LMFDB - Embedded newform 1840.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1840.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:	$108.563514411$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{3} -5.00000 q^{5} +32.0000 q^{7} -11.0000 q^{9} +O(q^{10})$

$q-4.00000 q^{3} -5.00000 q^{5} +32.0000 q^{7} -11.0000 q^{9} -40.0000 q^{11} -66.0000 q^{13} +20.0000 q^{15} +130.000 q^{17} +88.0000 q^{19} -128.000 q^{21} -23.0000 q^{23} +25.0000 q^{25} +152.000 q^{27} -130.000 q^{29} -40.0000 q^{31} +160.000 q^{33} -160.000 q^{35} -334.000 q^{37} +264.000 q^{39} -22.0000 q^{41} +272.000 q^{43} +55.0000 q^{45} -24.0000 q^{47} +681.000 q^{49} -520.000 q^{51} +258.000 q^{53} +200.000 q^{55} -352.000 q^{57} -612.000 q^{59} -366.000 q^{61} -352.000 q^{63} +330.000 q^{65} +496.000 q^{67} +92.0000 q^{69} -248.000 q^{71} +826.000 q^{73} -100.000 q^{75} -1280.00 q^{77} +296.000 q^{79} -311.000 q^{81} +1296.00 q^{83} -650.000 q^{85} +520.000 q^{87} -646.000 q^{89} -2112.00 q^{91} +160.000 q^{93} -440.000 q^{95} -1438.00 q^{97} +440.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−4.00000	−0.769800	−0.384900	−	0.922958i	$-0.625764\pi$
$3$	−4.00000	−0.769800	−0.384900	+	0.922958i	$0.625764\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	32.0000	1.72784	0.863919	−	0.503631i	$-0.168003\pi$
$7$	32.0000	1.72784	0.863919	+	0.503631i	$0.168003\pi$
$8$	0	0
$9$	−11.0000	−0.407407
$10$	0	0
$11$	−40.0000	−1.09640	−0.548202	−	0.836346i	$-0.684688\pi$
$11$	−40.0000	−1.09640	−0.548202	+	0.836346i	$0.684688\pi$
$12$	0	0
$13$	−66.0000	−1.40809	−0.704043	−	0.710158i	$-0.748624\pi$
$13$	−66.0000	−1.40809	−0.704043	+	0.710158i	$0.748624\pi$
$14$	0	0
$15$	20.0000	0.344265
$16$	0	0
$17$	130.000	1.85468	0.927342	−	0.374215i	$-0.122088\pi$
$17$	130.000	1.85468	0.927342	+	0.374215i	$0.122088\pi$
$18$	0	0
$19$	88.0000	1.06256	0.531279	−	0.847197i	$-0.321712\pi$
$19$	88.0000	1.06256	0.531279	+	0.847197i	$0.321712\pi$
$20$	0	0
$21$	−128.000	−1.33009
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	152.000	1.08342
$28$	0	0
$29$	−130.000	−0.832427	−0.416214	−	0.909267i	$-0.636643\pi$
$29$	−130.000	−0.832427	−0.416214	+	0.909267i	$0.636643\pi$
$30$	0	0
$31$	−40.0000	−0.231749	−0.115874	−	0.993264i	$-0.536967\pi$
$31$	−40.0000	−0.231749	−0.115874	+	0.993264i	$0.536967\pi$
$32$	0	0
$33$	160.000	0.844013
$34$	0	0
$35$	−160.000	−0.772712
$36$	0	0
$37$	−334.000	−1.48403	−0.742017	−	0.670381i	$-0.766131\pi$
$37$	−334.000	−1.48403	−0.742017	+	0.670381i	$0.766131\pi$
$38$	0	0
$39$	264.000	1.08394
$40$	0	0
$41$	−22.0000	−0.0838006	−0.0419003	−	0.999122i	$-0.513341\pi$
$41$	−22.0000	−0.0838006	−0.0419003	+	0.999122i	$0.513341\pi$
$42$	0	0
$43$	272.000	0.964642	0.482321	−	0.875995i	$-0.339794\pi$
$43$	272.000	0.964642	0.482321	+	0.875995i	$0.339794\pi$
$44$	0	0
$45$	55.0000	0.182198
$46$	0	0
$47$	−24.0000	−0.0744843	−0.0372421	−	0.999306i	$-0.511857\pi$
$47$	−24.0000	−0.0744843	−0.0372421	+	0.999306i	$0.511857\pi$
$48$	0	0
$49$	681.000	1.98542
$50$	0	0
$51$	−520.000	−1.42774
$52$	0	0
$53$	258.000	0.668661	0.334330	−	0.942456i	$-0.391490\pi$
$53$	258.000	0.668661	0.334330	+	0.942456i	$0.391490\pi$
$54$	0	0
$55$	200.000	0.490327
$56$	0	0
$57$	−352.000	−0.817957
$58$	0	0
$59$	−612.000	−1.35043	−0.675217	−	0.737619i	$-0.735950\pi$
$59$	−612.000	−1.35043	−0.675217	+	0.737619i	$0.735950\pi$
$60$	0	0
$61$	−366.000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−366.000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	−352.000	−0.703934
$64$	0	0
$65$	330.000	0.629715
$66$	0	0
$67$	496.000	0.904419	0.452209	−	0.891912i	$-0.350636\pi$
$67$	496.000	0.904419	0.452209	+	0.891912i	$0.350636\pi$
$68$	0	0
$69$	92.0000	0.160514
$70$	0	0
$71$	−248.000	−0.414538	−0.207269	−	0.978284i	$-0.566458\pi$
$71$	−248.000	−0.414538	−0.207269	+	0.978284i	$0.566458\pi$
$72$	0	0
$73$	826.000	1.32433	0.662164	−	0.749359i	$-0.269638\pi$
$73$	826.000	1.32433	0.662164	+	0.749359i	$0.269638\pi$
$74$	0	0
$75$	−100.000	−0.153960
$76$	0	0
$77$	−1280.00	−1.89441
$78$	0	0
$79$	296.000	0.421552	0.210776	−	0.977534i	$-0.432401\pi$
$79$	296.000	0.421552	0.210776	+	0.977534i	$0.432401\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	1296.00	1.71391	0.856955	−	0.515392i	$-0.172354\pi$
$83$	1296.00	1.71391	0.856955	+	0.515392i	$0.172354\pi$
$84$	0	0
$85$	−650.000	−0.829440
$86$	0	0
