LMFDB - Embedded newform 740.4.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [740,4,Mod(1,740)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("740.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(740, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 740 = 2^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	740.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$43.6614134042$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-8.00000 q^{3} +5.00000 q^{5} +4.00000 q^{7} +37.0000 q^{9} -20.0000 q^{11} -10.0000 q^{13} -40.0000 q^{15} -62.0000 q^{17} +8.00000 q^{19} -32.0000 q^{21} +192.000 q^{23} +25.0000 q^{25} -80.0000 q^{27} -154.000 q^{29} +124.000 q^{31} +160.000 q^{33} +20.0000 q^{35} +37.0000 q^{37} +80.0000 q^{39} +186.000 q^{41} +92.0000 q^{43} +185.000 q^{45} +476.000 q^{47} -327.000 q^{49} +496.000 q^{51} -258.000 q^{53} -100.000 q^{55} -64.0000 q^{57} -176.000 q^{59} -458.000 q^{61} +148.000 q^{63} -50.0000 q^{65} +336.000 q^{67} -1536.00 q^{69} -232.000 q^{71} -470.000 q^{73} -200.000 q^{75} -80.0000 q^{77} -676.000 q^{79} -359.000 q^{81} -608.000 q^{83} -310.000 q^{85} +1232.00 q^{87} -102.000 q^{89} -40.0000 q^{91} -992.000 q^{93} +40.0000 q^{95} -30.0000 q^{97} -740.000 q^{99} +O(q^{100})$

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
$3$	−8.00000	−1.53960	−0.769800	−	0.638285i	$-0.779644\pi$
$3$	−8.00000	−1.53960	−0.769800	+	0.638285i	$0.779644\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	4.00000	0.215980	0.107990	−	0.994152i	$-0.465559\pi$
$7$	4.00000	0.215980	0.107990	+	0.994152i	$0.465559\pi$
$8$	0	0
$9$	37.0000	1.37037
$10$	0	0
$11$	−20.0000	−0.548202	−0.274101	−	0.961701i	$-0.588380\pi$
$11$	−20.0000	−0.548202	−0.274101	+	0.961701i	$0.588380\pi$
$12$	0	0
$13$	−10.0000	−0.213346	−0.106673	−	0.994294i	$-0.534020\pi$
$13$	−10.0000	−0.213346	−0.106673	+	0.994294i	$0.534020\pi$
$14$	0	0
$15$	−40.0000	−0.688530
$16$	0	0
$17$	−62.0000	−0.884542	−0.442271	−	0.896882i	$-0.645827\pi$
$17$	−62.0000	−0.884542	−0.442271	+	0.896882i	$0.645827\pi$
$18$	0	0
$19$	8.00000	0.0965961	0.0482980	−	0.998833i	$-0.484620\pi$
$19$	8.00000	0.0965961	0.0482980	+	0.998833i	$0.484620\pi$
$20$	0	0
$21$	−32.0000	−0.332522
$22$	0	0
$23$	192.000	1.74064	0.870321	−	0.492485i	$-0.163911\pi$
$23$	192.000	1.74064	0.870321	+	0.492485i	$0.163911\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−80.0000	−0.570222
$28$	0	0
$29$	−154.000	−0.986106	−0.493053	−	0.869999i	$-0.664119\pi$
$29$	−154.000	−0.986106	−0.493053	+	0.869999i	$0.664119\pi$
$30$	0	0
$31$	124.000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	124.000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	160.000	0.844013
$34$	0	0
$35$	20.0000	0.0965891
$36$	0	0
$37$	37.0000	0.164399
$37$	37.0000	0.164399
$38$	0	0
$39$	80.0000	0.328468
$40$	0	0
$41$	186.000	0.708496	0.354248	−	0.935152i	$-0.384737\pi$
$41$	186.000	0.708496	0.354248	+	0.935152i	$0.384737\pi$
$42$	0	0
$43$	92.0000	0.326276	0.163138	−	0.986603i	$-0.447838\pi$
$43$	92.0000	0.326276	0.163138	+	0.986603i	$0.447838\pi$
$44$	0	0
$45$	185.000	0.612848
$46$	0	0
$47$	476.000	1.47727	0.738635	−	0.674105i	$-0.235471\pi$
$47$	476.000	1.47727	0.738635	+	0.674105i	$0.235471\pi$
$48$	0	0
$49$	−327.000	−0.953353
$50$	0	0
$51$	496.000	1.36184
$52$	0	0
$53$	−258.000	−0.668661	−0.334330	−	0.942456i	$-0.608510\pi$
$53$	−258.000	−0.668661	−0.334330	+	0.942456i	$0.608510\pi$
$54$	0	0
$55$	−100.000	−0.245164
$56$	0	0
$57$	−64.0000	−0.148719
$58$	0	0
$59$	−176.000	−0.388360	−0.194180	−	0.980966i	$-0.562205\pi$
$59$	−176.000	−0.388360	−0.194180	+	0.980966i	$0.562205\pi$
$60$	0	0
$61$	−458.000	−0.961326	−0.480663	−	0.876905i	$-0.659604\pi$
$61$	−458.000	−0.961326	−0.480663	+	0.876905i	$0.659604\pi$
$62$	0	0
$63$	148.000	0.295972
$64$	0	0
$65$	−50.0000	−0.0954113
$66$	0	0
$67$	336.000	0.612671	0.306335	−	0.951924i	$-0.400897\pi$
$67$	336.000	0.612671	0.306335	+	0.951924i	$0.400897\pi$
$68$	0	0
$69$	−1536.00	−2.67989
$70$	0	0
$71$	−232.000	−0.387793	−0.193897	−	0.981022i	$-0.562113\pi$
$71$	−232.000	−0.387793	−0.193897	+	0.981022i	$0.562113\pi$
$72$	0	0
$73$	−470.000	−0.753553	−0.376776	−	0.926304i	$-0.622967\pi$
$73$	−470.000	−0.753553	−0.376776	+	0.926304i	$0.622967\pi$
$74$	0	0
$75$	−200.000	−0.307920
$76$	0	0
$77$	−80.0000	−0.118401
$78$	0	0
$79$	−676.000	−0.962733	−0.481367	−	0.876519i	$-0.659859\pi$
