LMFDB - Embedded newform 2040.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2040,2,Mod(1,2040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("2040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2040.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:	$16.2894820123$
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	$\chi$	$=$	2040.1

$q$ -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^{3} -1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} -2.00000 q^{13} -1.00000 q^{15} -1.00000 q^{17} -5.00000 q^{19} -3.00000 q^{21} -6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.00000 q^{29} +2.00000 q^{31} +5.00000 q^{33} +3.00000 q^{35} -7.00000 q^{37} -2.00000 q^{39} +1.00000 q^{41} -4.00000 q^{43} -1.00000 q^{45} +3.00000 q^{47} +2.00000 q^{49} -1.00000 q^{51} -9.00000 q^{53} -5.00000 q^{55} -5.00000 q^{57} +4.00000 q^{59} +14.0000 q^{61} -3.00000 q^{63} +2.00000 q^{65} +8.00000 q^{67} -6.00000 q^{69} -4.00000 q^{71} -9.00000 q^{73} +1.00000 q^{75} -15.0000 q^{77} -14.0000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +1.00000 q^{85} -3.00000 q^{87} -12.0000 q^{89} +6.00000 q^{91} +2.00000 q^{93} +5.00000 q^{95} -10.0000 q^{97} +5.00000 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	0	0
$55$	−5.00000	−0.674200
$56$	0	0
$57$	−5.00000	−0.662266
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	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
$63$	−3.00000	−0.377964
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−9.00000	−1.05337	−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000	−1.05337	−0.526685	+	0.850060i	$0.676565\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−15.0000	−1.70941
$78$	0	0
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	−3.00000	−0.321634
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	2.00000	0.207390
$94$	0	0
$95$	5.00000	0.512989
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	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
$111$	−7.00000	−0.664411
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	1.00000	0.0901670
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	18.0000	1.59724	0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000	1.59724	0.798621	+	0.601834i	$0.205563\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	15.0000	1.30066
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	0	0
$139$	22.0000	1.86602	0.933008	−	0.359856i	$-0.117174\pi$
$139$	22.0000	1.86602	0.933008	+	0.359856i	$0.117174\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	0	0
$143$	−10.0000	−0.836242
$144$	0	0
$145$	3.00000	0.249136
$146$	0	0
$147$	2.00000	0.164957
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	3.00000	0.244137	0.122068	−	0.992522i	$-0.461047\pi$
$151$	3.00000	0.244137	0.122068	+	0.992522i	$0.461047\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	24.0000	1.91541	0.957704	−	0.287754i	$-0.0929087\pi$
$157$	24.0000	1.91541	0.957704	+	0.287754i	$0.0929087\pi$
$158$	0	0
$159$	−9.00000	−0.713746
$160$	0	0
$161$	18.0000	1.41860
$162$	0	0
$163$	3.00000	0.234978	0.117489	−	0.993074i	$-0.462515\pi$
$163$	3.00000	0.234978	0.117489	+	0.993074i	$0.462515\pi$
$164$	0	0
$165$	−5.00000	−0.389249
$166$	0	0
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−5.00000	−0.382360
$172$	0	0
$173$	−8.00000	−0.608229	−0.304114	−	0.952636i	$-0.598361\pi$
$173$	−8.00000	−0.608229	−0.304114	+	0.952636i	$0.598361\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	0	0
$185$	7.00000	0.514650
$186$	0	0
$187$	−5.00000	−0.365636
$188$	0	0
$189$	−3.00000	−0.218218
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	9.00000	0.631676
$204$	0	0
$205$	−1.00000	−0.0698430
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−25.0000	−1.72929