$87$	520.000	0.640803
$88$	0	0
$89$	−646.000	−0.769392	−0.384696	−	0.923043i	$-0.625694\pi$
$89$	−646.000	−0.769392	−0.384696	+	0.923043i	$0.625694\pi$
$90$	0	0
$91$	−2112.00	−2.43294
$92$	0	0
$93$	160.000	0.178400
$94$	0	0
$95$	−440.000	−0.475190
$96$	0	0
$97$	−1438.00	−1.50522	−0.752612	−	0.658464i	$-0.771206\pi$
$97$	−1438.00	−1.50522	−0.752612	+	0.658464i	$0.771206\pi$
$98$	0	0
$99$	440.000	0.446683
$100$	0	0
$101$	−850.000	−0.837408	−0.418704	−	0.908123i	$-0.637515\pi$
$101$	−850.000	−0.837408	−0.418704	+	0.908123i	$0.637515\pi$
$102$	0	0
$103$	1320.00	1.26275	0.631376	−	0.775477i	$-0.282490\pi$
$103$	1320.00	1.26275	0.631376	+	0.775477i	$0.282490\pi$
$104$	0	0
$105$	640.000	0.594834
$106$	0	0
$107$	−72.0000	−0.0650514	−0.0325257	−	0.999471i	$-0.510355\pi$
$107$	−72.0000	−0.0650514	−0.0325257	+	0.999471i	$0.510355\pi$
$108$	0	0
$109$	242.000	0.212655	0.106328	−	0.994331i	$-0.466091\pi$
$109$	242.000	0.212655	0.106328	+	0.994331i	$0.466091\pi$
$110$	0	0
$111$	1336.00	1.14241
$112$	0	0
$113$	1018.00	0.847481	0.423741	−	0.905784i	$-0.360717\pi$
$113$	1018.00	0.847481	0.423741	+	0.905784i	$0.360717\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	726.000	0.573664
$118$	0	0
$119$	4160.00	3.20459
$120$	0	0
$121$	269.000	0.202104
$122$	0	0
$123$	88.0000	0.0645097
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−120.000	−0.0838447	−0.0419224	−	0.999121i	$-0.513348\pi$
$127$	−120.000	−0.0838447	−0.0419224	+	0.999121i	$0.513348\pi$
$128$	0	0
$129$	−1088.00	−0.742582
$130$	0	0
$131$	−1340.00	−0.893713	−0.446856	−	0.894606i	$-0.647456\pi$
$131$	−1340.00	−0.893713	−0.446856	+	0.894606i	$0.647456\pi$
$132$	0	0
$133$	2816.00	1.83593
$134$	0	0
$135$	−760.000	−0.484521
$136$	0	0
$137$	−22.0000	−0.0137196	−0.00685981	−	0.999976i	$-0.502184\pi$
$137$	−22.0000	−0.0137196	−0.00685981	+	0.999976i	$0.502184\pi$
$138$	0	0
$139$	−2596.00	−1.58410	−0.792050	−	0.610456i	$-0.790986\pi$
$139$	−2596.00	−1.58410	−0.792050	+	0.610456i	$0.790986\pi$
$140$	0	0
$141$	96.0000	0.0573380
$142$	0	0
$143$	2640.00	1.54383
$144$	0	0
$145$	650.000	0.372273
$146$	0	0
$147$	−2724.00	−1.52838
$148$	0	0
$149$	1410.00	0.775246	0.387623	−	0.921818i	$-0.373296\pi$
$149$	1410.00	0.775246	0.387623	+	0.921818i	$0.373296\pi$
$150$	0	0
$151$	352.000	0.189704	0.0948522	−	0.995491i	$-0.469762\pi$
$151$	352.000	0.189704	0.0948522	+	0.995491i	$0.469762\pi$
$152$	0	0
$153$	−1430.00	−0.755612
$154$	0	0
$155$	200.000	0.103641
$156$	0	0
$157$	−374.000	−0.190118	−0.0950588	−	0.995472i	$-0.530304\pi$
$157$	−374.000	−0.190118	−0.0950588	+	0.995472i	$0.530304\pi$
$158$	0	0
$159$	−1032.00	−0.514735
$160$	0	0
$161$	−736.000	−0.360279
$162$	0	0
$163$	−308.000	−0.148003	−0.0740013	−	0.997258i	$-0.523577\pi$
$163$	−308.000	−0.148003	−0.0740013	+	0.997258i	$0.523577\pi$
$164$	0	0
$165$	−800.000	−0.377454
$166$	0	0
$167$	1072.00	0.496730	0.248365	−	0.968667i	$-0.420107\pi$
$167$	1072.00	0.496730	0.248365	+	0.968667i	$0.420107\pi$
$168$	0	0
$169$	2159.00	0.982704
$170$	0	0
$171$	−968.000	−0.432894
$172$	0	0
$173$	−3178.00	−1.39664	−0.698320	−	0.715785i	$-0.746069\pi$
$173$	−3178.00	−1.39664	−0.698320	+	0.715785i	$0.746069\pi$
$174$	0	0
$175$	800.000	0.345568
$176$	0	0
$177$	2448.00	1.03956
$178$	0	0
$179$	396.000	0.165354	0.0826772	−	0.996576i	$-0.473653\pi$
$179$	396.000	0.165354	0.0826772	+	0.996576i	$0.473653\pi$
$180$	0	0
$181$	2986.00	1.22623	0.613115	−	0.789994i	$-0.289916\pi$
$181$	2986.00	1.22623	0.613115	+	0.789994i	$0.289916\pi$
$182$	0	0
$183$	1464.00	0.591377
$184$	0	0
$185$	1670.00	0.663680
$186$	0	0
$187$	−5200.00	−2.03348
$188$	0	0
$189$	4864.00	1.87198
$190$	0	0
$191$	−4392.00	−1.66384	−0.831921	−	0.554894i	$-0.812759\pi$
$191$	−4392.00	−1.66384	−0.831921	+	0.554894i	$0.812759\pi$
$192$	0	0
$193$	−3054.00	−1.13902	−0.569512	−	0.821983i	$-0.692868\pi$
$193$	−3054.00	−1.13902	−0.569512	+	0.821983i	$0.692868\pi$
$194$	0	0
$195$	−1320.00	−0.484755
$196$	0	0
$197$	−3090.00	−1.11753	−0.558765	−	0.829326i	$-0.688725\pi$
$197$	−3090.00	−1.11753	−0.558765	+	0.829326i	$0.688725\pi$
$198$	0	0
$199$	−4608.00	−1.64147	−0.820735	−	0.571309i	$-0.806436\pi$
$199$	−4608.00	−1.64147	−0.820735	+	0.571309i	$0.806436\pi$
$200$	0	0
$201$	−1984.00	−0.696222
$202$	0	0
$203$	−4160.00	−1.43830
$204$	0	0
$205$	110.000	0.0374767
$206$	0	0
$207$	253.000	0.0849503
$208$	0	0
$209$	−3520.00	−1.16499