$79$	−676.000	−0.962733	−0.481367	+	0.876519i	$0.659859\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	−608.000	−0.804056	−0.402028	−	0.915627i	$-0.631695\pi$
$83$	−608.000	−0.804056	−0.402028	+	0.915627i	$0.631695\pi$
$84$	0	0
$85$	−310.000	−0.395579
$86$	0	0
$87$	1232.00	1.51821
$88$	0	0
$89$	−102.000	−0.121483	−0.0607415	−	0.998154i	$-0.519347\pi$
$89$	−102.000	−0.121483	−0.0607415	+	0.998154i	$0.519347\pi$
$90$	0	0
$91$	−40.0000	−0.0460785
$92$	0	0
$93$	−992.000	−1.10608
$94$	0	0
$95$	40.0000	0.0431991
$96$	0	0
$97$	−30.0000	−0.0314025	−0.0157012	−	0.999877i	$-0.504998\pi$
$97$	−30.0000	−0.0314025	−0.0157012	+	0.999877i	$0.504998\pi$
$98$	0	0
$99$	−740.000	−0.751240
$100$	0	0
$101$	−738.000	−0.727067	−0.363533	−	0.931581i	$-0.618430\pi$
$101$	−738.000	−0.727067	−0.363533	+	0.931581i	$0.618430\pi$
$102$	0	0
$103$	−504.000	−0.482142	−0.241071	−	0.970508i	$-0.577499\pi$
$103$	−504.000	−0.482142	−0.241071	+	0.970508i	$0.577499\pi$
$104$	0	0
$105$	−160.000	−0.148709
$106$	0	0
$107$	−848.000	−0.766161	−0.383081	−	0.923715i	$-0.625137\pi$
$107$	−848.000	−0.766161	−0.383081	+	0.923715i	$0.625137\pi$
$108$	0	0
$109$	230.000	0.202110	0.101055	−	0.994881i	$-0.467778\pi$
$109$	230.000	0.202110	0.101055	+	0.994881i	$0.467778\pi$
$110$	0	0
$111$	−296.000	−0.253109
$112$	0	0
$113$	66.0000	0.0549448	0.0274724	−	0.999623i	$-0.491254\pi$
$113$	66.0000	0.0549448	0.0274724	+	0.999623i	$0.491254\pi$
$114$	0	0
$115$	960.000	0.778439
$116$	0	0
$117$	−370.000	−0.292363
$118$	0	0
$119$	−248.000	−0.191043
$120$	0	0
$121$	−931.000	−0.699474
$122$	0	0
$123$	−1488.00	−1.09080
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−2100.00	−1.46728	−0.733641	−	0.679537i	$-0.762181\pi$
$127$	−2100.00	−1.46728	−0.733641	+	0.679537i	$0.762181\pi$
$128$	0	0
$129$	−736.000	−0.502335
$130$	0	0
$131$	−1448.00	−0.965743	−0.482872	−	0.875691i	$-0.660406\pi$
$131$	−1448.00	−0.965743	−0.482872	+	0.875691i	$0.660406\pi$
$132$	0	0
$133$	32.0000	0.0208628
$134$	0	0
$135$	−400.000	−0.255011
$136$	0	0
$137$	1146.00	0.714667	0.357334	−	0.933977i	$-0.383686\pi$
$137$	1146.00	0.714667	0.357334	+	0.933977i	$0.383686\pi$
$138$	0	0
$139$	−140.000	−0.0854291	−0.0427146	−	0.999087i	$-0.513601\pi$
$139$	−140.000	−0.0854291	−0.0427146	+	0.999087i	$0.513601\pi$
$140$	0	0
$141$	−3808.00	−2.27441
$142$	0	0
$143$	200.000	0.116957
$144$	0	0
$145$	−770.000	−0.441000
$146$	0	0
$147$	2616.00	1.46778
$148$	0	0
$149$	−1778.00	−0.977580	−0.488790	−	0.872401i	$-0.662562\pi$
$149$	−1778.00	−0.977580	−0.488790	+	0.872401i	$0.662562\pi$
$150$	0	0
$151$	2224.00	1.19859	0.599293	−	0.800530i	$-0.295448\pi$
$151$	2224.00	1.19859	0.599293	+	0.800530i	$0.295448\pi$
$152$	0	0
$153$	−2294.00	−1.21215
$154$	0	0
$155$	620.000	0.321288
$156$	0	0
$157$	2614.00	1.32879	0.664395	−	0.747382i	$-0.268689\pi$
$157$	2614.00	1.32879	0.664395	+	0.747382i	$0.268689\pi$
$158$	0	0
$159$	2064.00	1.02947
$160$	0	0
$161$	768.000	0.375943
$162$	0	0
$163$	−2308.00	−1.10906	−0.554529	−	0.832164i	$-0.687102\pi$
$163$	−2308.00	−1.10906	−0.554529	+	0.832164i	$0.687102\pi$
$164$	0	0
$165$	800.000	0.377454
$166$	0	0
$167$	−968.000	−0.448539	−0.224270	−	0.974527i	$-0.572000\pi$
$167$	−968.000	−0.448539	−0.224270	+	0.974527i	$0.572000\pi$
$168$	0	0
$169$	−2097.00	−0.954483
$170$	0	0
$171$	296.000	0.132372
$172$	0	0
$173$	614.000	0.269836	0.134918	−	0.990857i	$-0.456923\pi$
$173$	614.000	0.269836	0.134918	+	0.990857i	$0.456923\pi$
$174$	0	0
$175$	100.000	0.0431959
$176$	0	0
$177$	1408.00	0.597920
$178$	0	0
$179$	528.000	0.220472	0.110236	−	0.993905i	$-0.464839\pi$
$179$	528.000	0.220472	0.110236	+	0.993905i	$0.464839\pi$
$180$	0	0
$181$	−2994.00	−1.22952	−0.614758	−	0.788716i	$-0.710746\pi$
$181$	−2994.00	−1.22952	−0.614758	+	0.788716i	$0.710746\pi$
$182$	0	0
$183$	3664.00	1.48006
$184$	0	0
$185$	185.000	0.0735215
$186$	0	0
$187$	1240.00	0.484908
$188$	0	0
$189$	−320.000	−0.123156
$190$	0	0
$191$	−2844.00	−1.07741	−0.538703	−	0.842496i	$-0.681086\pi$
$191$	−2844.00	−1.07741	−0.538703	+	0.842496i	$0.681086\pi$
$192$	0	0
$193$	−3678.00	−1.37175	−0.685876	−	0.727718i	$-0.740581\pi$
$193$	−3678.00	−1.37175	−0.685876	+	0.727718i	$0.740581\pi$
$194$	0	0
$195$	400.000	0.146895
$196$	0	0
$197$	318.000	0.115008	0.0575040	−	0.998345i	$-0.481686\pi$
$197$	318.000	0.115008	0.0575040	+	0.998345i	$0.481686\pi$
