$832$	0	0
$833$	21.5692	0.747329
$834$	0	0
$835$	6.39230	0.221215
$836$	0	0
$837$	−40.7846	−1.40972
$838$	0	0
$839$	−18.9282	−0.653474	−0.326737	−	0.945115i	$-0.605949\pi$
$839$	−18.9282	−0.653474	−0.326737	+	0.945115i	$0.605949\pi$
$840$	0	0
$841$	26.7128	0.921131
$842$	0	0
$843$	17.4641	0.601496
$844$	0	0
$845$	3.00000	0.103203
$846$	0	0
$847$	23.9090	0.821522
$848$	0	0
$849$	−23.3205	−0.800358
$850$	0	0
$851$	−8.92820	−0.306055
$852$	0	0
$853$	45.7128	1.56518	0.782588	−	0.622539i	$-0.213899\pi$
$853$	45.7128	1.56518	0.782588	+	0.622539i	$0.213899\pi$
$854$	0	0
$855$	27.6603	0.945961
$856$	0	0
$857$	4.14359	0.141542	0.0707712	−	0.997493i	$-0.477454\pi$
$857$	4.14359	0.141542	0.0707712	+	0.997493i	$0.477454\pi$
$858$	0	0
$859$	−17.4115	−0.594074	−0.297037	−	0.954866i	$-0.595999\pi$
$859$	−17.4115	−0.594074	−0.297037	+	0.954866i	$0.595999\pi$
$860$	0	0
$861$	−17.0718	−0.581805
$862$	0	0
$863$	−33.3731	−1.13603	−0.568016	−	0.823017i	$-0.692289\pi$
$863$	−33.3731	−1.13603	−0.568016	+	0.823017i	$0.692289\pi$
$864$	0	0
$865$	−4.53590	−0.154225
$866$	0	0
$867$	2.73205	0.0927853
$868$	0	0
$869$	−33.8564	−1.14850
$870$	0	0
$871$	−21.0718	−0.713991
$872$	0	0
$873$	−62.4974	−2.11522
$874$	0	0
$875$	1.26795	0.0428645
$876$	0	0
$877$	20.5359	0.693448	0.346724	−	0.937967i	$-0.387294\pi$
$877$	20.5359	0.693448	0.346724	+	0.937967i	$0.387294\pi$
$878$	0	0
$879$	−2.53590	−0.0855337
$880$	0	0
$881$	−47.3205	−1.59427	−0.797134	−	0.603802i	$-0.793652\pi$
$881$	−47.3205	−1.59427	−0.797134	+	0.603802i	$0.793652\pi$
$882$	0	0
$883$	39.1769	1.31841	0.659204	−	0.751964i	$-0.270893\pi$
$883$	39.1769	1.31841	0.659204	+	0.751964i	$0.270893\pi$
$884$	0	0
$885$	−10.0000	−0.336146
$886$	0	0
$887$	−11.9090	−0.399864	−0.199932	−	0.979810i	$-0.564072\pi$
$887$	−11.9090	−0.399864	−0.199932	+	0.979810i	$0.564072\pi$
$888$	0	0
$889$	−2.96668	−0.0994992
$890$	0	0
$891$	13.4641	0.451064
$892$	0	0
$893$	7.85641	0.262905
$894$	0	0
$895$	4.73205	0.158175
$896$	0	0
$897$	97.5692	3.25774
$898$	0	0
$899$	76.1051	2.53825
$900$	0	0
$901$	−51.7128	−1.72280
$902$	0	0
$903$	−20.7846	−0.691669
$904$	0	0
$905$	−15.3205	−0.509271
$906$	0	0
$907$	29.3205	0.973571	0.486786	−	0.873521i	$-0.338169\pi$
$907$	29.3205	0.973571	0.486786	+	0.873521i	$0.338169\pi$
$908$	0	0
$909$	11.3205	0.375478
$910$	0	0
$911$	−13.1244	−0.434829	−0.217415	−	0.976079i	$-0.569762\pi$
$911$	−13.1244	−0.434829	−0.217415	+	0.976079i	$0.569762\pi$
$912$	0	0
$913$	−9.07180	−0.300233
$914$	0	0
$915$	−38.2487	−1.26446
$916$	0	0
$917$	−21.2154	−0.700594
$918$	0	0
$919$	−33.1244	−1.09267	−0.546336	−	0.837566i	$-0.683978\pi$
$919$	−33.1244	−1.09267	−0.546336	+	0.837566i	$0.683978\pi$
$920$	0	0
$921$	−3.46410	−0.114146
$922$	0	0
$923$	32.0000	1.05329
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	28.5359	0.937242
$928$	0	0
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	−33.4115	−1.09502
$932$	0	0
$933$	−71.5692	−2.34307
$934$	0	0
$935$	21.8564	0.714781
$936$	0	0
$937$	13.3205	0.435162	0.217581	−	0.976042i	$-0.430183\pi$
$937$	13.3205	0.435162	0.217581	+	0.976042i	$0.430183\pi$
$938$	0	0
$939$	32.3923	1.05708
$940$	0	0
$941$	52.3923	1.70794	0.853970	−	0.520322i	$-0.174188\pi$
$941$	52.3923	1.70794	0.853970	+	0.520322i	$0.174188\pi$
$942$	0	0
$943$	−44.0000	−1.43284
$944$	0	0
$945$	−5.07180	−0.164986
$946$	0	0
$947$	−58.1051	−1.88816	−0.944081	−	0.329713i	$-0.893048\pi$
$947$	−58.1051	−1.88816	−0.944081	+	0.329713i	$0.893048\pi$
$948$	0	0
$949$	−41.5692	−1.34939
$950$	0	0
$951$	−42.2487	−1.37001
$952$	0	0
$953$	−3.07180	−0.0995053	−0.0497526	−	0.998762i	$-0.515843\pi$