$210$	0	0
$211$	4324.00	1.41079	0.705394	−	0.708815i	$-0.250770\pi$
$211$	4324.00	1.41079	0.705394	+	0.708815i	$0.250770\pi$
$212$	0	0
$213$	992.000	0.319111
$214$	0	0
$215$	−1360.00	−0.431401
$216$	0	0
$217$	−1280.00	−0.400424
$218$	0	0
$219$	−3304.00	−1.01947
$220$	0	0
$221$	−8580.00	−2.61155
$222$	0	0
$223$	−2592.00	−0.778355	−0.389177	−	0.921163i	$-0.627241\pi$
$223$	−2592.00	−0.778355	−0.389177	+	0.921163i	$0.627241\pi$
$224$	0	0
$225$	−275.000	−0.0814815
$226$	0	0
$227$	−120.000	−0.0350867	−0.0175433	−	0.999846i	$-0.505585\pi$
$227$	−120.000	−0.0350867	−0.0175433	+	0.999846i	$0.505585\pi$
$228$	0	0
$229$	−5094.00	−1.46996	−0.734980	−	0.678088i	$-0.762809\pi$
$229$	−5094.00	−1.46996	−0.734980	+	0.678088i	$0.762809\pi$
$230$	0	0
$231$	5120.00	1.45832
$232$	0	0
$233$	−1478.00	−0.415567	−0.207783	−	0.978175i	$-0.566625\pi$
$233$	−1478.00	−0.415567	−0.207783	+	0.978175i	$0.566625\pi$
$234$	0	0
$235$	120.000	0.0333104
$236$	0	0
$237$	−1184.00	−0.324511
$238$	0	0
$239$	−6296.00	−1.70399	−0.851997	−	0.523547i	$-0.824608\pi$
$239$	−6296.00	−1.70399	−0.851997	+	0.523547i	$0.824608\pi$
$240$	0	0
$241$	−4454.00	−1.19049	−0.595243	−	0.803545i	$-0.702944\pi$
$241$	−4454.00	−1.19049	−0.595243	+	0.803545i	$0.702944\pi$
$242$	0	0
$243$	−2860.00	−0.755017
$244$	0	0
$245$	−3405.00	−0.887908
$246$	0	0
$247$	−5808.00	−1.49617
$248$	0	0
$249$	−5184.00	−1.31937
$250$	0	0
$251$	−528.000	−0.132777	−0.0663886	−	0.997794i	$-0.521148\pi$
$251$	−528.000	−0.132777	−0.0663886	+	0.997794i	$0.521148\pi$
$252$	0	0
$253$	920.000	0.228616
$254$	0	0
$255$	2600.00	0.638503
$256$	0	0
$257$	−1134.00	−0.275241	−0.137621	−	0.990485i	$-0.543945\pi$
$257$	−1134.00	−0.275241	−0.137621	+	0.990485i	$0.543945\pi$
$258$	0	0
$259$	−10688.0	−2.56417
$260$	0	0
$261$	1430.00	0.339137
$262$	0	0
$263$	−5304.00	−1.24357	−0.621785	−	0.783188i	$-0.713592\pi$
$263$	−5304.00	−1.24357	−0.621785	+	0.783188i	$0.713592\pi$
$264$	0	0
$265$	−1290.00	−0.299034
$266$	0	0
$267$	2584.00	0.592278
$268$	0	0
$269$	−50.0000	−0.0113329	−0.00566646	−	0.999984i	$-0.501804\pi$
$269$	−50.0000	−0.0113329	−0.00566646	+	0.999984i	$0.501804\pi$
$270$	0	0
$271$	2000.00	0.448308	0.224154	−	0.974554i	$-0.428038\pi$
$271$	2000.00	0.448308	0.224154	+	0.974554i	$0.428038\pi$
$272$	0	0
$273$	8448.00	1.87288
$274$	0	0
$275$	−1000.00	−0.219281
$276$	0	0
$277$	−2338.00	−0.507136	−0.253568	−	0.967318i	$-0.581604\pi$
$277$	−2338.00	−0.507136	−0.253568	+	0.967318i	$0.581604\pi$
$278$	0	0
$279$	440.000	0.0944162
$280$	0	0
$281$	−5142.00	−1.09162	−0.545811	−	0.837908i	$-0.683778\pi$
$281$	−5142.00	−1.09162	−0.545811	+	0.837908i	$0.683778\pi$
$282$	0	0
$283$	3888.00	0.816670	0.408335	−	0.912832i	$-0.366110\pi$
$283$	3888.00	0.816670	0.408335	+	0.912832i	$0.366110\pi$
$284$	0	0
$285$	1760.00	0.365801
$286$	0	0
$287$	−704.000	−0.144794
$288$	0	0
$289$	11987.0	2.43985
$290$	0	0
$291$	5752.00	1.15872
$292$	0	0
$293$	−2782.00	−0.554697	−0.277348	−	0.960769i	$-0.589456\pi$
$293$	−2782.00	−0.554697	−0.277348	+	0.960769i	$0.589456\pi$
$294$	0	0
$295$	3060.00	0.603933
$296$	0	0
$297$	−6080.00	−1.18787
$298$	0	0
$299$	1518.00	0.293606
$300$	0	0
$301$	8704.00	1.66674
$302$	0	0
$303$	3400.00	0.644637
$304$	0	0
$305$	1830.00	0.343559
$306$	0	0
$307$	−7172.00	−1.33331	−0.666657	−	0.745364i	$-0.732276\pi$
$307$	−7172.00	−1.33331	−0.666657	+	0.745364i	$0.732276\pi$
$308$	0	0
$309$	−5280.00	−0.972067
$310$	0	0
$311$	−6144.00	−1.12024	−0.560119	−	0.828412i	$-0.689245\pi$
$311$	−6144.00	−1.12024	−0.560119	+	0.828412i	$0.689245\pi$
$312$	0	0
$313$	−6598.00	−1.19150	−0.595752	−	0.803168i	$-0.703146\pi$
$313$	−6598.00	−1.19150	−0.595752	+	0.803168i	$0.703146\pi$
$314$	0	0
$315$	1760.00	0.314809
$316$	0	0
$317$	3558.00	0.630401	0.315201	−	0.949025i	$-0.397928\pi$
$317$	3558.00	0.630401	0.315201	+	0.949025i	$0.397928\pi$
$318$	0	0
$319$	5200.00	0.912677
$320$	0	0
$321$	288.000	0.0500766
$322$	0	0
$323$	11440.0	1.97071
$324$	0	0
$325$	−1650.00	−0.281617
$326$	0	0
$327$	−968.000	−0.163702
$328$	0	0
$329$	−768.000	−0.128697
$330$	0	0
$331$	780.000	0.129525	0.0647624	−	0.997901i	$-0.479371\pi$
$331$	780.000	0.129525	0.0647624	+	0.997901i	$0.479371\pi$
$332$	0	0
$333$	3674.00	0.604606
$334$	0	0
$335$	−2480.00	−0.404468
$336$	0	0
$337$	8218.00	1.32838	0.664188	−	0.747565i	$-0.268777\pi$
$337$	8218.00	1.32838	0.664188	+	0.747565i	$0.268777\pi$