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	−6.00000	−0.407307
$218$	0	0
$219$	−9.00000	−0.608164
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	−21.0000	−1.38772	−0.693860	−	0.720110i	$-0.744091\pi$
$229$	−21.0000	−1.38772	−0.693860	+	0.720110i	$0.744091\pi$
$230$	0	0
$231$	−15.0000	−0.986928
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−3.00000	−0.195698
$236$	0	0
$237$	−14.0000	−0.909398
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.00000	−0.127775
$246$	0	0
$247$	10.0000	0.636285
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−30.0000	−1.88608
$254$	0	0
$255$	1.00000	0.0626224
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	21.0000	1.30488
$260$	0	0
$261$	−3.00000	−0.185695
$262$	0	0
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	9.00000	0.552866
$266$	0	0
$267$	−12.0000	−0.734388
$268$	0	0
$269$	23.0000	1.40233	0.701167	−	0.712997i	$-0.252663\pi$
$269$	23.0000	1.40233	0.701167	+	0.712997i	$0.252663\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	5.00000	0.301511
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	−32.0000	−1.90896	−0.954480	−	0.298275i	$-0.903589\pi$
$281$	−32.0000	−1.90896	−0.954480	+	0.298275i	$0.903589\pi$
$282$	0	0
$283$	29.0000	1.72387	0.861936	−	0.507018i	$-0.169252\pi$
$283$	29.0000	1.72387	0.861936	+	0.507018i	$0.169252\pi$
$284$	0	0
$285$	5.00000	0.296174
$286$	0	0
$287$	−3.00000	−0.177084
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	−29.0000	−1.69420	−0.847099	−	0.531435i	$-0.821653\pi$
$293$	−29.0000	−1.69420	−0.847099	+	0.531435i	$0.821653\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	12.0000	0.693978
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	−18.0000	−1.03407
$304$	0	0
$305$	−14.0000	−0.801638
$306$	0	0
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	0	0
$313$	23.0000	1.30004	0.650018	−	0.759918i	$-0.274761\pi$
$313$	23.0000	1.30004	0.650018	+	0.759918i	$0.274761\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	−15.0000	−0.839839
$320$	0	0
$321$	−8.00000	−0.446516
$322$	0	0
$323$	5.00000	0.278207
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	2.00000	0.110600
$328$	0	0
$329$	−9.00000	−0.496186
$330$	0	0
$331$	7.00000	0.384755	0.192377	−	0.981321i	$-0.438380\pi$
$331$	7.00000	0.384755	0.192377	+	0.981321i	$0.438380\pi$
$332$	0	0
$333$	−7.00000	−0.383598
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	0	0
$339$	−14.0000	−0.760376
$340$	0	0
$341$	10.0000	0.541530
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	6.00000	0.323029
$346$	0	0
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	0	0
$349$	−13.0000	−0.695874	−0.347937	−	0.937518i	$-0.613118\pi$
$349$	−13.0000	−0.695874	−0.347937	+	0.937518i	$0.613118\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	−15.0000	−0.798369	−0.399185	−	0.916871i	$-0.630707\pi$
$353$	−15.0000	−0.798369	−0.399185	+	0.916871i	$0.630707\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	0	0
$357$	3.00000	0.158777
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	14.0000	0.734809
$364$	0	0
$365$	9.00000	0.471082
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	1.00000	0.0520579
$370$	0	0
$371$	27.0000	1.40177
$372$	0	0
$373$	12.0000	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	0	0
$383$	1.00000	0.0510976	0.0255488	−	0.999674i	$-0.491867\pi$
$383$	1.00000	0.0510976	0.0255488	+	0.999674i	$0.491867\pi$
$384$	0	0
$385$	15.0000	0.764471
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	0	0
$394$	0	0
$395$	14.0000	0.704416
$396$	0	0
$397$	21.0000	1.05396	0.526980	−	0.849878i	$-0.323324\pi$
$397$	21.0000	1.05396	0.526980	+	0.849878i	$0.323324\pi$
$398$	0	0
$399$	15.0000	0.750939
$400$	0	0
$401$	7.00000	0.349563	0.174782	−	0.984607i	$-0.444078\pi$
