LMFDB - Embedded newform 1200.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(1,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, 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("1200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1200 = 2^{4} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1200.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:	$70.8022920069$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 600)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1200.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-3.00000 q^{3} +19.0000 q^{7} +9.00000 q^{9} -22.0000 q^{11} +1.00000 q^{13} -58.0000 q^{17} +53.0000 q^{19} -57.0000 q^{21} -58.0000 q^{23} -27.0000 q^{27} +22.0000 q^{29} +35.0000 q^{31} +66.0000 q^{33} -270.000 q^{37} -3.00000 q^{39} -468.000 q^{41} +431.000 q^{43} +230.000 q^{47} +18.0000 q^{49} +174.000 q^{51} -159.000 q^{57} -446.000 q^{59} +127.000 q^{61} +171.000 q^{63} +811.000 q^{67} +174.000 q^{69} -36.0000 q^{71} +522.000 q^{73} -418.000 q^{77} -1368.00 q^{79} +81.0000 q^{81} +1138.00 q^{83} -66.0000 q^{87} +144.000 q^{89} +19.0000 q^{91} -105.000 q^{93} -1079.00 q^{97} -198.000 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	19.0000	1.02590	0.512952	−	0.858417i	$-0.328552\pi$
$7$	19.0000	1.02590	0.512952	+	0.858417i	$0.328552\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−22.0000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−22.0000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	1.00000	0.0213346	0.0106673	−	0.999943i	$-0.496604\pi$
$13$	1.00000	0.0213346	0.0106673	+	0.999943i	$0.496604\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−58.0000	−0.827474	−0.413737	−	0.910396i	$-0.635777\pi$
$17$	−58.0000	−0.827474	−0.413737	+	0.910396i	$0.635777\pi$
$18$	0	0
$19$	53.0000	0.639949	0.319975	−	0.947426i	$-0.396326\pi$
$19$	53.0000	0.639949	0.319975	+	0.947426i	$0.396326\pi$
$20$	0	0
$21$	−57.0000	−0.592306
$22$	0	0
$23$	−58.0000	−0.525819	−0.262909	−	0.964821i	$-0.584682\pi$
$23$	−58.0000	−0.525819	−0.262909	+	0.964821i	$0.584682\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	22.0000	0.140872	0.0704362	−	0.997516i	$-0.477561\pi$
$29$	22.0000	0.140872	0.0704362	+	0.997516i	$0.477561\pi$
$30$	0	0
$31$	35.0000	0.202780	0.101390	−	0.994847i	$-0.467671\pi$
$31$	35.0000	0.202780	0.101390	+	0.994847i	$0.467671\pi$
$32$	0	0
$33$	66.0000	0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−270.000	−1.19967	−0.599834	−	0.800124i	$-0.704767\pi$
$37$	−270.000	−1.19967	−0.599834	+	0.800124i	$0.704767\pi$
$38$	0	0
$39$	−3.00000	−0.0123176
$40$	0	0
$41$	−468.000	−1.78267	−0.891333	−	0.453349i	$-0.850229\pi$
$41$	−468.000	−1.78267	−0.891333	+	0.453349i	$0.850229\pi$
$42$	0	0
$43$	431.000	1.52853	0.764266	−	0.644901i	$-0.223101\pi$
$43$	431.000	1.52853	0.764266	+	0.644901i	$0.223101\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	230.000	0.713807	0.356904	−	0.934141i	$-0.383832\pi$
$47$	230.000	0.713807	0.356904	+	0.934141i	$0.383832\pi$
$48$	0	0
$49$	18.0000	0.0524781
$50$	0	0
$51$	174.000	0.477743
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−159.000	−0.369475
$58$	0	0
$59$	−446.000	−0.984140	−0.492070	−	0.870556i	$-0.663760\pi$
$59$	−446.000	−0.984140	−0.492070	+	0.870556i	$0.663760\pi$
$60$	0	0
$61$	127.000	0.266569	0.133284	−	0.991078i	$-0.457448\pi$
$61$	127.000	0.266569	0.133284	+	0.991078i	$0.457448\pi$
$62$	0	0
$63$	171.000	0.341968
$64$	0	0
$65$	0	0
$66$	0	0
$67$	811.000	1.47880	0.739399	−	0.673268i	$-0.235110\pi$