$338$	0	0
$339$	−4072.00	−0.652391
$340$	0	0
$341$	1600.00	0.254090
$342$	0	0
$343$	10816.0	1.70265
$344$	0	0
$345$	−460.000	−0.0717843
$346$	0	0
$347$	−2844.00	−0.439982	−0.219991	−	0.975502i	$-0.570603\pi$
$347$	−2844.00	−0.439982	−0.219991	+	0.975502i	$0.570603\pi$
$348$	0	0
$349$	5582.00	0.856154	0.428077	−	0.903742i	$-0.359191\pi$
$349$	5582.00	0.856154	0.428077	+	0.903742i	$0.359191\pi$
$350$	0	0
$351$	−10032.0	−1.52555
$352$	0	0
$353$	5970.00	0.900145	0.450072	−	0.892992i	$-0.351398\pi$
$353$	5970.00	0.900145	0.450072	+	0.892992i	$0.351398\pi$
$354$	0	0
$355$	1240.00	0.185387
$356$	0	0
$357$	−16640.0	−2.46690
$358$	0	0
$359$	2464.00	0.362242	0.181121	−	0.983461i	$-0.442027\pi$
$359$	2464.00	0.362242	0.181121	+	0.983461i	$0.442027\pi$
$360$	0	0
$361$	885.000	0.129028
$362$	0	0
$363$	−1076.00	−0.155579
$364$	0	0
$365$	−4130.00	−0.592258
$366$	0	0
$367$	6344.00	0.902327	0.451164	−	0.892441i	$-0.351009\pi$
$367$	6344.00	0.902327	0.451164	+	0.892441i	$0.351009\pi$
$368$	0	0
$369$	242.000	0.0341410
$370$	0	0
$371$	8256.00	1.15534
$372$	0	0
$373$	−11102.0	−1.54113	−0.770563	−	0.637364i	$-0.780025\pi$
$373$	−11102.0	−1.54113	−0.770563	+	0.637364i	$0.780025\pi$
$374$	0	0
$375$	500.000	0.0688530
$376$	0	0
$377$	8580.00	1.17213
$378$	0	0
$379$	10016.0	1.35749	0.678743	−	0.734376i	$-0.262525\pi$
$379$	10016.0	1.35749	0.678743	+	0.734376i	$0.262525\pi$
$380$	0	0
$381$	480.000	0.0645437
$382$	0	0
$383$	−9776.00	−1.30426	−0.652128	−	0.758109i	$-0.726124\pi$
$383$	−9776.00	−1.30426	−0.652128	+	0.758109i	$0.726124\pi$
$384$	0	0
$385$	6400.00	0.847206
$386$	0	0
$387$	−2992.00	−0.393002
$388$	0	0
$389$	−1622.00	−0.211410	−0.105705	−	0.994398i	$-0.533710\pi$
$389$	−1622.00	−0.211410	−0.105705	+	0.994398i	$0.533710\pi$
$390$	0	0
$391$	−2990.00	−0.386728
$392$	0	0
$393$	5360.00	0.687980
$394$	0	0
$395$	−1480.00	−0.188524
$396$	0	0
$397$	−11202.0	−1.41615	−0.708076	−	0.706136i	$-0.750437\pi$
$397$	−11202.0	−1.41615	−0.708076	+	0.706136i	$0.750437\pi$
$398$	0	0
$399$	−11264.0	−1.41330
$400$	0	0
$401$	−10198.0	−1.26998	−0.634992	−	0.772518i	$-0.718997\pi$
$401$	−10198.0	−1.26998	−0.634992	+	0.772518i	$0.718997\pi$
$402$	0	0
$403$	2640.00	0.326322
$404$	0	0
$405$	1555.00	0.190787
$406$	0	0
$407$	13360.0	1.62710
$408$	0	0
$409$	12154.0	1.46938	0.734690	−	0.678403i	$-0.237328\pi$
$409$	12154.0	1.46938	0.734690	+	0.678403i	$0.237328\pi$
$410$	0	0
$411$	88.0000	0.0105614
$412$	0	0
$413$	−19584.0	−2.33333
$414$	0	0
$415$	−6480.00	−0.766484
$416$	0	0
$417$	10384.0	1.21944
$418$	0	0
$419$	−6152.00	−0.717291	−0.358645	−	0.933474i	$-0.616761\pi$
$419$	−6152.00	−0.717291	−0.358645	+	0.933474i	$0.616761\pi$
$420$	0	0
$421$	2450.00	0.283624	0.141812	−	0.989894i	$-0.454707\pi$
$421$	2450.00	0.283624	0.141812	+	0.989894i	$0.454707\pi$
$422$	0	0
$423$	264.000	0.0303454
$424$	0	0
$425$	3250.00	0.370937
$426$	0	0
$427$	−11712.0	−1.32736
$428$	0	0
$429$	−10560.0	−1.18844
$430$	0	0
$431$	−9048.00	−1.01120	−0.505600	−	0.862768i	$-0.668729\pi$
$431$	−9048.00	−1.01120	−0.505600	+	0.862768i	$0.668729\pi$
$432$	0	0
$433$	−14302.0	−1.58732	−0.793661	−	0.608361i	$-0.791827\pi$
$433$	−14302.0	−1.58732	−0.793661	+	0.608361i	$0.791827\pi$
$434$	0	0
$435$	−2600.00	−0.286576
$436$	0	0
$437$	−2024.00	−0.221558
$438$	0	0
$439$	7432.00	0.807995	0.403998	−	0.914760i	$-0.367620\pi$
$439$	7432.00	0.807995	0.403998	+	0.914760i	$0.367620\pi$
$440$	0	0
$441$	−7491.00	−0.808876
$442$	0	0
$443$	−1452.00	−0.155726	−0.0778630	−	0.996964i	$-0.524810\pi$
$443$	−1452.00	−0.155726	−0.0778630	+	0.996964i	$0.524810\pi$
$444$	0	0
$445$	3230.00	0.344082
$446$	0	0
$447$	−5640.00	−0.596785
$448$	0	0
$449$	−4910.00	−0.516074	−0.258037	−	0.966135i	$-0.583076\pi$
$449$	−4910.00	−0.516074	−0.258037	+	0.966135i	$0.583076\pi$
$450$	0	0
$451$	880.000	0.0918793
$452$	0	0
$453$	−1408.00	−0.146034
$454$	0	0
$455$	10560.0	1.08804
$456$	0	0
$457$	−12582.0	−1.28788	−0.643940	−	0.765076i	$-0.722701\pi$
$457$	−12582.0	−1.28788	−0.643940	+	0.765076i	$0.722701\pi$
$458$	0	0
$459$	19760.0	2.00941
$460$	0	0
$461$	9182.00	0.927654	0.463827	−	0.885926i	$-0.346476\pi$
$461$	9182.00	0.927654	0.463827	+	0.885926i	$0.346476\pi$
$462$	0	0
$463$	6728.00	0.675328	0.337664	−	0.941267i	$-0.390363\pi$
$463$	6728.00	0.675328	0.337664	+	0.941267i	$0.390363\pi$
$464$	0	0
$465$	−800.000	−0.0797830
$466$	0	0