$198$	0	0
$199$	−884.000	−0.314900	−0.157450	−	0.987527i	$-0.550327\pi$
$199$	−884.000	−0.314900	−0.157450	+	0.987527i	$0.550327\pi$
$200$	0	0
$201$	−2688.00	−0.943268
$202$	0	0
$203$	−616.000	−0.212979
$204$	0	0
$205$	930.000	0.316849
$206$	0	0
$207$	7104.00	2.38532
$208$	0	0
$209$	−160.000	−0.0529542
$210$	0	0
$211$	3292.00	1.07408	0.537039	−	0.843557i	$-0.319543\pi$
$211$	3292.00	1.07408	0.537039	+	0.843557i	$0.319543\pi$
$212$	0	0
$213$	1856.00	0.597047
$214$	0	0
$215$	460.000	0.145915
$216$	0	0
$217$	496.000	0.155164
$218$	0	0
$219$	3760.00	1.16017
$220$	0	0
$221$	620.000	0.188714
$222$	0	0
$223$	20.0000	0.00600583	0.00300291	−	0.999995i	$-0.499044\pi$
$223$	20.0000	0.00600583	0.00300291	+	0.999995i	$0.499044\pi$
$224$	0	0
$225$	925.000	0.274074
$226$	0	0
$227$	1308.00	0.382445	0.191222	−	0.981547i	$-0.438755\pi$
$227$	1308.00	0.382445	0.191222	+	0.981547i	$0.438755\pi$
$228$	0	0
$229$	−610.000	−0.176026	−0.0880130	−	0.996119i	$-0.528052\pi$
$229$	−610.000	−0.176026	−0.0880130	+	0.996119i	$0.528052\pi$
$230$	0	0
$231$	640.000	0.182290
$232$	0	0
$233$	1610.00	0.452681	0.226340	−	0.974048i	$-0.427324\pi$
$233$	1610.00	0.452681	0.226340	+	0.974048i	$0.427324\pi$
$234$	0	0
$235$	2380.00	0.660656
$236$	0	0
$237$	5408.00	1.48223
$238$	0	0
$239$	−1268.00	−0.343180	−0.171590	−	0.985168i	$-0.554890\pi$
$239$	−1268.00	−0.343180	−0.171590	+	0.985168i	$0.554890\pi$
$240$	0	0
$241$	3490.00	0.932824	0.466412	−	0.884568i	$-0.345546\pi$
$241$	3490.00	0.932824	0.466412	+	0.884568i	$0.345546\pi$
$242$	0	0
$243$	5032.00	1.32841
$244$	0	0
$245$	−1635.00	−0.426352
$246$	0	0
$247$	−80.0000	−0.0206084
$248$	0	0
$249$	4864.00	1.23793
$250$	0	0
$251$	−960.000	−0.241413	−0.120706	−	0.992688i	$-0.538516\pi$
$251$	−960.000	−0.241413	−0.120706	+	0.992688i	$0.538516\pi$
$252$	0	0
$253$	−3840.00	−0.954224
$254$	0	0
$255$	2480.00	0.609034
$256$	0	0
$257$	6834.00	1.65873	0.829364	−	0.558708i	$-0.188703\pi$
$257$	6834.00	1.65873	0.829364	+	0.558708i	$0.188703\pi$
$258$	0	0
$259$	148.000	0.0355068
$260$	0	0
$261$	−5698.00	−1.35133
$262$	0	0
$263$	4436.00	1.04006	0.520029	−	0.854148i	$-0.325921\pi$
$263$	4436.00	1.04006	0.520029	+	0.854148i	$0.325921\pi$
$264$	0	0
$265$	−1290.00	−0.299034
$266$	0	0
$267$	816.000	0.187035
$268$	0	0
$269$	−7434.00	−1.68498	−0.842489	−	0.538714i	$-0.818910\pi$
$269$	−7434.00	−1.68498	−0.842489	+	0.538714i	$0.818910\pi$
$270$	0	0
$271$	−704.000	−0.157804	−0.0789021	−	0.996882i	$-0.525141\pi$
$271$	−704.000	−0.157804	−0.0789021	+	0.996882i	$0.525141\pi$
$272$	0	0
$273$	320.000	0.0709424
$274$	0	0
$275$	−500.000	−0.109640
$276$	0	0
$277$	7150.00	1.55091	0.775455	−	0.631403i	$-0.217521\pi$
$277$	7150.00	1.55091	0.775455	+	0.631403i	$0.217521\pi$
$278$	0	0
$279$	4588.00	0.984503
$280$	0	0
$281$	−4614.00	−0.979531	−0.489765	−	0.871854i	$-0.662918\pi$
$281$	−4614.00	−0.979531	−0.489765	+	0.871854i	$0.662918\pi$
$282$	0	0
$283$	−5548.00	−1.16535	−0.582676	−	0.812705i	$-0.697994\pi$
$283$	−5548.00	−1.16535	−0.582676	+	0.812705i	$0.697994\pi$
$284$	0	0
$285$	−320.000	−0.0665093
$286$	0	0
$287$	744.000	0.153021
$288$	0	0
$289$	−1069.00	−0.217586
$290$	0	0
$291$	240.000	0.0483472
$292$	0	0
$293$	−6626.00	−1.32114	−0.660572	−	0.750763i	$-0.729686\pi$
$293$	−6626.00	−1.32114	−0.660572	+	0.750763i	$0.729686\pi$
$294$	0	0
$295$	−880.000	−0.173680
$296$	0	0
$297$	1600.00	0.312597
$298$	0	0
$299$	−1920.00	−0.371359
$300$	0	0
$301$	368.000	0.0704690
$302$	0	0
$303$	5904.00	1.11939
$304$	0	0
$305$	−2290.00	−0.429918
$306$	0	0
$307$	−96.0000	−0.0178469	−0.00892347	−	0.999960i	$-0.502840\pi$
$307$	−96.0000	−0.0178469	−0.00892347	+	0.999960i	$0.502840\pi$
$308$	0	0
$309$	4032.00	0.742306
$310$	0	0
$311$	−1372.00	−0.250157	−0.125079	−	0.992147i	$-0.539918\pi$
$311$	−1372.00	−0.250157	−0.125079	+	0.992147i	$0.539918\pi$
$312$	0	0
$313$	−214.000	−0.0386454	−0.0193227	−	0.999813i	$-0.506151\pi$
$313$	−214.000	−0.0386454	−0.0193227	+	0.999813i	$0.506151\pi$
$314$	0	0
$315$	740.000	0.132363
$316$	0	0
$317$	−7962.00	−1.41070	−0.705348	−	0.708861i	$-0.749209\pi$
$317$	−7962.00	−1.41070	−0.705348	+	0.708861i	$0.749209\pi$
$318$	0	0
$319$	3080.00	0.540586
$320$	0	0
$321$	6784.00	1.17958
$322$	0	0
$323$	−496.000	−0.0854433
$324$	0	0
$325$	−250.000	−0.0426692
$326$	0	0
$327$	−1840.00	−0.311169
$328$	0	0
$329$	1904.00	0.319061
$330$	0	0