$67$	811.000	1.47880	0.739399	+	0.673268i	$0.235110\pi$
$68$	0	0
$69$	174.000	0.303582
$70$	0	0
$71$	−36.0000	−0.0601748	−0.0300874	−	0.999547i	$-0.509579\pi$
$71$	−36.0000	−0.0601748	−0.0300874	+	0.999547i	$0.509579\pi$
$72$	0	0
$73$	522.000	0.836924	0.418462	−	0.908234i	$-0.362569\pi$
$73$	522.000	0.836924	0.418462	+	0.908234i	$0.362569\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−418.000	−0.618643
$78$	0	0
$79$	−1368.00	−1.94825	−0.974127	−	0.226002i	$-0.927434\pi$
$79$	−1368.00	−1.94825	−0.974127	+	0.226002i	$0.927434\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1138.00	1.50496	0.752480	−	0.658615i	$-0.228857\pi$
$83$	1138.00	1.50496	0.752480	+	0.658615i	$0.228857\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−66.0000	−0.0813327
$88$	0	0
$89$	144.000	0.171505	0.0857526	−	0.996316i	$-0.472671\pi$
$89$	144.000	0.171505	0.0857526	+	0.996316i	$0.472671\pi$
$90$	0	0
$91$	19.0000	0.0218873
$92$	0	0
$93$	−105.000	−0.117075
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1079.00	−1.12944	−0.564721	−	0.825282i	$-0.691016\pi$
$97$	−1079.00	−1.12944	−0.564721	+	0.825282i	$0.691016\pi$
$98$	0	0
$99$	−198.000	−0.201008
$100$	0	0
$101$	−1440.00	−1.41867	−0.709333	−	0.704873i	$-0.751004\pi$
$101$	−1440.00	−1.41867	−0.709333	+	0.704873i	$0.751004\pi$
$102$	0	0
$103$	−124.000	−0.118622	−0.0593111	−	0.998240i	$-0.518890\pi$
$103$	−124.000	−0.118622	−0.0593111	+	0.998240i	$0.518890\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	432.000	0.390309	0.195154	−	0.980773i	$-0.437479\pi$
$107$	432.000	0.390309	0.195154	+	0.980773i	$0.437479\pi$
$108$	0	0
$109$	701.000	0.615997	0.307998	−	0.951387i	$-0.400341\pi$
$109$	701.000	0.615997	0.307998	+	0.951387i	$0.400341\pi$
$110$	0	0
$111$	810.000	0.692629
$112$	0	0
$113$	−1044.00	−0.869126	−0.434563	−	0.900641i	$-0.643097\pi$
$113$	−1044.00	−0.869126	−0.434563	+	0.900641i	$0.643097\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	9.00000	0.00711154
$118$	0	0
$119$	−1102.00	−0.848909
$120$	0	0
$121$	−847.000	−0.636364
$122$	0	0
$123$	1404.00	1.02922
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−504.000	−0.352148	−0.176074	−	0.984377i	$-0.556340\pi$
$127$	−504.000	−0.352148	−0.176074	+	0.984377i	$0.556340\pi$
$128$	0	0
$129$	−1293.00	−0.882498
$130$	0	0
$131$	−72.0000	−0.0480204	−0.0240102	−	0.999712i	$-0.507643\pi$
$131$	−72.0000	−0.0480204	−0.0240102	+	0.999712i	$0.507643\pi$
$132$	0	0
$133$	1007.00	0.656526
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1850.00	−1.15369	−0.576847	−	0.816852i	$-0.695717\pi$
$137$	−1850.00	−1.15369	−0.576847	+	0.816852i	$0.695717\pi$
$138$	0	0
$139$	−1836.00	−1.12034	−0.560171	−	0.828377i	$-0.689265\pi$
$139$	−1836.00	−1.12034	−0.560171	+	0.828377i	$0.689265\pi$
$140$	0	0
$141$	−690.000	−0.412117
$142$	0	0
$143$	−22.0000	−0.0128653
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−54.0000	−0.0302983
$148$	0	0
$149$	−2398.00	−1.31847	−0.659234	−	0.751938i	$-0.729119\pi$
$149$	−2398.00	−1.31847	−0.659234	+	0.751938i	$0.729119\pi$
$150$	0	0
$151$	−1871.00	−1.00834	−0.504172	−	0.863604i	$-0.668202\pi$
$151$	−1871.00	−1.00834	−0.504172	+	0.863604i	$0.668202\pi$
$152$	0	0
$153$	−522.000	−0.275825
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3293.00	−1.67395	−0.836975	−	0.547242i	$-0.815678\pi$
$157$	−3293.00	−1.67395	−0.836975	+	0.547242i	$0.815678\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1102.00	−0.539440
$162$	0	0
$163$	−883.000	−0.424306	−0.212153	−	0.977236i	$-0.568048\pi$