$467$	−16544.0	−1.63933	−0.819663	−	0.572846i	$-0.805839\pi$
$467$	−16544.0	−1.63933	−0.819663	+	0.572846i	$0.805839\pi$
$468$	0	0
$469$	15872.0	1.56269
$470$	0	0
$471$	1496.00	0.146353
$472$	0	0
$473$	−10880.0	−1.05764
$474$	0	0
$475$	2200.00	0.212511
$476$	0	0
$477$	−2838.00	−0.272417
$478$	0	0
$479$	4352.00	0.415131	0.207566	−	0.978221i	$-0.433446\pi$
$479$	4352.00	0.415131	0.207566	+	0.978221i	$0.433446\pi$
$480$	0	0
$481$	22044.0	2.08965
$482$	0	0
$483$	2944.00	0.277343
$484$	0	0
$485$	7190.00	0.673157
$486$	0	0
$487$	6336.00	0.589551	0.294776	−	0.955566i	$-0.404755\pi$
$487$	6336.00	0.589551	0.294776	+	0.955566i	$0.404755\pi$
$488$	0	0
$489$	1232.00	0.113932
$490$	0	0
$491$	2036.00	0.187135	0.0935676	−	0.995613i	$-0.470173\pi$
$491$	2036.00	0.187135	0.0935676	+	0.995613i	$0.470173\pi$
$492$	0	0
$493$	−16900.0	−1.54389
$494$	0	0
$495$	−2200.00	−0.199763
$496$	0	0
$497$	−7936.00	−0.716254
$498$	0	0
$499$	9404.00	0.843649	0.421825	−	0.906677i	$-0.361390\pi$
$499$	9404.00	0.843649	0.421825	+	0.906677i	$0.361390\pi$
$500$	0	0
$501$	−4288.00	−0.382383
$502$	0	0
$503$	−7792.00	−0.690712	−0.345356	−	0.938472i	$-0.612242\pi$
$503$	−7792.00	−0.690712	−0.345356	+	0.938472i	$0.612242\pi$
$504$	0	0
$505$	4250.00	0.374500
$506$	0	0
$507$	−8636.00	−0.756486
$508$	0	0
$509$	2014.00	0.175381	0.0876906	−	0.996148i	$-0.472051\pi$
$509$	2014.00	0.175381	0.0876906	+	0.996148i	$0.472051\pi$
$510$	0	0
$511$	26432.0	2.28822
$512$	0	0
$513$	13376.0	1.15120
$514$	0	0
$515$	−6600.00	−0.564720
$516$	0	0
$517$	960.000	0.0816649
$518$	0	0
$519$	12712.0	1.07513
$520$	0	0
$521$	4530.00	0.380927	0.190463	−	0.981694i	$-0.439001\pi$
$521$	4530.00	0.380927	0.190463	+	0.981694i	$0.439001\pi$
$522$	0	0
$523$	−104.000	−0.00869522	−0.00434761	−	0.999991i	$-0.501384\pi$
$523$	−104.000	−0.00869522	−0.00434761	+	0.999991i	$0.501384\pi$
$524$	0	0
$525$	−3200.00	−0.266018
$526$	0	0
$527$	−5200.00	−0.429821
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	6732.00	0.550177
$532$	0	0
$533$	1452.00	0.117998
$534$	0	0
$535$	360.000	0.0290919
$536$	0	0
$537$	−1584.00	−0.127290
$538$	0	0
$539$	−27240.0	−2.17683
$540$	0	0
$541$	−4066.00	−0.323126	−0.161563	−	0.986862i	$-0.551653\pi$
$541$	−4066.00	−0.323126	−0.161563	+	0.986862i	$0.551653\pi$
$542$	0	0
$543$	−11944.0	−0.943952
$544$	0	0
$545$	−1210.00	−0.0951022
$546$	0	0
$547$	12532.0	0.979579	0.489789	−	0.871841i	$-0.337074\pi$
$547$	12532.0	0.979579	0.489789	+	0.871841i	$0.337074\pi$
$548$	0	0
$549$	4026.00	0.312979
$550$	0	0
$551$	−11440.0	−0.884502
$552$	0	0
$553$	9472.00	0.728373
$554$	0	0
$555$	−6680.00	−0.510901
$556$	0	0
$557$	−14014.0	−1.06605	−0.533027	−	0.846098i	$-0.678946\pi$
$557$	−14014.0	−1.06605	−0.533027	+	0.846098i	$0.678946\pi$
$558$	0	0
$559$	−17952.0	−1.35830
$560$	0	0
$561$	20800.0	1.56538
$562$	0	0
$563$	3720.00	0.278471	0.139236	−	0.990259i	$-0.455535\pi$
$563$	3720.00	0.278471	0.139236	+	0.990259i	$0.455535\pi$
$564$	0	0
$565$	−5090.00	−0.379005
$566$	0	0
$567$	−9952.00	−0.737116
$568$	0	0
$569$	−7758.00	−0.571586	−0.285793	−	0.958291i	$-0.592257\pi$
$569$	−7758.00	−0.571586	−0.285793	+	0.958291i	$0.592257\pi$
$570$	0	0
$571$	−3488.00	−0.255636	−0.127818	−	0.991798i	$-0.540797\pi$
$571$	−3488.00	−0.255636	−0.127818	+	0.991798i	$0.540797\pi$
$572$	0	0
$573$	17568.0	1.28083
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−3614.00	−0.260750	−0.130375	−	0.991465i	$-0.541618\pi$
$577$	−3614.00	−0.260750	−0.130375	+	0.991465i	$0.541618\pi$
$578$	0	0
$579$	12216.0	0.876821
$580$	0	0
$581$	41472.0	2.96136
$582$	0	0
$583$	−10320.0	−0.733123
$584$	0	0
$585$	−3630.00	−0.256550
$586$	0	0
$587$	1596.00	0.112221	0.0561107	−	0.998425i	$-0.482130\pi$
$587$	1596.00	0.112221	0.0561107	+	0.998425i	$0.482130\pi$
$588$	0	0
$589$	−3520.00	−0.246246
$590$	0	0
$591$	12360.0	0.860275
$592$	0	0
$593$	15490.0	1.07268	0.536339	−	0.844003i	$-0.319807\pi$
$593$	15490.0	1.07268	0.536339	+	0.844003i	$0.319807\pi$
$594$	0	0
$595$	−20800.0	−1.43314
$596$	0	0
$597$	18432.0	1.26360
$598$	0	0
$599$	7232.00	0.493308	0.246654	−	0.969104i	$-0.420669\pi$
$599$	7232.00	0.493308	0.246654	+	0.969104i	$0.420669\pi$
$600$	0	0
$601$	3130.00	0.212438	0.106219	−	0.994343i	$-0.466126\pi$
$601$	3130.00	0.212438	0.106219	+	0.994343i	$0.466126\pi$
$602$	0	0
$603$	−5456.00	−0.368467
$604$	0	0
$605$	−1345.00	−0.0903835
$606$	0	0