$331$	−520.000	−0.0863498	−0.0431749	−	0.999068i	$-0.513747\pi$
$331$	−520.000	−0.0863498	−0.0431749	+	0.999068i	$0.513747\pi$
$332$	0	0
$333$	1369.00	0.225288
$334$	0	0
$335$	1680.00	0.273995
$336$	0	0
$337$	2930.00	0.473612	0.236806	−	0.971557i	$-0.423899\pi$
$337$	2930.00	0.473612	0.236806	+	0.971557i	$0.423899\pi$
$338$	0	0
$339$	−528.000	−0.0845930
$340$	0	0
$341$	−2480.00	−0.393840
$342$	0	0
$343$	−2680.00	−0.421885
$344$	0	0
$345$	−7680.00	−1.19848
$346$	0	0
$347$	−8052.00	−1.24569	−0.622844	−	0.782346i	$-0.714023\pi$
$347$	−8052.00	−1.24569	−0.622844	+	0.782346i	$0.714023\pi$
$348$	0	0
$349$	−11578.0	−1.77581	−0.887903	−	0.460031i	$-0.847838\pi$
$349$	−11578.0	−1.77581	−0.887903	+	0.460031i	$0.847838\pi$
$350$	0	0
$351$	800.000	0.121655
$352$	0	0
$353$	6690.00	1.00870	0.504352	−	0.863498i	$-0.331731\pi$
$353$	6690.00	1.00870	0.504352	+	0.863498i	$0.331731\pi$
$354$	0	0
$355$	−1160.00	−0.173427
$356$	0	0
$357$	1984.00	0.294130
$358$	0	0
$359$	−2448.00	−0.359890	−0.179945	−	0.983677i	$-0.557592\pi$
$359$	−2448.00	−0.359890	−0.179945	+	0.983677i	$0.557592\pi$
$360$	0	0
$361$	−6795.00	−0.990669
$362$	0	0
$363$	7448.00	1.07691
$364$	0	0
$365$	−2350.00	−0.336999
$366$	0	0
$367$	3196.00	0.454577	0.227289	−	0.973827i	$-0.427014\pi$
$367$	3196.00	0.454577	0.227289	+	0.973827i	$0.427014\pi$
$368$	0	0
$369$	6882.00	0.970901
$370$	0	0
$371$	−1032.00	−0.144417
$372$	0	0
$373$	−7394.00	−1.02640	−0.513199	−	0.858269i	$-0.671540\pi$
$373$	−7394.00	−1.02640	−0.513199	+	0.858269i	$0.671540\pi$
$374$	0	0
$375$	−1000.00	−0.137706
$376$	0	0
$377$	1540.00	0.210382
$378$	0	0
$379$	−10580.0	−1.43393	−0.716963	−	0.697111i	$-0.754468\pi$
$379$	−10580.0	−1.43393	−0.716963	+	0.697111i	$0.754468\pi$
$380$	0	0
$381$	16800.0	2.25903
$382$	0	0
$383$	9128.00	1.21780	0.608902	−	0.793245i	$-0.291610\pi$
$383$	9128.00	1.21780	0.608902	+	0.793245i	$0.291610\pi$
$384$	0	0
$385$	−400.000	−0.0529504
$386$	0	0
$387$	3404.00	0.447119
$388$	0	0
$389$	2670.00	0.348006	0.174003	−	0.984745i	$-0.444330\pi$
$389$	2670.00	0.348006	0.174003	+	0.984745i	$0.444330\pi$
$390$	0	0
$391$	−11904.0	−1.53967
$392$	0	0
$393$	11584.0	1.48686
$394$	0	0
$395$	−3380.00	−0.430547
$396$	0	0
$397$	2118.00	0.267757	0.133878	−	0.990998i	$-0.457257\pi$
$397$	2118.00	0.267757	0.133878	+	0.990998i	$0.457257\pi$
$398$	0	0
$399$	−256.000	−0.0321204
$400$	0	0
$401$	8994.00	1.12005	0.560024	−	0.828477i	$-0.310792\pi$
$401$	8994.00	1.12005	0.560024	+	0.828477i	$0.310792\pi$
$402$	0	0
$403$	−1240.00	−0.153272
$404$	0	0
$405$	−1795.00	−0.220233
$406$	0	0
$407$	−740.000	−0.0901239
$408$	0	0
$409$	6026.00	0.728525	0.364262	−	0.931296i	$-0.381321\pi$
$409$	6026.00	0.728525	0.364262	+	0.931296i	$0.381321\pi$
$410$	0	0
$411$	−9168.00	−1.10030
$412$	0	0
$413$	−704.000	−0.0838779
$414$	0	0
$415$	−3040.00	−0.359585
$416$	0	0
$417$	1120.00	0.131527
$418$	0	0
$419$	15876.0	1.85106	0.925529	−	0.378677i	$-0.123621\pi$
$419$	15876.0	1.85106	0.925529	+	0.378677i	$0.123621\pi$
$420$	0	0
$421$	−2258.00	−0.261397	−0.130699	−	0.991422i	$-0.541722\pi$
$421$	−2258.00	−0.261397	−0.130699	+	0.991422i	$0.541722\pi$
$422$	0	0
$423$	17612.0	2.02441
$424$	0	0
$425$	−1550.00	−0.176908
$426$	0	0
$427$	−1832.00	−0.207627
$428$	0	0
$429$	−1600.00	−0.180067
$430$	0	0
$431$	5372.00	0.600372	0.300186	−	0.953881i	$-0.402951\pi$
$431$	5372.00	0.600372	0.300186	+	0.953881i	$0.402951\pi$
$432$	0	0
$433$	5058.00	0.561367	0.280684	−	0.959800i	$-0.409439\pi$
$433$	5058.00	0.561367	0.280684	+	0.959800i	$0.409439\pi$
$434$	0	0
$435$	6160.00	0.678964
$436$	0	0
$437$	1536.00	0.168139
$438$	0	0
$439$	7900.00	0.858876	0.429438	−	0.903096i	$-0.358712\pi$
$439$	7900.00	0.858876	0.429438	+	0.903096i	$0.358712\pi$
$440$	0	0
$441$	−12099.0	−1.30645
$442$	0	0
$443$	6288.00	0.674384	0.337192	−	0.941436i	$-0.390523\pi$
$443$	6288.00	0.674384	0.337192	+	0.941436i	$0.390523\pi$
$444$	0	0
$445$	−510.000	−0.0543288
$446$	0	0
$447$	14224.0	1.50508
$448$	0	0
$449$	5586.00	0.587126	0.293563	−	0.955940i	$-0.405159\pi$
$449$	5586.00	0.587126	0.293563	+	0.955940i	$0.405159\pi$
$450$	0	0
$451$	−3720.00	−0.388399
$452$	0	0
$453$	−17792.0	−1.84534
$454$	0	0
$455$	−200.000	−0.0206069
$456$	0	0
$457$	9370.00	0.959103	0.479552	−	0.877514i	$-0.340799\pi$
$457$	9370.00	0.959103	0.479552	+	0.877514i	$0.340799\pi$
$458$	0	0
$459$	4960.00	0.504386
$460$	0	0