$163$	−883.000	−0.424306	−0.212153	+	0.977236i	$0.568048\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4104.00	−1.90166	−0.950830	−	0.309715i	$-0.899766\pi$
$167$	−4104.00	−1.90166	−0.950830	+	0.309715i	$0.899766\pi$
$168$	0	0
$169$	−2196.00	−0.999545
$170$	0	0
$171$	477.000	0.213316
$172$	0	0
$173$	1706.00	0.749739	0.374869	−	0.927078i	$-0.377688\pi$
$173$	1706.00	0.749739	0.374869	+	0.927078i	$0.377688\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1338.00	0.568193
$178$	0	0
$179$	−662.000	−0.276426	−0.138213	−	0.990403i	$-0.544136\pi$
$179$	−662.000	−0.276426	−0.138213	+	0.990403i	$0.544136\pi$
$180$	0	0
$181$	4121.00	1.69233	0.846164	−	0.532922i	$-0.178906\pi$
$181$	4121.00	1.69233	0.846164	+	0.532922i	$0.178906\pi$
$182$	0	0
$183$	−381.000	−0.153903
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1276.00	0.498986
$188$	0	0
$189$	−513.000	−0.197435
$190$	0	0
$191$	958.000	0.362924	0.181462	−	0.983398i	$-0.441917\pi$
$191$	958.000	0.362924	0.181462	+	0.983398i	$0.441917\pi$
$192$	0	0
$193$	3187.00	1.18863	0.594314	−	0.804233i	$-0.297424\pi$
$193$	3187.00	1.18863	0.594314	+	0.804233i	$0.297424\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2282.00	−0.825308	−0.412654	−	0.910888i	$-0.635398\pi$
$197$	−2282.00	−0.825308	−0.412654	+	0.910888i	$0.635398\pi$
$198$	0	0
$199$	−1043.00	−0.371539	−0.185770	−	0.982593i	$-0.559478\pi$
$199$	−1043.00	−0.371539	−0.185770	+	0.982593i	$0.559478\pi$
$200$	0	0
$201$	−2433.00	−0.853784
$202$	0	0
$203$	418.000	0.144521
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−522.000	−0.175273
$208$	0	0
$209$	−1166.00	−0.385904
$210$	0	0
$211$	−4139.00	−1.35043	−0.675214	−	0.737621i	$-0.735949\pi$
$211$	−4139.00	−1.35043	−0.675214	+	0.737621i	$0.735949\pi$
$212$	0	0
$213$	108.000	0.0347420
$214$	0	0
$215$	0	0
$216$	0	0
$217$	665.000	0.208033
$218$	0	0
$219$	−1566.00	−0.483199
$220$	0	0
$221$	−58.0000	−0.0176539
$222$	0	0
$223$	−413.000	−0.124020	−0.0620101	−	0.998076i	$-0.519751\pi$
$223$	−413.000	−0.124020	−0.0620101	+	0.998076i	$0.519751\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5652.00	−1.65258	−0.826292	−	0.563242i	$-0.809554\pi$
$227$	−5652.00	−1.65258	−0.826292	+	0.563242i	$0.809554\pi$
$228$	0	0
$229$	−4391.00	−1.26710	−0.633549	−	0.773703i	$-0.718403\pi$
$229$	−4391.00	−1.26710	−0.633549	+	0.773703i	$0.718403\pi$
$230$	0	0
$231$	1254.00	0.357174
$232$	0	0
$233$	−2052.00	−0.576957	−0.288479	−	0.957486i	$-0.593149\pi$
$233$	−2052.00	−0.576957	−0.288479	+	0.957486i	$0.593149\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4104.00	1.12482
$238$	0	0
$239$	4320.00	1.16919	0.584597	−	0.811324i	$-0.301252\pi$
$239$	4320.00	1.16919	0.584597	+	0.811324i	$0.301252\pi$
$240$	0	0
$241$	4265.00	1.13997	0.569985	−	0.821655i	$-0.306949\pi$
$241$	4265.00	1.13997	0.569985	+	0.821655i	$0.306949\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	53.0000	0.0136531
$248$	0	0
$249$	−3414.00	−0.868889
$250$	0	0
$251$	2412.00	0.606550	0.303275	−	0.952903i	$-0.401920\pi$
$251$	2412.00	0.606550	0.303275	+	0.952903i	$0.401920\pi$
$252$	0	0
$253$	1276.00	0.317081
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6948.00	1.68640	0.843199	−	0.537601i	$-0.180669\pi$
$257$	6948.00	1.68640	0.843199	+	0.537601i	$0.180669\pi$
$258$	0	0
$259$	−5130.00	−1.23074
$260$	0	0
$261$	198.000	0.0469574
$262$	0	0
$263$	3564.00	0.835611	0.417805	−	0.908537i	$-0.362799\pi$
$263$	3564.00	0.835611	0.417805	+	0.908537i	$0.362799\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−432.000	−0.0990186
$268$	0	0
$269$	−1534.00	−0.347694	−0.173847	−	0.984773i	$-0.555620\pi$