$607$	17744.0	1.18650	0.593251	−	0.805018i	$-0.297844\pi$
$607$	17744.0	1.18650	0.593251	+	0.805018i	$0.297844\pi$
$608$	0	0
$609$	16640.0	1.10720
$610$	0	0
$611$	1584.00	0.104880
$612$	0	0
$613$	18042.0	1.18876	0.594380	−	0.804185i	$-0.297398\pi$
$613$	18042.0	1.18876	0.594380	+	0.804185i	$0.297398\pi$
$614$	0	0
$615$	−440.000	−0.0288496
$616$	0	0
$617$	−8526.00	−0.556311	−0.278155	−	0.960536i	$-0.589723\pi$
$617$	−8526.00	−0.556311	−0.278155	+	0.960536i	$0.589723\pi$
$618$	0	0
$619$	12280.0	0.797375	0.398687	−	0.917087i	$-0.369466\pi$
$619$	12280.0	0.797375	0.398687	+	0.917087i	$0.369466\pi$
$620$	0	0
$621$	−3496.00	−0.225909
$622$	0	0
$623$	−20672.0	−1.32938
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	14080.0	0.896812
$628$	0	0
$629$	−43420.0	−2.75241
$630$	0	0
$631$	−3560.00	−0.224598	−0.112299	−	0.993674i	$-0.535821\pi$
$631$	−3560.00	−0.224598	−0.112299	+	0.993674i	$0.535821\pi$
$632$	0	0
$633$	−17296.0	−1.08603
$634$	0	0
$635$	600.000	0.0374965
$636$	0	0
$637$	−44946.0	−2.79564
$638$	0	0
$639$	2728.00	0.168886
$640$	0	0
$641$	−17414.0	−1.07303	−0.536515	−	0.843891i	$-0.680259\pi$
$641$	−17414.0	−1.07303	−0.536515	+	0.843891i	$0.680259\pi$
$642$	0	0
$643$	7016.00	0.430302	0.215151	−	0.976581i	$-0.430976\pi$
$643$	7016.00	0.430302	0.215151	+	0.976581i	$0.430976\pi$
$644$	0	0
$645$	5440.00	0.332093
$646$	0	0
$647$	−26696.0	−1.62215	−0.811073	−	0.584945i	$-0.801116\pi$
$647$	−26696.0	−1.62215	−0.811073	+	0.584945i	$0.801116\pi$
$648$	0	0
$649$	24480.0	1.48062
$650$	0	0
$651$	5120.00	0.308247
$652$	0	0
$653$	−66.0000	−0.00395525	−0.00197763	−	0.999998i	$-0.500629\pi$
$653$	−66.0000	−0.00395525	−0.00197763	+	0.999998i	$0.500629\pi$
$654$	0	0
$655$	6700.00	0.399680
$656$	0	0
$657$	−9086.00	−0.539541
$658$	0	0
$659$	1112.00	0.0657320	0.0328660	−	0.999460i	$-0.489537\pi$
$659$	1112.00	0.0657320	0.0328660	+	0.999460i	$0.489537\pi$
$660$	0	0
$661$	−17686.0	−1.04070	−0.520352	−	0.853952i	$-0.674199\pi$
$661$	−17686.0	−1.04070	−0.520352	+	0.853952i	$0.674199\pi$
$662$	0	0
$663$	34320.0	2.01037
$664$	0	0
$665$	−14080.0	−0.821051
$666$	0	0
$667$	2990.00	0.173573
$668$	0	0
$669$	10368.0	0.599178
$670$	0	0
$671$	14640.0	0.842282
$672$	0	0
$673$	−5182.00	−0.296807	−0.148404	−	0.988927i	$-0.547414\pi$
$673$	−5182.00	−0.296807	−0.148404	+	0.988927i	$0.547414\pi$
$674$	0	0
$675$	3800.00	0.216685
$676$	0	0
$677$	25506.0	1.44797	0.723985	−	0.689816i	$-0.242309\pi$
$677$	25506.0	1.44797	0.723985	+	0.689816i	$0.242309\pi$
$678$	0	0
$679$	−46016.0	−2.60078
$680$	0	0
$681$	480.000	0.0270098
$682$	0	0
$683$	−5012.00	−0.280789	−0.140394	−	0.990096i	$-0.544837\pi$
$683$	−5012.00	−0.280789	−0.140394	+	0.990096i	$0.544837\pi$
$684$	0	0
$685$	110.000	0.00613560
$686$	0	0
$687$	20376.0	1.13158
$688$	0	0
$689$	−17028.0	−0.941531
$690$	0	0
$691$	19988.0	1.10040	0.550202	−	0.835032i	$-0.314551\pi$
$691$	19988.0	1.10040	0.550202	+	0.835032i	$0.314551\pi$
$692$	0	0
$693$	14080.0	0.771796
$694$	0	0
$695$	12980.0	0.708431
$696$	0	0
$697$	−2860.00	−0.155424
$698$	0	0
$699$	5912.00	0.319903
$700$	0	0
$701$	−5918.00	−0.318858	−0.159429	−	0.987209i	$-0.550965\pi$
$701$	−5918.00	−0.318858	−0.159429	+	0.987209i	$0.550965\pi$
$702$	0	0
$703$	−29392.0	−1.57687
$704$	0	0
$705$	−480.000	−0.0256423
$706$	0	0
$707$	−27200.0	−1.44690
$708$	0	0
$709$	34626.0	1.83414	0.917071	−	0.398724i	$-0.130547\pi$
$709$	34626.0	1.83414	0.917071	+	0.398724i	$0.130547\pi$
$710$	0	0
$711$	−3256.00	−0.171743
$712$	0	0
$713$	920.000	0.0483230
$714$	0	0
$715$	−13200.0	−0.690422
$716$	0	0
$717$	25184.0	1.31173
$718$	0	0
$719$	31080.0	1.61208	0.806042	−	0.591858i	$-0.201606\pi$
$719$	31080.0	1.61208	0.806042	+	0.591858i	$0.201606\pi$
$720$	0	0
$721$	42240.0	2.18183
$722$	0	0
$723$	17816.0	0.916437
$724$	0	0
$725$	−3250.00	−0.166485
$726$	0	0
$727$	−8016.00	−0.408937	−0.204468	−	0.978873i	$-0.565547\pi$
$727$	−8016.00	−0.408937	−0.204468	+	0.978873i	$0.565547\pi$
$728$	0	0
$729$	19837.0	1.00782
$730$	0	0
$731$	35360.0	1.78911
$732$	0	0
$733$	−28806.0	−1.45153	−0.725767	−	0.687941i	$-0.758515\pi$
$733$	−28806.0	−1.45153	−0.725767	+	0.687941i	$0.758515\pi$
$734$	0	0
$735$	13620.0	0.683512
$736$	0	0
$737$	−19840.0	−0.991609
$738$	0	0
$739$	11052.0	0.550141	0.275071	−	0.961424i	$-0.411299\pi$
$739$	11052.0	0.550141	0.275071	+	0.961424i	$0.411299\pi$
$740$	0	0
$741$	23232.0	1.15175