$461$	−10330.0	−1.04364	−0.521818	−	0.853057i	$-0.674746\pi$
$461$	−10330.0	−1.04364	−0.521818	+	0.853057i	$0.674746\pi$
$462$	0	0
$463$	−10488.0	−1.05274	−0.526370	−	0.850256i	$-0.676447\pi$
$463$	−10488.0	−1.05274	−0.526370	+	0.850256i	$0.676447\pi$
$464$	0	0
$465$	−4960.00	−0.494655
$466$	0	0
$467$	16732.0	1.65795	0.828977	−	0.559283i	$-0.188923\pi$
$467$	16732.0	1.65795	0.828977	+	0.559283i	$0.188923\pi$
$468$	0	0
$469$	1344.00	0.132324
$470$	0	0
$471$	−20912.0	−2.04580
$472$	0	0
$473$	−1840.00	−0.178865
$474$	0	0
$475$	200.000	0.0193192
$476$	0	0
$477$	−9546.00	−0.916313
$478$	0	0
$479$	−884.000	−0.0843236	−0.0421618	−	0.999111i	$-0.513424\pi$
$479$	−884.000	−0.0843236	−0.0421618	+	0.999111i	$0.513424\pi$
$480$	0	0
$481$	−370.000	−0.0350739
$482$	0	0
$483$	−6144.00	−0.578803
$484$	0	0
$485$	−150.000	−0.0140436
$486$	0	0
$487$	−17016.0	−1.58330	−0.791652	−	0.610973i	$-0.790778\pi$
$487$	−17016.0	−1.58330	−0.791652	+	0.610973i	$0.790778\pi$
$488$	0	0
$489$	18464.0	1.70751
$490$	0	0
$491$	−6892.00	−0.633466	−0.316733	−	0.948515i	$-0.602586\pi$
$491$	−6892.00	−0.633466	−0.316733	+	0.948515i	$0.602586\pi$
$492$	0	0
$493$	9548.00	0.872252
$494$	0	0
$495$	−3700.00	−0.335965
$496$	0	0
$497$	−928.000	−0.0837555
$498$	0	0
$499$	−1616.00	−0.144974	−0.0724871	−	0.997369i	$-0.523094\pi$
$499$	−1616.00	−0.144974	−0.0724871	+	0.997369i	$0.523094\pi$
$500$	0	0
$501$	7744.00	0.690572
$502$	0	0
$503$	−6144.00	−0.544627	−0.272314	−	0.962209i	$-0.587789\pi$
$503$	−6144.00	−0.544627	−0.272314	+	0.962209i	$0.587789\pi$
$504$	0	0
$505$	−3690.00	−0.325154
$506$	0	0
$507$	16776.0	1.46952
$508$	0	0
$509$	13926.0	1.21269	0.606345	−	0.795202i	$-0.292635\pi$
$509$	13926.0	1.21269	0.606345	+	0.795202i	$0.292635\pi$
$510$	0	0
$511$	−1880.00	−0.162752
$512$	0	0
$513$	−640.000	−0.0550813
$514$	0	0
$515$	−2520.00	−0.215620
$516$	0	0
$517$	−9520.00	−0.809844
$518$	0	0
$519$	−4912.00	−0.415439
$520$	0	0
$521$	14922.0	1.25479	0.627394	−	0.778702i	$-0.284122\pi$
$521$	14922.0	1.25479	0.627394	+	0.778702i	$0.284122\pi$
$522$	0	0
$523$	572.000	0.0478237	0.0239119	−	0.999714i	$-0.492388\pi$
$523$	572.000	0.0478237	0.0239119	+	0.999714i	$0.492388\pi$
$524$	0	0
$525$	−800.000	−0.0665045
$526$	0	0
$527$	−7688.00	−0.635474
$528$	0	0
$529$	24697.0	2.02983
$530$	0	0
$531$	−6512.00	−0.532197
$532$	0	0
$533$	−1860.00	−0.151155
$534$	0	0
$535$	−4240.00	−0.342638
$536$	0	0
$537$	−4224.00	−0.339440
$538$	0	0
$539$	6540.00	0.522630
$540$	0	0
$541$	−14794.0	−1.17568	−0.587841	−	0.808977i	$-0.700022\pi$
$541$	−14794.0	−1.17568	−0.587841	+	0.808977i	$0.700022\pi$
$542$	0	0
$543$	23952.0	1.89296
$544$	0	0
$545$	1150.00	0.0903864
$546$	0	0
$547$	−2204.00	−0.172278	−0.0861392	−	0.996283i	$-0.527453\pi$
$547$	−2204.00	−0.172278	−0.0861392	+	0.996283i	$0.527453\pi$
$548$	0	0
$549$	−16946.0	−1.31737
$550$	0	0
$551$	−1232.00	−0.0952540
$552$	0	0
$553$	−2704.00	−0.207931
$554$	0	0
$555$	−1480.00	−0.113194
$556$	0	0
$557$	−1546.00	−0.117605	−0.0588026	−	0.998270i	$-0.518728\pi$
$557$	−1546.00	−0.117605	−0.0588026	+	0.998270i	$0.518728\pi$
$558$	0	0
$559$	−920.000	−0.0696098
$560$	0	0
$561$	−9920.00	−0.746565
$562$	0	0
$563$	−3212.00	−0.240443	−0.120222	−	0.992747i	$-0.538361\pi$
$563$	−3212.00	−0.240443	−0.120222	+	0.992747i	$0.538361\pi$
$564$	0	0
$565$	330.000	0.0245720
$566$	0	0
$567$	−1436.00	−0.106360
$568$	0	0
$569$	−4790.00	−0.352913	−0.176456	−	0.984308i	$-0.556463\pi$
$569$	−4790.00	−0.352913	−0.176456	+	0.984308i	$0.556463\pi$
$570$	0	0
$571$	356.000	0.0260913	0.0130457	−	0.999915i	$-0.495847\pi$
$571$	356.000	0.0260913	0.0130457	+	0.999915i	$0.495847\pi$
$572$	0	0
$573$	22752.0	1.65878
$574$	0	0
$575$	4800.00	0.348128
$576$	0	0
$577$	−1102.00	−0.0795093	−0.0397546	−	0.999209i	$-0.512658\pi$
$577$	−1102.00	−0.0795093	−0.0397546	+	0.999209i	$0.512658\pi$
$578$	0	0
$579$	29424.0	2.11195
$580$	0	0
$581$	−2432.00	−0.173660
$582$	0	0
$583$	5160.00	0.366562
$584$	0	0
$585$	−1850.00	−0.130749
$586$	0	0
$587$	652.000	0.0458448	0.0229224	−	0.999737i	$-0.492703\pi$
$587$	652.000	0.0458448	0.0229224	+	0.999737i	$0.492703\pi$
$588$	0	0
$589$	992.000	0.0693967
$590$	0	0
$591$	−2544.00	−0.177066
$592$	0	0
$593$	−23662.0	−1.63859	−0.819293	−	0.573375i	$-0.805634\pi$
$593$	−23662.0	−1.63859	−0.819293	+	0.573375i	$0.805634\pi$
$594$	0	0
$595$	−1240.00	−0.0854370
$596$	0	0
$597$	7072.00	0.484820