$269$	−1534.00	−0.347694	−0.173847	+	0.984773i	$0.555620\pi$
$270$	0	0
$271$	−7704.00	−1.72688	−0.863440	−	0.504451i	$-0.831695\pi$
$271$	−7704.00	−1.72688	−0.863440	+	0.504451i	$0.831695\pi$
$272$	0	0
$273$	−57.0000	−0.0126366
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4877.00	1.05787	0.528936	−	0.848662i	$-0.322591\pi$
$277$	4877.00	1.05787	0.528936	+	0.848662i	$0.322591\pi$
$278$	0	0
$279$	315.000	0.0675934
$280$	0	0
$281$	−3758.00	−0.797806	−0.398903	−	0.916993i	$-0.630609\pi$
$281$	−3758.00	−0.797806	−0.398903	+	0.916993i	$0.630609\pi$
$282$	0	0
$283$	−935.000	−0.196396	−0.0981978	−	0.995167i	$-0.531308\pi$
$283$	−935.000	−0.196396	−0.0981978	+	0.995167i	$0.531308\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8892.00	−1.82884
$288$	0	0
$289$	−1549.00	−0.315286
$290$	0	0
$291$	3237.00	0.652084
$292$	0	0
$293$	6214.00	1.23900	0.619498	−	0.784998i	$-0.287336\pi$
$293$	6214.00	1.23900	0.619498	+	0.784998i	$0.287336\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	594.000	0.116052
$298$	0	0
$299$	−58.0000	−0.0112181
$300$	0	0
$301$	8189.00	1.56813
$302$	0	0
$303$	4320.00	0.819068
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3905.00	0.725961	0.362981	−	0.931797i	$-0.381759\pi$
$307$	3905.00	0.725961	0.362981	+	0.931797i	$0.381759\pi$
$308$	0	0
$309$	372.000	0.0684865
$310$	0	0
$311$	−9662.00	−1.76168	−0.880839	−	0.473416i	$-0.843021\pi$
$311$	−9662.00	−1.76168	−0.880839	+	0.473416i	$0.843021\pi$
$312$	0	0
$313$	5147.00	0.929475	0.464737	−	0.885449i	$-0.346149\pi$
$313$	5147.00	0.929475	0.464737	+	0.885449i	$0.346149\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9216.00	−1.63288	−0.816439	−	0.577432i	$-0.804055\pi$
$317$	−9216.00	−1.63288	−0.816439	+	0.577432i	$0.804055\pi$
$318$	0	0
$319$	−484.000	−0.0849492
$320$	0	0
$321$	−1296.00	−0.225345
$322$	0	0
$323$	−3074.00	−0.529542
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2103.00	−0.355646
$328$	0	0
$329$	4370.00	0.732298
$330$	0	0
$331$	−2196.00	−0.364662	−0.182331	−	0.983237i	$-0.558364\pi$
$331$	−2196.00	−0.364662	−0.182331	+	0.983237i	$0.558364\pi$
$332$	0	0
$333$	−2430.00	−0.399889
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2521.00	0.407500	0.203750	−	0.979023i	$-0.434687\pi$
$337$	2521.00	0.407500	0.203750	+	0.979023i	$0.434687\pi$
$338$	0	0
$339$	3132.00	0.501790
$340$	0	0
$341$	−770.000	−0.122281
$342$	0	0
$343$	−6175.00	−0.972066
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7034.00	−1.08820	−0.544099	−	0.839021i	$-0.683129\pi$
$347$	−7034.00	−1.08820	−0.544099	+	0.839021i	$0.683129\pi$
$348$	0	0
$349$	7362.00	1.12917	0.564583	−	0.825376i	$-0.309037\pi$
$349$	7362.00	1.12917	0.564583	+	0.825376i	$0.309037\pi$
$350$	0	0
$351$	−27.0000	−0.00410585
$352$	0	0
$353$	−9382.00	−1.41460	−0.707300	−	0.706914i	$-0.750087\pi$
$353$	−9382.00	−1.41460	−0.707300	+	0.706914i	$0.750087\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3306.00	0.490118
$358$	0	0
$359$	10116.0	1.48719	0.743596	−	0.668629i	$-0.233119\pi$
$359$	10116.0	1.48719	0.743596	+	0.668629i	$0.233119\pi$
$360$	0	0
$361$	−4050.00	−0.590465
$362$	0	0
$363$	2541.00	0.367405
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3277.00	0.466098	0.233049	−	0.972465i	$-0.425130\pi$
$367$	3277.00	0.466098	0.233049	+	0.972465i	$0.425130\pi$
$368$	0	0
$369$	−4212.00	−0.594222
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10891.0	1.51184	0.755918	−	0.654667i	$-0.227191\pi$
$373$	10891.0	1.51184	0.755918	+	0.654667i	$0.227191\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	22.0000	0.00300546