$742$	0	0
$743$	29384.0	1.45087	0.725434	−	0.688292i	$-0.241639\pi$
$743$	29384.0	1.45087	0.725434	+	0.688292i	$0.241639\pi$
$744$	0	0
$745$	−7050.00	−0.346701
$746$	0	0
$747$	−14256.0	−0.698259
$748$	0	0
$749$	−2304.00	−0.112398
$750$	0	0
$751$	13056.0	0.634381	0.317191	−	0.948362i	$-0.397261\pi$
$751$	13056.0	0.634381	0.317191	+	0.948362i	$0.397261\pi$
$752$	0	0
$753$	2112.00	0.102212
$754$	0	0
$755$	−1760.00	−0.0848384
$756$	0	0
$757$	29290.0	1.40629	0.703146	−	0.711045i	$-0.251778\pi$
$757$	29290.0	1.40629	0.703146	+	0.711045i	$0.251778\pi$
$758$	0	0
$759$	−3680.00	−0.175989
$760$	0	0
$761$	7770.00	0.370121	0.185061	−	0.982727i	$-0.440752\pi$
$761$	7770.00	0.370121	0.185061	+	0.982727i	$0.440752\pi$
$762$	0	0
$763$	7744.00	0.367433
$764$	0	0
$765$	7150.00	0.337920
$766$	0	0
$767$	40392.0	1.90153
$768$	0	0
$769$	−9030.00	−0.423446	−0.211723	−	0.977330i	$-0.567907\pi$
$769$	−9030.00	−0.423446	−0.211723	+	0.977330i	$0.567907\pi$
$770$	0	0
$771$	4536.00	0.211881
$772$	0	0
$773$	−19574.0	−0.910773	−0.455387	−	0.890294i	$-0.650499\pi$
$773$	−19574.0	−0.910773	−0.455387	+	0.890294i	$0.650499\pi$
$774$	0	0
$775$	−1000.00	−0.0463498
$776$	0	0
$777$	42752.0	1.97390
$778$	0	0
$779$	−1936.00	−0.0890429
$780$	0	0
$781$	9920.00	0.454501
$782$	0	0
$783$	−19760.0	−0.901871
$784$	0	0
$785$	1870.00	0.0850231
$786$	0	0
$787$	−40136.0	−1.81791	−0.908954	−	0.416896i	$-0.863118\pi$
$787$	−40136.0	−1.81791	−0.908954	+	0.416896i	$0.863118\pi$
$788$	0	0
$789$	21216.0	0.957300
$790$	0	0
$791$	32576.0	1.46431
$792$	0	0
$793$	24156.0	1.08172
$794$	0	0
$795$	5160.00	0.230197
$796$	0	0
$797$	7010.00	0.311552	0.155776	−	0.987792i	$-0.450212\pi$
$797$	7010.00	0.311552	0.155776	+	0.987792i	$0.450212\pi$
$798$	0	0
$799$	−3120.00	−0.138145
$800$	0	0
$801$	7106.00	0.313456
$802$	0	0
$803$	−33040.0	−1.45200
$804$	0	0
$805$	3680.00	0.161122
$806$	0	0
$807$	200.000	0.00872408
$808$	0	0
$809$	970.000	0.0421550	0.0210775	−	0.999778i	$-0.493290\pi$
$809$	970.000	0.0421550	0.0210775	+	0.999778i	$0.493290\pi$
$810$	0	0
$811$	−17156.0	−0.742822	−0.371411	−	0.928469i	$-0.621126\pi$
$811$	−17156.0	−0.742822	−0.371411	+	0.928469i	$0.621126\pi$
$812$	0	0
$813$	−8000.00	−0.345107
$814$	0	0
$815$	1540.00	0.0661888
$816$	0	0
$817$	23936.0	1.02499
$818$	0	0
$819$	23232.0	0.991199
$820$	0	0
$821$	−34850.0	−1.48145	−0.740727	−	0.671806i	$-0.765519\pi$
$821$	−34850.0	−1.48145	−0.740727	+	0.671806i	$0.765519\pi$
$822$	0	0
$823$	−3112.00	−0.131807	−0.0659037	−	0.997826i	$-0.520993\pi$
$823$	−3112.00	−0.131807	−0.0659037	+	0.997826i	$0.520993\pi$
$824$	0	0
$825$	4000.00	0.168803
$826$	0	0
$827$	21832.0	0.917984	0.458992	−	0.888440i	$-0.348211\pi$
$827$	21832.0	0.917984	0.458992	+	0.888440i	$0.348211\pi$
$828$	0	0
$829$	−12002.0	−0.502831	−0.251415	−	0.967879i	$-0.580896\pi$
$829$	−12002.0	−0.502831	−0.251415	+	0.967879i	$0.580896\pi$
$830$	0	0
$831$	9352.00	0.390394
$832$	0	0
$833$	88530.0	3.68233
$834$	0	0
$835$	−5360.00	−0.222144
$836$	0	0
$837$	−6080.00	−0.251082
$838$	0	0
$839$	−936.000	−0.0385153	−0.0192576	−	0.999815i	$-0.506130\pi$
$839$	−936.000	−0.0385153	−0.0192576	+	0.999815i	$0.506130\pi$
$840$	0	0
$841$	−7489.00	−0.307065
$842$	0	0
$843$	20568.0	0.840332
$844$	0	0
$845$	−10795.0	−0.439478
$846$	0	0
$847$	8608.00	0.349202
$848$	0	0
$849$	−15552.0	−0.628673
$850$	0	0
$851$	7682.00	0.309443
$852$	0	0
$853$	1366.00	0.0548311	0.0274156	−	0.999624i	$-0.491272\pi$
$853$	1366.00	0.0548311	0.0274156	+	0.999624i	$0.491272\pi$
$854$	0	0
$855$	4840.00	0.193596
$856$	0	0
$857$	−5638.00	−0.224726	−0.112363	−	0.993667i	$-0.535842\pi$
$857$	−5638.00	−0.224726	−0.112363	+	0.993667i	$0.535842\pi$
$858$	0	0
$859$	−30812.0	−1.22386	−0.611928	−	0.790914i	$-0.709606\pi$
$859$	−30812.0	−1.22386	−0.611928	+	0.790914i	$0.709606\pi$
$860$	0	0
$861$	2816.00	0.111462
$862$	0	0
$863$	−9184.00	−0.362256	−0.181128	−	0.983460i	$-0.557975\pi$
$863$	−9184.00	−0.362256	−0.181128	+	0.983460i	$0.557975\pi$
$864$	0	0
$865$	15890.0	0.624597
$866$	0	0
$867$	−47948.0	−1.87820
$868$	0	0
$869$	−11840.0	−0.462192
$870$	0	0
$871$	−32736.0	−1.27350
$872$	0	0
$873$	15818.0	0.613240
$874$	0	0
$875$	−4000.00	−0.154542
$876$	0	0
$877$	−22746.0	−0.875801	−0.437901	−	0.899023i	$-0.644278\pi$
$877$	−22746.0	−0.875801	−0.437901	+	0.899023i	$0.644278\pi$
$878$	0	0
$879$	11128.0	0.427006
$880$	0	0