$598$	0	0
$599$	−3952.00	−0.269573	−0.134787	−	0.990875i	$-0.543035\pi$
$599$	−3952.00	−0.269573	−0.134787	+	0.990875i	$0.543035\pi$
$600$	0	0
$601$	−14726.0	−0.999478	−0.499739	−	0.866176i	$-0.666571\pi$
$601$	−14726.0	−0.999478	−0.499739	+	0.866176i	$0.666571\pi$
$602$	0	0
$603$	12432.0	0.839586
$604$	0	0
$605$	−4655.00	−0.312814
$606$	0	0
$607$	−17896.0	−1.19667	−0.598333	−	0.801248i	$-0.704170\pi$
$607$	−17896.0	−1.19667	−0.598333	+	0.801248i	$0.704170\pi$
$608$	0	0
$609$	4928.00	0.327903
$610$	0	0
$611$	−4760.00	−0.315170
$612$	0	0
$613$	−8562.00	−0.564137	−0.282068	−	0.959394i	$-0.591021\pi$
$613$	−8562.00	−0.564137	−0.282068	+	0.959394i	$0.591021\pi$
$614$	0	0
$615$	−7440.00	−0.487821
$616$	0	0
$617$	−4886.00	−0.318805	−0.159403	−	0.987214i	$-0.550957\pi$
$617$	−4886.00	−0.318805	−0.159403	+	0.987214i	$0.550957\pi$
$618$	0	0
$619$	23316.0	1.51397	0.756986	−	0.653431i	$-0.226671\pi$
$619$	23316.0	1.51397	0.756986	+	0.653431i	$0.226671\pi$
$620$	0	0
$621$	−15360.0	−0.992553
$622$	0	0
$623$	−408.000	−0.0262378
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	1280.00	0.0815284
$628$	0	0
$629$	−2294.00	−0.145418
$630$	0	0
$631$	852.000	0.0537521	0.0268761	−	0.999639i	$-0.491444\pi$
$631$	852.000	0.0537521	0.0268761	+	0.999639i	$0.491444\pi$
$632$	0	0
$633$	−26336.0	−1.65365
$634$	0	0
$635$	−10500.0	−0.656189
$636$	0	0
$637$	3270.00	0.203394
$638$	0	0
$639$	−8584.00	−0.531421
$640$	0	0
$641$	14482.0	0.892363	0.446181	−	0.894943i	$-0.352784\pi$
$641$	14482.0	0.892363	0.446181	+	0.894943i	$0.352784\pi$
$642$	0	0
$643$	−4.00000	−0.000245326 0	−0.000122663	−	1.00000i	$-0.500039\pi$
$643$	−4.00000	−0.000245326 0	−0.000122663		1.00000i	$0.500039\pi$
$644$	0	0
$645$	−3680.00	−0.224651
$646$	0	0
$647$	7808.00	0.474442	0.237221	−	0.971456i	$-0.423763\pi$
$647$	7808.00	0.474442	0.237221	+	0.971456i	$0.423763\pi$
$648$	0	0
$649$	3520.00	0.212900
$650$	0	0
$651$	−3968.00	−0.238891
$652$	0	0
$653$	−28714.0	−1.72077	−0.860387	−	0.509641i	$-0.829778\pi$
$653$	−28714.0	−1.72077	−0.860387	+	0.509641i	$0.829778\pi$
$654$	0	0
$655$	−7240.00	−0.431893
$656$	0	0
$657$	−17390.0	−1.03265
$658$	0	0
$659$	26172.0	1.54707	0.773533	−	0.633756i	$-0.218488\pi$
$659$	26172.0	1.54707	0.773533	+	0.633756i	$0.218488\pi$
$660$	0	0
$661$	−14978.0	−0.881357	−0.440678	−	0.897665i	$-0.645262\pi$
$661$	−14978.0	−0.881357	−0.440678	+	0.897665i	$0.645262\pi$
$662$	0	0
$663$	−4960.00	−0.290544
$664$	0	0
$665$	160.000	0.00933013
$666$	0	0
$667$	−29568.0	−1.71646
$668$	0	0
$669$	−160.000	−0.00924657
$670$	0	0
$671$	9160.00	0.527001
$672$	0	0
$673$	−3886.00	−0.222577	−0.111288	−	0.993788i	$-0.535498\pi$
$673$	−3886.00	−0.222577	−0.111288	+	0.993788i	$0.535498\pi$
$674$	0	0
$675$	−2000.00	−0.114044
$676$	0	0
$677$	−12866.0	−0.730400	−0.365200	−	0.930929i	$-0.618999\pi$
$677$	−12866.0	−0.730400	−0.365200	+	0.930929i	$0.618999\pi$
$678$	0	0
$679$	−120.000	−0.00678229
$680$	0	0
$681$	−10464.0	−0.588813
$682$	0	0
$683$	−6996.00	−0.391939	−0.195970	−	0.980610i	$-0.562785\pi$
$683$	−6996.00	−0.391939	−0.195970	+	0.980610i	$0.562785\pi$
$684$	0	0
$685$	5730.00	0.319609
$686$	0	0
$687$	4880.00	0.271010
$688$	0	0
$689$	2580.00	0.142656
$690$	0	0
$691$	−10108.0	−0.556478	−0.278239	−	0.960512i	$-0.589751\pi$
$691$	−10108.0	−0.556478	−0.278239	+	0.960512i	$0.589751\pi$
$692$	0	0
$693$	−2960.00	−0.162253
$694$	0	0
$695$	−700.000	−0.0382051
$696$	0	0
$697$	−11532.0	−0.626694
$698$	0	0
$699$	−12880.0	−0.696948
$700$	0	0
$701$	12822.0	0.690842	0.345421	−	0.938448i	$-0.387736\pi$
$701$	12822.0	0.690842	0.345421	+	0.938448i	$0.387736\pi$
$702$	0	0
$703$	296.000	0.0158803
$704$	0	0
$705$	−19040.0	−1.01715
$706$	0	0
$707$	−2952.00	−0.157032
$708$	0	0
$709$	−27650.0	−1.46462	−0.732311	−	0.680970i	$-0.761558\pi$
$709$	−27650.0	−1.46462	−0.732311	+	0.680970i	$0.761558\pi$
$710$	0	0
$711$	−25012.0	−1.31930
$712$	0	0
$713$	23808.0	1.25051
$714$	0	0
$715$	1000.00	0.0523047
$716$	0	0
$717$	10144.0	0.528361
$718$	0	0
$719$	9448.00	0.490057	0.245028	−	0.969516i	$-0.421203\pi$
$719$	9448.00	0.490057	0.245028	+	0.969516i	$0.421203\pi$
$720$	0	0
$721$	−2016.00	−0.104133
$722$	0	0
$723$	−27920.0	−1.43618
$724$	0	0
$725$	−3850.00	−0.197221
$726$	0	0
$727$	7248.00	0.369757	0.184879	−	0.982761i	$-0.440811\pi$
$727$	7248.00	0.369757	0.184879	+	0.982761i	$0.440811\pi$
$728$	0	0
$729$	−30563.0	−1.55276