$378$	0	0
$379$	2591.00	0.351163	0.175581	−	0.984465i	$-0.443819\pi$
$379$	2591.00	0.351163	0.175581	+	0.984465i	$0.443819\pi$
$380$	0	0
$381$	1512.00	0.203313
$382$	0	0
$383$	612.000	0.0816494	0.0408247	−	0.999166i	$-0.487001\pi$
$383$	612.000	0.0816494	0.0408247	+	0.999166i	$0.487001\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3879.00	0.509511
$388$	0	0
$389$	3708.00	0.483298	0.241649	−	0.970364i	$-0.422312\pi$
$389$	3708.00	0.483298	0.241649	+	0.970364i	$0.422312\pi$
$390$	0	0
$391$	3364.00	0.435102
$392$	0	0
$393$	216.000	0.0277246
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3833.00	−0.484566	−0.242283	−	0.970206i	$-0.577896\pi$
$397$	−3833.00	−0.484566	−0.242283	+	0.970206i	$0.577896\pi$
$398$	0	0
$399$	−3021.00	−0.379046
$400$	0	0
$401$	−9288.00	−1.15666	−0.578330	−	0.815803i	$-0.696295\pi$
$401$	−9288.00	−1.15666	−0.578330	+	0.815803i	$0.696295\pi$
$402$	0	0
$403$	35.0000	0.00432624
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5940.00	0.723427
$408$	0	0
$409$	−755.000	−0.0912771	−0.0456386	−	0.998958i	$-0.514532\pi$
$409$	−755.000	−0.0912771	−0.0456386	+	0.998958i	$0.514532\pi$
$410$	0	0
$411$	5550.00	0.666086
$412$	0	0
$413$	−8474.00	−1.00963
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5508.00	0.646830
$418$	0	0
$419$	9576.00	1.11651	0.558256	−	0.829669i	$-0.311471\pi$
$419$	9576.00	1.11651	0.558256	+	0.829669i	$0.311471\pi$
$420$	0	0
$421$	9414.00	1.08981	0.544905	−	0.838498i	$-0.316566\pi$
$421$	9414.00	1.08981	0.544905	+	0.838498i	$0.316566\pi$
$422$	0	0
$423$	2070.00	0.237936
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2413.00	0.273474
$428$	0	0
$429$	66.0000	0.00742776
$430$	0	0
$431$	2254.00	0.251906	0.125953	−	0.992036i	$-0.459801\pi$
$431$	2254.00	0.251906	0.125953	+	0.992036i	$0.459801\pi$
$432$	0	0
$433$	−6301.00	−0.699323	−0.349661	−	0.936876i	$-0.613703\pi$
$433$	−6301.00	−0.699323	−0.349661	+	0.936876i	$0.613703\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3074.00	−0.336497
$438$	0	0
$439$	3779.00	0.410847	0.205423	−	0.978673i	$-0.434143\pi$
$439$	3779.00	0.410847	0.205423	+	0.978673i	$0.434143\pi$
$440$	0	0
$441$	162.000	0.0174927
$442$	0	0
$443$	−7632.00	−0.818527	−0.409263	−	0.912416i	$-0.634214\pi$
$443$	−7632.00	−0.818527	−0.409263	+	0.912416i	$0.634214\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	7194.00	0.761218
$448$	0	0
$449$	2988.00	0.314059	0.157029	−	0.987594i	$-0.449808\pi$
$449$	2988.00	0.314059	0.157029	+	0.987594i	$0.449808\pi$
$450$	0	0
$451$	10296.0	1.07499
$452$	0	0
$453$	5613.00	0.582167
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1370.00	−0.140232	−0.0701159	−	0.997539i	$-0.522337\pi$
$457$	−1370.00	−0.140232	−0.0701159	+	0.997539i	$0.522337\pi$
$458$	0	0
$459$	1566.00	0.159248
$460$	0	0
$461$	−7992.00	−0.807429	−0.403714	−	0.914885i	$-0.632281\pi$
$461$	−7992.00	−0.807429	−0.403714	+	0.914885i	$0.632281\pi$
$462$	0	0
$463$	3096.00	0.310763	0.155382	−	0.987855i	$-0.450339\pi$
$463$	3096.00	0.310763	0.155382	+	0.987855i	$0.450339\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8338.00	0.826203	0.413101	−	0.910685i	$-0.364446\pi$
$467$	8338.00	0.826203	0.413101	+	0.910685i	$0.364446\pi$
$468$	0	0
$469$	15409.0	1.51710
$470$	0	0
$471$	9879.00	0.966455
$472$	0	0
$473$	−9482.00	−0.921740
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4918.00	−0.469121	−0.234561	−	0.972101i	$-0.575365\pi$
$479$	−4918.00	−0.469121	−0.234561	+	0.972101i	$0.575365\pi$
$480$	0	0
$481$	−270.000	−0.0255945