$881$	−30846.0	−1.17960	−0.589800	−	0.807549i	$-0.700794\pi$
$881$	−30846.0	−1.17960	−0.589800	+	0.807549i	$0.700794\pi$
$882$	0	0
$883$	3380.00	0.128818	0.0644089	−	0.997924i	$-0.479484\pi$
$883$	3380.00	0.128818	0.0644089	+	0.997924i	$0.479484\pi$
$884$	0	0
$885$	−12240.0	−0.464907
$886$	0	0
$887$	−13296.0	−0.503310	−0.251655	−	0.967817i	$-0.580975\pi$
$887$	−13296.0	−0.503310	−0.251655	+	0.967817i	$0.580975\pi$
$888$	0	0
$889$	−3840.00	−0.144870
$890$	0	0
$891$	12440.0	0.467739
$892$	0	0
$893$	−2112.00	−0.0791438
$894$	0	0
$895$	−1980.00	−0.0739487
$896$	0	0
$897$	−6072.00	−0.226018
$898$	0	0
$899$	5200.00	0.192914
$900$	0	0
$901$	33540.0	1.24015
$902$	0	0
$903$	−34816.0	−1.28306
$904$	0	0
$905$	−14930.0	−0.548387
$906$	0	0
$907$	19168.0	0.701723	0.350862	−	0.936427i	$-0.385889\pi$
$907$	19168.0	0.701723	0.350862	+	0.936427i	$0.385889\pi$
$908$	0	0
$909$	9350.00	0.341166
$910$	0	0
$911$	−7528.00	−0.273780	−0.136890	−	0.990586i	$-0.543711\pi$
$911$	−7528.00	−0.273780	−0.136890	+	0.990586i	$0.543711\pi$
$912$	0	0
$913$	−51840.0	−1.87914
$914$	0	0
$915$	−7320.00	−0.264472
$916$	0	0
$917$	−42880.0	−1.54419
$918$	0	0
$919$	1184.00	0.0424990	0.0212495	−	0.999774i	$-0.493236\pi$
$919$	1184.00	0.0424990	0.0212495	+	0.999774i	$0.493236\pi$
$920$	0	0
$921$	28688.0	1.02639
$922$	0	0
$923$	16368.0	0.583705
$924$	0	0
$925$	−8350.00	−0.296807
$926$	0	0
$927$	−14520.0	−0.514455
$928$	0	0
$929$	33810.0	1.19405	0.597024	−	0.802224i	$-0.296350\pi$
$929$	33810.0	1.19405	0.597024	+	0.802224i	$0.296350\pi$
$930$	0	0
$931$	59928.0	2.10962
$932$	0	0
$933$	24576.0	0.862360
$934$	0	0
$935$	26000.0	0.909402
$936$	0	0
$937$	23842.0	0.831253	0.415627	−	0.909535i	$-0.363562\pi$
$937$	23842.0	0.831253	0.415627	+	0.909535i	$0.363562\pi$
$938$	0	0
$939$	26392.0	0.917221
$940$	0	0
$941$	−40598.0	−1.40644	−0.703218	−	0.710974i	$-0.748254\pi$
$941$	−40598.0	−1.40644	−0.703218	+	0.710974i	$0.748254\pi$
$942$	0	0
$943$	506.000	0.0174736
$944$	0	0
$945$	−24320.0	−0.837174
$946$	0	0
$947$	16268.0	0.558225	0.279112	−	0.960258i	$-0.409960\pi$
$947$	16268.0	0.558225	0.279112	+	0.960258i	$0.409960\pi$
$948$	0	0
$949$	−54516.0	−1.86477
$950$	0	0
$951$	−14232.0	−0.485283
$952$	0	0
$953$	−27702.0	−0.941612	−0.470806	−	0.882237i	$-0.656037\pi$
$953$	−27702.0	−0.941612	−0.470806	+	0.882237i	$0.656037\pi$
$954$	0	0
$955$	21960.0	0.744093
$956$	0	0
$957$	−20800.0	−0.702579
$958$	0	0
$959$	−704.000	−0.0237053
$960$	0	0
$961$	−28191.0	−0.946293
$962$	0	0
$963$	792.000	0.0265024
$964$	0	0
$965$	15270.0	0.509387
$966$	0	0
$967$	6824.00	0.226934	0.113467	−	0.993542i	$-0.463804\pi$
$967$	6824.00	0.226934	0.113467	+	0.993542i	$0.463804\pi$
$968$	0	0
$969$	−45760.0	−1.51705
$970$	0	0
$971$	−30184.0	−0.997580	−0.498790	−	0.866723i	$-0.666222\pi$
$971$	−30184.0	−0.997580	−0.498790	+	0.866723i	$0.666222\pi$
$972$	0	0
$973$	−83072.0	−2.73707
$974$	0	0
$975$	6600.00	0.216789
$976$	0	0
$977$	22258.0	0.728860	0.364430	−	0.931231i	$-0.381264\pi$
$977$	22258.0	0.728860	0.364430	+	0.931231i	$0.381264\pi$
$978$	0	0
$979$	25840.0	0.843565
$980$	0	0
$981$	−2662.00	−0.0866372
$982$	0	0
$983$	37168.0	1.20598	0.602988	−	0.797750i	$-0.293977\pi$
$983$	37168.0	1.20598	0.602988	+	0.797750i	$0.293977\pi$
$984$	0	0
$985$	15450.0	0.499775
$986$	0	0
$987$	3072.00	0.0990708
$988$	0	0
$989$	−6256.00	−0.201142
$990$	0	0
$991$	46936.0	1.50451	0.752256	−	0.658871i	$-0.228966\pi$
$991$	46936.0	1.50451	0.752256	+	0.658871i	$0.228966\pi$
$992$	0	0
$993$	−3120.00	−0.0997082
$994$	0	0
$995$	23040.0	0.734088
$996$	0	0
$997$	50350.0	1.59940	0.799699	−	0.600401i	$-0.204992\pi$
$997$	50350.0	1.59940	0.799699	+	0.600401i	$0.204992\pi$
$998$	0	0
$999$	−50768.0	−1.60784

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.c.1.1		1
4.3	odd	2		115.4.a.a.1.1	✓	1
12.11	even	2		1035.4.a.b.1.1		1
20.3	even	4		575.4.b.e.24.1		2
20.7	even	4		575.4.b.e.24.2		2
20.19	odd	2		575.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.a.a.1.1	✓	1	4.3	odd	2
575.4.a.b.1.1		1	20.19	odd	2
575.4.b.e.24.1		2	20.3	even	4
575.4.b.e.24.2		2	20.7	even	4
1035.4.a.b.1.1		1	12.11	even	2
1840.4.a.c.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

Introduction

L-functions

Modular forms

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Fields

Representations

Groups

Database

Properties

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