$401$	7.00000	0.349563	0.174782	+	0.984607i	$0.444078\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−35.0000	−1.73489
$408$	0	0
$409$	1.00000	0.0494468	0.0247234	−	0.999694i	$-0.492129\pi$
$409$	1.00000	0.0494468	0.0247234	+	0.999694i	$0.492129\pi$
$410$	0	0
$411$	9.00000	0.443937
$412$	0	0
$413$	−12.0000	−0.590481
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	22.0000	1.07734
$418$	0	0
$419$	3.00000	0.146560	0.0732798	−	0.997311i	$-0.476653\pi$
$419$	3.00000	0.146560	0.0732798	+	0.997311i	$0.476653\pi$
$420$	0	0
$421$	9.00000	0.438633	0.219317	−	0.975654i	$-0.429617\pi$
$421$	9.00000	0.438633	0.219317	+	0.975654i	$0.429617\pi$
$422$	0	0
$423$	3.00000	0.145865
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	−42.0000	−2.03252
$428$	0	0
$429$	−10.0000	−0.482805
$430$	0	0
$431$	27.0000	1.30054	0.650272	−	0.759701i	$-0.274655\pi$
$431$	27.0000	1.30054	0.650272	+	0.759701i	$0.274655\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	0	0
$435$	3.00000	0.143839
$436$	0	0
$437$	30.0000	1.43509
$438$	0	0
$439$	2.00000	0.0954548	0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000	0.0954548	0.0477274	+	0.998860i	$0.484802\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	0	0
$447$	−18.0000	−0.851371
$448$	0	0
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	5.00000	0.235441
$452$	0	0
$453$	3.00000	0.140952
$454$	0	0
$455$	−6.00000	−0.281284
$456$	0	0
$457$	−4.00000	−0.187112	−0.0935561	−	0.995614i	$-0.529823\pi$
$457$	−4.00000	−0.187112	−0.0935561	+	0.995614i	$0.529823\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	0	0
$463$	18.0000	0.836531	0.418265	−	0.908325i	$-0.362638\pi$
$463$	18.0000	0.836531	0.418265	+	0.908325i	$0.362638\pi$
$464$	0	0
$465$	−2.00000	−0.0927478
$466$	0	0
$467$	−19.0000	−0.879215	−0.439608	−	0.898190i	$-0.644882\pi$
$467$	−19.0000	−0.879215	−0.439608	+	0.898190i	$0.644882\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	0	0
$471$	24.0000	1.10586
$472$	0	0
$473$	−20.0000	−0.919601
$474$	0	0
$475$	−5.00000	−0.229416
$476$	0	0
$477$	−9.00000	−0.412082
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	14.0000	0.638345
$482$	0	0
$483$	18.0000	0.819028
$484$	0	0
$485$	10.0000	0.454077
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	3.00000	0.135665
$490$	0	0
$491$	−18.0000	−0.812329	−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000	−0.812329	−0.406164	+	0.913800i	$0.633134\pi$
$492$	0	0
$493$	3.00000	0.135113
$494$	0	0
$495$	−5.00000	−0.224733
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	2.00000	0.0893534
$502$	0	0
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	27.0000	1.19441
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	16.0000	0.705044
$516$	0	0
$517$	15.0000	0.659699
$518$	0	0
$519$	−8.00000	−0.351161
$520$	0	0
$521$	31.0000	1.35813	0.679067	−	0.734076i	$-0.262384\pi$
$521$	31.0000	1.35813	0.679067	+	0.734076i	$0.262384\pi$
$522$	0	0
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	0	0
$525$	−3.00000	−0.130931
$526$	0	0
$527$	−2.00000	−0.0871214
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	−2.00000	−0.0866296
$534$	0	0
$535$	8.00000	0.345870
$536$	0	0
$537$	10.0000	0.431532
$538$	0	0
$539$	10.0000	0.430730
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−23.0000	−0.983409	−0.491704	−	0.870762i	$-0.663626\pi$
$547$	−23.0000	−0.983409	−0.491704	+	0.870762i	$0.663626\pi$
$548$	0	0
$549$	14.0000	0.597505
$550$	0	0
$551$	15.0000	0.639021
$552$	0	0
$553$	42.0000	1.78602
$554$	0	0
$555$	7.00000	0.297133
$556$	0	0
$557$	−22.0000	−0.932170	−0.466085	−	0.884740i	$-0.654336\pi$
$557$	−22.0000	−0.932170	−0.466085	+	0.884740i	$0.654336\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	−5.00000	−0.211100
$562$	0	0
$563$	−45.0000	−1.89652	−0.948262	−	0.317489i	$-0.897160\pi$