$953$	−3.07180	−0.0995053	−0.0497526	+	0.998762i	$0.515843\pi$
$954$	0	0
$955$	14.5885	0.472071
$956$	0	0
$957$	111.426	3.60188
$958$	0	0
$959$	−27.0333	−0.872951
$960$	0	0
$961$	72.9615	2.35360
$962$	0	0
$963$	−0.875644	−0.0282172
$964$	0	0
$965$	8.92820	0.287409
$966$	0	0
$967$	−38.3923	−1.23461	−0.617307	−	0.786723i	$-0.711776\pi$
$967$	−38.3923	−1.23461	−0.617307	+	0.786723i	$0.711776\pi$
$968$	0	0
$969$	67.7128	2.17525
$970$	0	0
$971$	−16.0000	−0.513464	−0.256732	−	0.966483i	$-0.582646\pi$
$971$	−16.0000	−0.513464	−0.256732	+	0.966483i	$0.582646\pi$
$972$	0	0
$973$	−24.0000	−0.769405
$974$	0	0
$975$	−10.9282	−0.349983
$976$	0	0
$977$	34.7846	1.11286	0.556429	−	0.830895i	$-0.312171\pi$
$977$	34.7846	1.11286	0.556429	+	0.830895i	$0.312171\pi$
$978$	0	0
$979$	10.9282	0.349267
$980$	0	0
$981$	8.92820	0.285056
$982$	0	0
$983$	45.3731	1.44718	0.723588	−	0.690232i	$-0.242492\pi$
$983$	45.3731	1.44718	0.723588	+	0.690232i	$0.242492\pi$
$984$	0	0
$985$	9.32051	0.296976
$986$	0	0
$987$	−4.39230	−0.139809
$988$	0	0
$989$	−53.5692	−1.70340
$990$	0	0
$991$	−25.9090	−0.823025	−0.411513	−	0.911404i	$-0.634999\pi$
$991$	−25.9090	−0.823025	−0.411513	+	0.911404i	$0.634999\pi$
$992$	0	0
$993$	−31.8564	−1.01093
$994$	0	0
$995$	13.1244	0.416070
$996$	0	0
$997$	−21.7128	−0.687652	−0.343826	−	0.939033i	$-0.611723\pi$
$997$	−21.7128	−0.687652	−0.343826	+	0.939033i	$0.611723\pi$
$998$	0	0
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	740.2.a.d.1.1	✓	2
3.2	odd	2		6660.2.a.i.1.1		2
4.3	odd	2		2960.2.a.p.1.2		2
5.2	odd	4		3700.2.d.g.149.4		4
5.3	odd	4		3700.2.d.g.149.1		4
5.4	even	2		3700.2.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.a.d.1.1	✓	2	1.1	even	1	trivial
2960.2.a.p.1.2		2	4.3	odd	2
3700.2.a.h.1.2		2	5.4	even	2
3700.2.d.g.149.1		4	5.3	odd	4
3700.2.d.g.149.4		4	5.2	odd	4
6660.2.a.i.1.1		2	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:	\( N \)	\(=\)	\( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	740.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{3} +1.00000 q^{5} +1.26795 q^{7} +4.46410 q^{9} -5.46410 q^{11} +4.00000 q^{13} -2.73205 q^{15} -4.00000 q^{17} +6.19615 q^{19} -3.46410 q^{21} -8.92820 q^{23} +1.00000 q^{25} -4.00000 q^{27} +7.46410 q^{29} +10.1962 q^{31} +14.9282 q^{33} +1.26795 q^{35} +1.00000 q^{37} -10.9282 q^{39} +4.92820 q^{41} +6.00000 q^{43} +4.46410 q^{45} +1.26795 q^{47} -5.39230 q^{49} +10.9282 q^{51} +12.9282 q^{53} -5.46410 q^{55} -16.9282 q^{57} +3.66025 q^{59} +14.0000 q^{61} +5.66025 q^{63} +4.00000 q^{65} -5.26795 q^{67} +24.3923 q^{69} +8.00000 q^{71} -10.3923 q^{73} -2.73205 q^{75} -6.92820 q^{77} +6.19615 q^{79} -2.46410 q^{81} +1.66025 q^{83} -4.00000 q^{85} -20.3923 q^{87} -2.00000 q^{89} +5.07180 q^{91} -27.8564 q^{93} +6.19615 q^{95} -14.0000 q^{97} -24.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} - 4 q^{11} + 8 q^{13} - 2 q^{15} - 8 q^{17} + 2 q^{19} - 4 q^{23} + 2 q^{25} - 8 q^{27} + 8 q^{29} + 10 q^{31} + 16 q^{33} + 6 q^{35} + 2 q^{37} - 8 q^{39} - 4 q^{41}+ \cdots - 28 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 - 4 * q^11 + 8 * q^13 - 2 * q^15 - 8 * q^17 + 2 * q^19 - 4 * q^23 + 2 * q^25 - 8 * q^27 + 8 * q^29 + 10 * q^31 + 16 * q^33 + 6 * q^35 + 2 * q^37 - 8 * q^39 - 4 * q^41 + 12 * q^43 + 2 * q^45 + 6 * q^47 + 10 * q^49 + 8 * q^51 + 12 * q^53 - 4 * q^55 - 20 * q^57 - 10 * q^59 + 28 * q^61 - 6 * q^63 + 8 * q^65 - 14 * q^67 + 28 * q^69 + 16 * q^71 - 2 * q^75 + 2 * q^79 + 2 * q^81 - 14 * q^83 - 8 * q^85 - 20 * q^87 - 4 * q^89 + 24 * q^91 - 28 * q^93 + 2 * q^95 - 28 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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