$482$	0	0
$483$	3306.00	0.311446
$484$	0	0
$485$	0	0
$486$	0	0
$487$	19637.0	1.82718	0.913591	−	0.406635i	$-0.133298\pi$
$487$	19637.0	1.82718	0.913591	+	0.406635i	$0.133298\pi$
$488$	0	0
$489$	2649.00	0.244973
$490$	0	0
$491$	−3096.00	−0.284563	−0.142282	−	0.989826i	$-0.545444\pi$
$491$	−3096.00	−0.284563	−0.142282	+	0.989826i	$0.545444\pi$
$492$	0	0
$493$	−1276.00	−0.116568
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−684.000	−0.0617336
$498$	0	0
$499$	−6875.00	−0.616768	−0.308384	−	0.951262i	$-0.599788\pi$
$499$	−6875.00	−0.616768	−0.308384	+	0.951262i	$0.599788\pi$
$500$	0	0
$501$	12312.0	1.09792
$502$	0	0
$503$	−11268.0	−0.998838	−0.499419	−	0.866361i	$-0.666453\pi$
$503$	−11268.0	−0.998838	−0.499419	+	0.866361i	$0.666453\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6588.00	0.577087
$508$	0	0
$509$	16078.0	1.40009	0.700044	−	0.714100i	$-0.253164\pi$
$509$	16078.0	1.40009	0.700044	+	0.714100i	$0.253164\pi$
$510$	0	0
$511$	9918.00	0.858604
$512$	0	0
$513$	−1431.00	−0.123158
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5060.00	−0.430442
$518$	0	0
$519$	−5118.00	−0.432862
$520$	0	0
$521$	−17366.0	−1.46030	−0.730152	−	0.683285i	$-0.760551\pi$
$521$	−17366.0	−1.46030	−0.730152	+	0.683285i	$0.760551\pi$
$522$	0	0
$523$	4913.00	0.410766	0.205383	−	0.978682i	$-0.434156\pi$
$523$	4913.00	0.410766	0.205383	+	0.978682i	$0.434156\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2030.00	−0.167795
$528$	0	0
$529$	−8803.00	−0.723514
$530$	0	0
$531$	−4014.00	−0.328047
$532$	0	0
$533$	−468.000	−0.0380325
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1986.00	0.159594
$538$	0	0
$539$	−396.000	−0.0316455
$540$	0	0
$541$	17605.0	1.39907	0.699536	−	0.714597i	$-0.253390\pi$
$541$	17605.0	1.39907	0.699536	+	0.714597i	$0.253390\pi$
$542$	0	0
$543$	−12363.0	−0.977067
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14560.0	1.13810	0.569050	−	0.822303i	$-0.307311\pi$
$547$	14560.0	1.13810	0.569050	+	0.822303i	$0.307311\pi$
$548$	0	0
$549$	1143.00	0.0888562
$550$	0	0
$551$	1166.00	0.0901511
$552$	0	0
$553$	−25992.0	−1.99872
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−20614.0	−1.56812	−0.784060	−	0.620685i	$-0.786855\pi$
$557$	−20614.0	−1.56812	−0.784060	+	0.620685i	$0.786855\pi$
$558$	0	0
$559$	431.000	0.0326107
$560$	0	0
$561$	−3828.00	−0.288090
$562$	0	0
$563$	−3442.00	−0.257661	−0.128830	−	0.991667i	$-0.541122\pi$
$563$	−3442.00	−0.257661	−0.128830	+	0.991667i	$0.541122\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1539.00	0.113989
$568$	0	0
$569$	22082.0	1.62693	0.813467	−	0.581611i	$-0.197577\pi$
$569$	22082.0	1.62693	0.813467	+	0.581611i	$0.197577\pi$
$570$	0	0
$571$	−451.000	−0.0330539	−0.0165269	−	0.999863i	$-0.505261\pi$
$571$	−451.000	−0.0330539	−0.0165269	+	0.999863i	$0.505261\pi$
$572$	0	0
$573$	−2874.00	−0.209534
$574$	0	0
$575$	0	0
$576$	0	0
$577$	20987.0	1.51421	0.757106	−	0.653292i	$-0.226613\pi$
$577$	20987.0	1.51421	0.757106	+	0.653292i	$0.226613\pi$
$578$	0	0
$579$	−9561.00	−0.686255
$580$	0	0
$581$	21622.0	1.54394
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3888.00	−0.273381	−0.136691	−	0.990614i	$-0.543647\pi$
$587$	−3888.00	−0.273381	−0.136691	+	0.990614i	$0.543647\pi$
$588$	0	0
$589$	1855.00	0.129769
$590$	0	0
$591$	6846.00	0.476492
$592$	0	0
$593$	−11268.0	−0.780306	−0.390153	−	0.920750i	$-0.627578\pi$
$593$	−11268.0	−0.780306	−0.390153	+	0.920750i	$0.627578\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3129.00	0.214508
$598$	0	0
$599$	3996.00	0.272575	0.136287	−	0.990669i	$-0.456483\pi$