$563$	−45.0000	−1.89652	−0.948262	+	0.317489i	$0.897160\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	0	0
$571$	−26.0000	−1.08807	−0.544033	−	0.839064i	$-0.683103\pi$
$571$	−26.0000	−1.08807	−0.544033	+	0.839064i	$0.683103\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	−6.00000	−0.250217
$576$	0	0
$577$	−12.0000	−0.499567	−0.249783	−	0.968302i	$-0.580359\pi$
$577$	−12.0000	−0.499567	−0.249783	+	0.968302i	$0.580359\pi$
$578$	0	0
$579$	22.0000	0.914289
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	−45.0000	−1.86371
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	−23.0000	−0.949312	−0.474656	−	0.880172i	$-0.657427\pi$
$587$	−23.0000	−0.949312	−0.474656	+	0.880172i	$0.657427\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	0	0
$595$	−3.00000	−0.122988
$596$	0	0
$597$	10.0000	0.409273
$598$	0	0
$599$	−42.0000	−1.71607	−0.858037	−	0.513588i	$-0.828316\pi$
$599$	−42.0000	−1.71607	−0.858037	+	0.513588i	$0.828316\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	−14.0000	−0.569181
$606$	0	0
$607$	−35.0000	−1.42061	−0.710303	−	0.703896i	$-0.751442\pi$
$607$	−35.0000	−1.42061	−0.710303	+	0.703896i	$0.751442\pi$
$608$	0	0
$609$	9.00000	0.364698
$610$	0	0
$611$	−6.00000	−0.242734
$612$	0	0
$613$	8.00000	0.323117	0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000	0.323117	0.161558	+	0.986863i	$0.448348\pi$
$614$	0	0
$615$	−1.00000	−0.0403239
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	0	0
$623$	36.0000	1.44231
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−25.0000	−0.998404
$628$	0	0
$629$	7.00000	0.279108
$630$	0	0
$631$	−29.0000	−1.15447	−0.577236	−	0.816577i	$-0.695869\pi$
$631$	−29.0000	−1.15447	−0.577236	+	0.816577i	$0.695869\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−18.0000	−0.714308
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	−4.00000	−0.158238
$640$	0	0
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	0	0
$643$	−7.00000	−0.276053	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	−7.00000	−0.276053	−0.138027	+	0.990429i	$0.544076\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	−6.00000	−0.235159
$652$	0	0
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−9.00000	−0.351123
$658$	0	0
$659$	−34.0000	−1.32445	−0.662226	−	0.749304i	$-0.730388\pi$
$659$	−34.0000	−1.32445	−0.662226	+	0.749304i	$0.730388\pi$
$660$	0	0
$661$	−1.00000	−0.0388955	−0.0194477	−	0.999811i	$-0.506191\pi$
$661$	−1.00000	−0.0388955	−0.0194477	+	0.999811i	$0.506191\pi$
$662$	0	0
$663$	2.00000	0.0776736
$664$	0	0
$665$	−15.0000	−0.581675
$666$	0	0
$667$	18.0000	0.696963
$668$	0	0
$669$	10.0000	0.386622
$670$	0	0
$671$	70.0000	2.70232
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	0	0
$679$	30.0000	1.15129
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	−9.00000	−0.343872
$686$	0	0
$687$	−21.0000	−0.801200
$688$	0	0
$689$	18.0000	0.685745
$690$	0	0
$691$	24.0000	0.913003	0.456502	−	0.889723i	$-0.349102\pi$
$691$	24.0000	0.913003	0.456502	+	0.889723i	$0.349102\pi$
$692$	0	0
$693$	−15.0000	−0.569803
$694$	0	0
$695$	−22.0000	−0.834508
$696$	0	0
$697$	−1.00000	−0.0378777
$698$	0	0
$699$	10.0000	0.378235
$700$	0	0
$701$	−44.0000	−1.66186	−0.830929	−	0.556379i	$-0.812190\pi$
$701$	−44.0000	−1.66186	−0.830929	+	0.556379i	$0.812190\pi$
$702$	0	0
$703$	35.0000	1.32005
$704$	0	0
$705$	−3.00000	−0.112987
$706$	0	0
$707$	54.0000	2.03088
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	−14.0000	−0.525041
$712$	0	0
$713$	−12.0000	−0.449404
$714$	0	0
$715$	10.0000	0.373979
$716$	0	0
$717$	6.00000	0.224074
$718$	0	0
$719$	27.0000	1.00693	0.503465	−	0.864016i	$-0.332058\pi$
$719$	27.0000	1.00693	0.503465	+	0.864016i	$0.332058\pi$
$720$	0	0
$721$	48.0000	1.78761
$722$	0	0
$723$	−6.00000	−0.223142
$724$	0	0
$725$	−3.00000	−0.111417
$726$	0	0