$730$	0	0
$731$	−5704.00	−0.288605
$732$	0	0
$733$	−13498.0	−0.680164	−0.340082	−	0.940396i	$-0.610455\pi$
$733$	−13498.0	−0.680164	−0.340082	+	0.940396i	$0.610455\pi$
$734$	0	0
$735$	13080.0	0.656412
$736$	0	0
$737$	−6720.00	−0.335868
$738$	0	0
$739$	7516.00	0.374128	0.187064	−	0.982348i	$-0.440103\pi$
$739$	7516.00	0.374128	0.187064	+	0.982348i	$0.440103\pi$
$740$	0	0
$741$	640.000	0.0317287
$742$	0	0
$743$	34068.0	1.68214	0.841072	−	0.540922i	$-0.181925\pi$
$743$	34068.0	1.68214	0.841072	+	0.540922i	$0.181925\pi$
$744$	0	0
$745$	−8890.00	−0.437187
$746$	0	0
$747$	−22496.0	−1.10185
$748$	0	0
$749$	−3392.00	−0.165475
$750$	0	0
$751$	−1608.00	−0.0781315	−0.0390657	−	0.999237i	$-0.512438\pi$
$751$	−1608.00	−0.0781315	−0.0390657	+	0.999237i	$0.512438\pi$
$752$	0	0
$753$	7680.00	0.371680
$754$	0	0
$755$	11120.0	0.536024
$756$	0	0
$757$	7278.00	0.349436	0.174718	−	0.984618i	$-0.444099\pi$
$757$	7278.00	0.349436	0.174718	+	0.984618i	$0.444099\pi$
$758$	0	0
$759$	30720.0	1.46912
$760$	0	0
$761$	874.000	0.0416327	0.0208163	−	0.999783i	$-0.493373\pi$
$761$	874.000	0.0416327	0.0208163	+	0.999783i	$0.493373\pi$
$762$	0	0
$763$	920.000	0.0436517
$764$	0	0
$765$	−11470.0	−0.542090
$766$	0	0
$767$	1760.00	0.0828552
$768$	0	0
$769$	−13006.0	−0.609894	−0.304947	−	0.952369i	$-0.598639\pi$
$769$	−13006.0	−0.609894	−0.304947	+	0.952369i	$0.598639\pi$
$770$	0	0
$771$	−54672.0	−2.55378
$772$	0	0
$773$	−25506.0	−1.18679	−0.593394	−	0.804912i	$-0.702212\pi$
$773$	−25506.0	−1.18679	−0.593394	+	0.804912i	$0.702212\pi$
$774$	0	0
$775$	3100.00	0.143684
$776$	0	0
$777$	−1184.00	−0.0546664
$778$	0	0
$779$	1488.00	0.0684379
$780$	0	0
$781$	4640.00	0.212589
$782$	0	0
$783$	12320.0	0.562300
$784$	0	0
$785$	13070.0	0.594253
$786$	0	0
$787$	−14216.0	−0.643895	−0.321948	−	0.946757i	$-0.604338\pi$
$787$	−14216.0	−0.643895	−0.321948	+	0.946757i	$0.604338\pi$
$788$	0	0
$789$	−35488.0	−1.60128
$790$	0	0
$791$	264.000	0.0118670
$792$	0	0
$793$	4580.00	0.205095
$794$	0	0
$795$	10320.0	0.460393
$796$	0	0
$797$	16518.0	0.734125	0.367062	−	0.930196i	$-0.380364\pi$
$797$	16518.0	0.734125	0.367062	+	0.930196i	$0.380364\pi$
$798$	0	0
$799$	−29512.0	−1.30671
$800$	0	0
$801$	−3774.00	−0.166477
$802$	0	0
$803$	9400.00	0.413099
$804$	0	0
$805$	3840.00	0.168127
$806$	0	0
$807$	59472.0	2.59419
$808$	0	0
$809$	−20486.0	−0.890296	−0.445148	−	0.895457i	$-0.646849\pi$
$809$	−20486.0	−0.890296	−0.445148	+	0.895457i	$0.646849\pi$
$810$	0	0
$811$	−9100.00	−0.394013	−0.197006	−	0.980402i	$-0.563122\pi$
$811$	−9100.00	−0.394013	−0.197006	+	0.980402i	$0.563122\pi$
$812$	0	0
$813$	5632.00	0.242956
$814$	0	0
$815$	−11540.0	−0.495986
$816$	0	0
$817$	736.000	0.0315170
$818$	0	0
$819$	−1480.00	−0.0631445
$820$	0	0
$821$	36622.0	1.55678	0.778390	−	0.627781i	$-0.216037\pi$
$821$	36622.0	1.55678	0.778390	+	0.627781i	$0.216037\pi$
$822$	0	0
$823$	20572.0	0.871318	0.435659	−	0.900112i	$-0.356515\pi$
$823$	20572.0	0.871318	0.435659	+	0.900112i	$0.356515\pi$
$824$	0	0
$825$	4000.00	0.168803
$826$	0	0
$827$	35060.0	1.47419	0.737095	−	0.675789i	$-0.236197\pi$
$827$	35060.0	1.47419	0.737095	+	0.675789i	$0.236197\pi$
$828$	0	0
$829$	−5850.00	−0.245089	−0.122545	−	0.992463i	$-0.539105\pi$
$829$	−5850.00	−0.245089	−0.122545	+	0.992463i	$0.539105\pi$
$830$	0	0
$831$	−57200.0	−2.38778
$832$	0	0
$833$	20274.0	0.843280
$834$	0	0
$835$	−4840.00	−0.200593
$836$	0	0
$837$	−9920.00	−0.409660
$838$	0	0
$839$	−33720.0	−1.38754	−0.693769	−	0.720198i	$-0.744051\pi$
$839$	−33720.0	−1.38754	−0.693769	+	0.720198i	$0.744051\pi$
$840$	0	0
$841$	−673.000	−0.0275944
$842$	0	0
$843$	36912.0	1.50809
$844$	0	0
$845$	−10485.0	−0.426858
$846$	0	0
$847$	−3724.00	−0.151072
$848$	0	0
$849$	44384.0	1.79418
$850$	0	0
$851$	7104.00	0.286160
$852$	0	0
$853$	−28562.0	−1.14648	−0.573238	−	0.819389i	$-0.694313\pi$
$853$	−28562.0	−1.14648	−0.573238	+	0.819389i	$0.694313\pi$
$854$	0	0
$855$	1480.00	0.0591988
$856$	0	0
$857$	33994.0	1.35497	0.677487	−	0.735535i	$-0.263069\pi$
$857$	33994.0	1.35497	0.677487	+	0.735535i	$0.263069\pi$
$858$	0	0
$859$	18176.0	0.721952	0.360976	−	0.932575i	$-0.382444\pi$
$859$	18176.0	0.721952	0.360976	+	0.932575i	$0.382444\pi$
$860$	0	0
$861$	−5952.00	−0.235591
$862$	0	0
$863$	38116.0	1.50346	0.751729	−	0.659472i	$-0.229220\pi$
$863$	38116.0	1.50346	0.751729	+	0.659472i	$0.229220\pi$
$864$	0	0