$599$	3996.00	0.272575	0.136287	+	0.990669i	$0.456483\pi$
$600$	0	0
$601$	−24965.0	−1.69442	−0.847208	−	0.531262i	$-0.821718\pi$
$601$	−24965.0	−1.69442	−0.847208	+	0.531262i	$0.821718\pi$
$602$	0	0
$603$	7299.00	0.492932
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−4176.00	−0.279240	−0.139620	−	0.990205i	$-0.544588\pi$
$607$	−4176.00	−0.279240	−0.139620	+	0.990205i	$0.544588\pi$
$608$	0	0
$609$	−1254.00	−0.0834395
$610$	0	0
$611$	230.000	0.0152288
$612$	0	0
$613$	−9558.00	−0.629762	−0.314881	−	0.949131i	$-0.601965\pi$
$613$	−9558.00	−0.629762	−0.314881	+	0.949131i	$0.601965\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9000.00	−0.587239	−0.293619	−	0.955922i	$-0.594860\pi$
$617$	−9000.00	−0.587239	−0.293619	+	0.955922i	$0.594860\pi$
$618$	0	0
$619$	15625.0	1.01457	0.507287	−	0.861777i	$-0.330648\pi$
$619$	15625.0	1.01457	0.507287	+	0.861777i	$0.330648\pi$
$620$	0	0
$621$	1566.00	0.101194
$622$	0	0
$623$	2736.00	0.175948
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3498.00	0.222802
$628$	0	0
$629$	15660.0	0.992695
$630$	0	0
$631$	31175.0	1.96681	0.983405	−	0.181424i	$-0.0580705\pi$
$631$	31175.0	1.96681	0.983405	+	0.181424i	$0.0580705\pi$
$632$	0	0
$633$	12417.0	0.779671
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.0000	0.00111960
$638$	0	0
$639$	−324.000	−0.0200583
$640$	0	0
$641$	−6732.00	−0.414817	−0.207409	−	0.978254i	$-0.566503\pi$
$641$	−6732.00	−0.414817	−0.207409	+	0.978254i	$0.566503\pi$
$642$	0	0
$643$	6228.00	0.381973	0.190986	−	0.981593i	$-0.438831\pi$
$643$	6228.00	0.381973	0.190986	+	0.981593i	$0.438831\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	396.000	0.0240624	0.0120312	−	0.999928i	$-0.496170\pi$
$647$	396.000	0.0240624	0.0120312	+	0.999928i	$0.496170\pi$
$648$	0	0
$649$	9812.00	0.593459
$650$	0	0
$651$	−1995.00	−0.120108
$652$	0	0
$653$	−4702.00	−0.281782	−0.140891	−	0.990025i	$-0.544997\pi$
$653$	−4702.00	−0.281782	−0.140891	+	0.990025i	$0.544997\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4698.00	0.278975
$658$	0	0
$659$	22140.0	1.30873	0.654364	−	0.756180i	$-0.272936\pi$
$659$	22140.0	1.30873	0.654364	+	0.756180i	$0.272936\pi$
$660$	0	0
$661$	−13518.0	−0.795445	−0.397723	−	0.917506i	$-0.630199\pi$
$661$	−13518.0	−0.795445	−0.397723	+	0.917506i	$0.630199\pi$
$662$	0	0
$663$	174.000	0.0101925
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1276.00	−0.0740733
$668$	0	0
$669$	1239.00	0.0716032
$670$	0	0
$671$	−2794.00	−0.160747
$672$	0	0
$673$	−11250.0	−0.644362	−0.322181	−	0.946678i	$-0.604416\pi$
$673$	−11250.0	−0.644362	−0.322181	+	0.946678i	$0.604416\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15170.0	−0.861197	−0.430599	−	0.902544i	$-0.641698\pi$
$677$	−15170.0	−0.861197	−0.430599	+	0.902544i	$0.641698\pi$
$678$	0	0
$679$	−20501.0	−1.15870
$680$	0	0
$681$	16956.0	0.954119
$682$	0	0
$683$	−22680.0	−1.27061	−0.635305	−	0.772262i	$-0.719125\pi$
$683$	−22680.0	−1.27061	−0.635305	+	0.772262i	$0.719125\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	13173.0	0.731559
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−128.000	−0.00704682	−0.00352341	−	0.999994i	$-0.501122\pi$
$691$	−128.000	−0.00704682	−0.00352341	+	0.999994i	$0.501122\pi$
$692$	0	0
$693$	−3762.00	−0.206214
$694$	0	0
$695$	0	0
$696$	0	0
$697$	27144.0	1.47511
$698$	0	0
$699$	6156.00	0.333106
$700$	0	0
$701$	25682.0	1.38373	0.691866	−	0.722026i	$-0.256789\pi$
$701$	25682.0	1.38373	0.691866	+	0.722026i	$0.256789\pi$
$702$	0	0
$703$	−14310.0	−0.767727