$727$	−50.0000	−1.85440	−0.927199	−	0.374570i	$-0.877790\pi$
$727$	−50.0000	−1.85440	−0.927199	+	0.374570i	$0.877790\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.00000	0.147945
$732$	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
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	40.0000	1.47342
$738$	0	0
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	10.0000	0.367359
$742$	0	0
$743$	−34.0000	−1.24734	−0.623670	−	0.781688i	$-0.714359\pi$
$743$	−34.0000	−1.24734	−0.623670	+	0.781688i	$0.714359\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	24.0000	0.876941
$750$	0	0
$751$	−50.0000	−1.82453	−0.912263	−	0.409605i	$-0.865667\pi$
$751$	−50.0000	−1.82453	−0.912263	+	0.409605i	$0.865667\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	−3.00000	−0.109181
$756$	0	0
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	0	0
$759$	−30.0000	−1.08893
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	−6.00000	−0.217215
$764$	0	0
$765$	1.00000	0.0361551
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	9.00000	0.323708	0.161854	−	0.986815i	$-0.448253\pi$
$773$	9.00000	0.323708	0.161854	+	0.986815i	$0.448253\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	0	0
$777$	21.0000	0.753371
$778$	0	0
$779$	−5.00000	−0.179144
$780$	0	0
$781$	−20.0000	−0.715656
$782$	0	0
$783$	−3.00000	−0.107211
$784$	0	0
$785$	−24.0000	−0.856597
$786$	0	0
$787$	−27.0000	−0.962446	−0.481223	−	0.876598i	$-0.659807\pi$
$787$	−27.0000	−0.962446	−0.481223	+	0.876598i	$0.659807\pi$
$788$	0	0
$789$	21.0000	0.747620
$790$	0	0
$791$	42.0000	1.49335
$792$	0	0
$793$	−28.0000	−0.994309
$794$	0	0
$795$	9.00000	0.319197
$796$	0	0
$797$	41.0000	1.45229	0.726147	−	0.687539i	$-0.241309\pi$
$797$	41.0000	1.45229	0.726147	+	0.687539i	$0.241309\pi$
$798$	0	0
$799$	−3.00000	−0.106132
$800$	0	0
$801$	−12.0000	−0.423999
$802$	0	0
$803$	−45.0000	−1.58802
$804$	0	0
$805$	−18.0000	−0.634417
$806$	0	0
$807$	23.0000	0.809638
$808$	0	0
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	0	0
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	0	0
$815$	−3.00000	−0.105085
$816$	0	0
$817$	20.0000	0.699711
$818$	0	0
$819$	6.00000	0.209657
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	7.00000	0.244005	0.122002	−	0.992530i	$-0.461068\pi$
$823$	7.00000	0.244005	0.122002	+	0.992530i	$0.461068\pi$
$824$	0	0
$825$	5.00000	0.174078
$826$	0	0
$827$	56.0000	1.94731	0.973655	−	0.228024i	$-0.0732266\pi$
$827$	56.0000	1.94731	0.973655	+	0.228024i	$0.0732266\pi$
$828$	0	0
$829$	−27.0000	−0.937749	−0.468874	−	0.883265i	$-0.655340\pi$
$829$	−27.0000	−0.937749	−0.468874	+	0.883265i	$0.655340\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	−2.00000	−0.0692129
$836$	0	0
$837$	2.00000	0.0691301
$838$	0	0
$839$	−1.00000	−0.0345238	−0.0172619	−	0.999851i	$-0.505495\pi$
$839$	−1.00000	−0.0345238	−0.0172619	+	0.999851i	$0.505495\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	−32.0000	−1.10214
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	−42.0000	−1.44314
$848$	0	0
$849$	29.0000	0.995277
$850$	0	0
$851$	42.0000	1.43974
$852$	0	0
$853$	−6.00000	−0.205436	−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000	−0.205436	−0.102718	+	0.994711i	$0.532754\pi$
$854$	0	0
$855$	5.00000	0.170996
$856$	0	0
$857$	−28.0000	−0.956462	−0.478231	−	0.878234i	$-0.658722\pi$
$857$	−28.0000	−0.956462	−0.478231	+	0.878234i	$0.658722\pi$
$858$	0	0
$859$	−35.0000	−1.19418	−0.597092	−	0.802173i	$-0.703677\pi$
$859$	−35.0000	−1.19418	−0.597092	+	0.802173i	$0.703677\pi$
$860$	0	0
$861$	−3.00000	−0.102240
$862$	0	0
$863$	−3.00000	−0.102121	−0.0510606	−	0.998696i	$-0.516260\pi$
$863$	−3.00000	−0.102121	−0.0510606	+	0.998696i	$0.516260\pi$
$864$	0	0
$865$	8.00000	0.272008
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−70.0000	−2.37459