$865$	3070.00	0.120674
$866$	0	0
$867$	8552.00	0.334996
$868$	0	0
$869$	13520.0	0.527773
$870$	0	0
$871$	−3360.00	−0.130711
$872$	0	0
$873$	−1110.00	−0.0430330
$874$	0	0
$875$	500.000	0.0193178
$876$	0	0
$877$	36742.0	1.41470	0.707348	−	0.706865i	$-0.249891\pi$
$877$	36742.0	1.41470	0.707348	+	0.706865i	$0.249891\pi$
$878$	0	0
$879$	53008.0	2.03403
$880$	0	0
$881$	38498.0	1.47223	0.736113	−	0.676859i	$-0.236659\pi$
$881$	38498.0	1.47223	0.736113	+	0.676859i	$0.236659\pi$
$882$	0	0
$883$	−37060.0	−1.41242	−0.706211	−	0.708002i	$-0.749597\pi$
$883$	−37060.0	−1.41242	−0.706211	+	0.708002i	$0.749597\pi$
$884$	0	0
$885$	7040.00	0.267398
$886$	0	0
$887$	16988.0	0.643068	0.321534	−	0.946898i	$-0.395802\pi$
$887$	16988.0	0.643068	0.321534	+	0.946898i	$0.395802\pi$
$888$	0	0
$889$	−8400.00	−0.316903
$890$	0	0
$891$	7180.00	0.269965
$892$	0	0
$893$	3808.00	0.142699
$894$	0	0
$895$	2640.00	0.0985983
$896$	0	0
$897$	15360.0	0.571745
$898$	0	0
$899$	−19096.0	−0.708440
$900$	0	0
$901$	15996.0	0.591458
$902$	0	0
$903$	−2944.00	−0.108494
$904$	0	0
$905$	−14970.0	−0.549856
$906$	0	0
$907$	−19676.0	−0.720321	−0.360160	−	0.932890i	$-0.617278\pi$
$907$	−19676.0	−0.720321	−0.360160	+	0.932890i	$0.617278\pi$
$908$	0	0
$909$	−27306.0	−0.996351
$910$	0	0
$911$	−20668.0	−0.751659	−0.375830	−	0.926689i	$-0.622642\pi$
$911$	−20668.0	−0.751659	−0.375830	+	0.926689i	$0.622642\pi$
$912$	0	0
$913$	12160.0	0.440786
$914$	0	0
$915$	18320.0	0.661902
$916$	0	0
$917$	−5792.00	−0.208581
$918$	0	0
$919$	−33444.0	−1.20045	−0.600226	−	0.799830i	$-0.704923\pi$
$919$	−33444.0	−1.20045	−0.600226	+	0.799830i	$0.704923\pi$
$920$	0	0
$921$	768.000	0.0274772
$922$	0	0
$923$	2320.00	0.0827343
$924$	0	0
$925$	925.000	0.0328798
$926$	0	0
$927$	−18648.0	−0.660713
$928$	0	0
$929$	−6430.00	−0.227084	−0.113542	−	0.993533i	$-0.536220\pi$
$929$	−6430.00	−0.227084	−0.113542	+	0.993533i	$0.536220\pi$
$930$	0	0
$931$	−2616.00	−0.0920902
$932$	0	0
$933$	10976.0	0.385143
$934$	0	0
$935$	6200.00	0.216857
$936$	0	0
$937$	4586.00	0.159891	0.0799456	−	0.996799i	$-0.474525\pi$
$937$	4586.00	0.159891	0.0799456	+	0.996799i	$0.474525\pi$
$938$	0	0
$939$	1712.00	0.0594984
$940$	0	0
$941$	17078.0	0.591633	0.295817	−	0.955245i	$-0.404408\pi$
$941$	17078.0	0.591633	0.295817	+	0.955245i	$0.404408\pi$
$942$	0	0
$943$	35712.0	1.23324
$944$	0	0
$945$	−1600.00	−0.0550773
$946$	0	0
$947$	15804.0	0.542303	0.271152	−	0.962537i	$-0.412596\pi$
$947$	15804.0	0.542303	0.271152	+	0.962537i	$0.412596\pi$
$948$	0	0
$949$	4700.00	0.160768
$950$	0	0
$951$	63696.0	2.17191
$952$	0	0
$953$	24426.0	0.830258	0.415129	−	0.909763i	$-0.363736\pi$
$953$	24426.0	0.830258	0.415129	+	0.909763i	$0.363736\pi$
$954$	0	0
$955$	−14220.0	−0.481831
$956$	0	0
$957$	−24640.0	−0.832286
$958$	0	0
$959$	4584.00	0.154354
$960$	0	0
$961$	−14415.0	−0.483871
$962$	0	0
$963$	−31376.0	−1.04992
$964$	0	0
$965$	−18390.0	−0.613466
$966$	0	0
$967$	6872.00	0.228530	0.114265	−	0.993450i	$-0.463549\pi$
$967$	6872.00	0.228530	0.114265	+	0.993450i	$0.463549\pi$
$968$	0	0
$969$	3968.00	0.131549
$970$	0	0
$971$	−3708.00	−0.122549	−0.0612747	−	0.998121i	$-0.519517\pi$
$971$	−3708.00	−0.122549	−0.0612747	+	0.998121i	$0.519517\pi$
$972$	0	0
$973$	−560.000	−0.0184510
$974$	0	0
$975$	2000.00	0.0656936
$976$	0	0
$977$	42194.0	1.38168	0.690842	−	0.723006i	$-0.257240\pi$
$977$	42194.0	1.38168	0.690842	+	0.723006i	$0.257240\pi$
$978$	0	0
$979$	2040.00	0.0665972
$980$	0	0
$981$	8510.00	0.276966
$982$	0	0
$983$	−14444.0	−0.468659	−0.234330	−	0.972157i	$-0.575290\pi$
$983$	−14444.0	−0.468659	−0.234330	+	0.972157i	$0.575290\pi$
$984$	0	0
$985$	1590.00	0.0514331
$986$	0	0
$987$	−15232.0	−0.491226
$988$	0	0
$989$	17664.0	0.567930
$990$	0	0
$991$	38620.0	1.23795	0.618973	−	0.785412i	$-0.287549\pi$
$991$	38620.0	1.23795	0.618973	+	0.785412i	$0.287549\pi$
$992$	0	0
$993$	4160.00	0.132944
$994$	0	0
$995$	−4420.00	−0.140828
$996$	0	0
$997$	8574.00	0.272358	0.136179	−	0.990684i	$-0.456518\pi$
$997$	8574.00	0.272358	0.136179	+	0.990684i	$0.456518\pi$
$998$	0	0
$999$	−2960.00	−0.0937440

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	740.4.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.4.a.a.1.1	✓	1	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	740.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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