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	3.00000	0.101419
$876$	0	0
$877$	−23.0000	−0.776655	−0.388327	−	0.921521i	$-0.626947\pi$
$877$	−23.0000	−0.776655	−0.388327	+	0.921521i	$0.626947\pi$
$878$	0	0
$879$	−29.0000	−0.978146
$880$	0	0
$881$	45.0000	1.51609	0.758044	−	0.652203i	$-0.226155\pi$
$881$	45.0000	1.51609	0.758044	+	0.652203i	$0.226155\pi$
$882$	0	0
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	0	0
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	−54.0000	−1.81110
$890$	0	0
$891$	5.00000	0.167506
$892$	0	0
$893$	−15.0000	−0.501956
$894$	0	0
$895$	−10.0000	−0.334263
$896$	0	0
$897$	12.0000	0.400668
$898$	0	0
$899$	−6.00000	−0.200111
$900$	0	0
$901$	9.00000	0.299833
$902$	0	0
$903$	12.0000	0.399335
$904$	0	0
$905$	10.0000	0.332411
$906$	0	0
$907$	−27.0000	−0.896520	−0.448260	−	0.893903i	$-0.647956\pi$
$907$	−27.0000	−0.896520	−0.448260	+	0.893903i	$0.647956\pi$
$908$	0	0
$909$	−18.0000	−0.597022
$910$	0	0
$911$	9.00000	0.298183	0.149092	−	0.988823i	$-0.452365\pi$
$911$	9.00000	0.298183	0.149092	+	0.988823i	$0.452365\pi$
$912$	0	0
$913$	−20.0000	−0.661903
$914$	0	0
$915$	−14.0000	−0.462826
$916$	0	0
$917$	0	0
$918$	0	0
$919$	59.0000	1.94623	0.973115	−	0.230319i	$-0.0739769\pi$
$919$	59.0000	1.94623	0.973115	+	0.230319i	$0.0739769\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	0	0
$923$	8.00000	0.263323
$924$	0	0
$925$	−7.00000	−0.230159
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	5.00000	0.164045	0.0820223	−	0.996630i	$-0.473862\pi$
$929$	5.00000	0.164045	0.0820223	+	0.996630i	$0.473862\pi$
$930$	0	0
$931$	−10.0000	−0.327737
$932$	0	0
$933$	21.0000	0.687509
$934$	0	0
$935$	5.00000	0.163517
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	23.0000	0.750577
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	−6.00000	−0.195387
$944$	0	0
$945$	3.00000	0.0975900
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	18.0000	0.584305
$950$	0	0
$951$	−22.0000	−0.713399
$952$	0	0
$953$	−15.0000	−0.485898	−0.242949	−	0.970039i	$-0.578115\pi$
$953$	−15.0000	−0.485898	−0.242949	+	0.970039i	$0.578115\pi$
$954$	0	0
$955$	−8.00000	−0.258874
$956$	0	0
$957$	−15.0000	−0.484881
$958$	0	0
$959$	−27.0000	−0.871875
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−8.00000	−0.257796
$964$	0	0
$965$	−22.0000	−0.708205
$966$	0	0
$967$	12.0000	0.385894	0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000	0.385894	0.192947	+	0.981209i	$0.438195\pi$
$968$	0	0
$969$	5.00000	0.160623
$970$	0	0
$971$	−56.0000	−1.79713	−0.898563	−	0.438845i	$-0.855388\pi$
$971$	−56.0000	−1.79713	−0.898563	+	0.438845i	$0.855388\pi$
$972$	0	0
$973$	−66.0000	−2.11586
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	−60.0000	−1.91761
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	−26.0000	−0.829271	−0.414636	−	0.909988i	$-0.636091\pi$
$983$	−26.0000	−0.829271	−0.414636	+	0.909988i	$0.636091\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	0	0
$987$	−9.00000	−0.286473
$988$	0	0
$989$	24.0000	0.763156
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	0	0
$993$	7.00000	0.222138
$994$	0	0
$995$	−10.0000	−0.317021
$996$	0	0
$997$	31.0000	0.981780	0.490890	−	0.871222i	$-0.336672\pi$
$997$	31.0000	0.981780	0.490890	+	0.871222i	$0.336672\pi$
$998$	0	0
$999$	−7.00000	−0.221470

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2040.2.a.i.1.1	✓	1
3.2	odd	2		6120.2.a.p.1.1		1
4.3	odd	2		4080.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.2.a.i.1.1	✓	1	1.1	even	1	trivial
4080.2.a.h.1.1		1	4.3	odd	2
6120.2.a.p.1.1		1	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2040.2.a.i.1.1

Level	$2040$
Weight	$2$
Character	2040.1
Self dual	yes
Analytic conductor	$16.289$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$