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−27360.0	−1.45542
$708$	0	0
$709$	−4951.00	−0.262255	−0.131127	−	0.991366i	$-0.541860\pi$
$709$	−4951.00	−0.262255	−0.131127	+	0.991366i	$0.541860\pi$
$710$	0	0
$711$	−12312.0	−0.649418
$712$	0	0
$713$	−2030.00	−0.106626
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−12960.0	−0.675035
$718$	0	0
$719$	22190.0	1.15097	0.575485	−	0.817812i	$-0.304813\pi$
$719$	22190.0	1.15097	0.575485	+	0.817812i	$0.304813\pi$
$720$	0	0
$721$	−2356.00	−0.121695
$722$	0	0
$723$	−12795.0	−0.658162
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−7685.00	−0.392051	−0.196025	−	0.980599i	$-0.562803\pi$
$727$	−7685.00	−0.392051	−0.196025	+	0.980599i	$0.562803\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−24998.0	−1.26482
$732$	0	0
$733$	29574.0	1.49023	0.745116	−	0.666934i	$-0.232394\pi$
$733$	29574.0	1.49023	0.745116	+	0.666934i	$0.232394\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−17842.0	−0.891748
$738$	0	0
$739$	32580.0	1.62175	0.810876	−	0.585218i	$-0.198991\pi$
$739$	32580.0	1.62175	0.810876	+	0.585218i	$0.198991\pi$
$740$	0	0
$741$	−159.000	−0.00788261
$742$	0	0
$743$	3060.00	0.151091	0.0755454	−	0.997142i	$-0.475930\pi$
$743$	3060.00	0.151091	0.0755454	+	0.997142i	$0.475930\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10242.0	0.501654
$748$	0	0
$749$	8208.00	0.400419
$750$	0	0
$751$	7992.00	0.388325	0.194163	−	0.980969i	$-0.437801\pi$
$751$	7992.00	0.388325	0.194163	+	0.980969i	$0.437801\pi$
$752$	0	0
$753$	−7236.00	−0.350192
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22841.0	1.09666	0.548329	−	0.836263i	$-0.315264\pi$
$757$	22841.0	1.09666	0.548329	+	0.836263i	$0.315264\pi$
$758$	0	0
$759$	−3828.00	−0.183067
$760$	0	0
$761$	−17172.0	−0.817982	−0.408991	−	0.912538i	$-0.634119\pi$
$761$	−17172.0	−0.817982	−0.408991	+	0.912538i	$0.634119\pi$
$762$	0	0
$763$	13319.0	0.631953
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−446.000	−0.0209963
$768$	0	0
$769$	−30869.0	−1.44755	−0.723774	−	0.690037i	$-0.757594\pi$
$769$	−30869.0	−1.44755	−0.723774	+	0.690037i	$0.757594\pi$
$770$	0	0
$771$	−20844.0	−0.973642
$772$	0	0
$773$	34884.0	1.62314	0.811572	−	0.584252i	$-0.198612\pi$
$773$	34884.0	1.62314	0.811572	+	0.584252i	$0.198612\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	15390.0	0.710570
$778$	0	0
$779$	−24804.0	−1.14082
$780$	0	0
$781$	792.000	0.0362868
$782$	0	0
$783$	−594.000	−0.0271109
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−5039.00	−0.228235	−0.114118	−	0.993467i	$-0.536404\pi$
$787$	−5039.00	−0.228235	−0.114118	+	0.993467i	$0.536404\pi$
$788$	0	0
$789$	−10692.0	−0.482440
$790$	0	0
$791$	−19836.0	−0.891640
$792$	0	0
$793$	127.000	0.00568714
$794$	0	0
$795$	0	0
$796$	0	0
$797$	540.000	0.0239997	0.0119999	−	0.999928i	$-0.496180\pi$
$797$	540.000	0.0239997	0.0119999	+	0.999928i	$0.496180\pi$
$798$	0	0
$799$	−13340.0	−0.590657
$800$	0	0
$801$	1296.00	0.0571684
$802$	0	0
$803$	−11484.0	−0.504684
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4602.00	0.200741
$808$	0	0
$809$	−41328.0	−1.79606	−0.898032	−	0.439931i	$-0.855003\pi$
$809$	−41328.0	−1.79606	−0.898032	+	0.439931i	$0.855003\pi$
$810$	0	0
$811$	12853.0	0.556510	0.278255	−	0.960507i	$-0.410244\pi$
$811$	12853.0	0.556510	0.278255	+	0.960507i	$0.410244\pi$
$812$	0	0
$813$	23112.0	0.997015
$814$	0	0
$815$	0	0
$816$	0	0
$817$	22843.0	0.978183
$818$	0	0
$819$	171.000	0.00729576
$820$	0	0
$821$	29470.0	1.25275	0.626376	−	0.779521i	$-0.284537\pi$
$821$	29470.0	1.25275	0.626376	+	0.779521i	$0.284537\pi$
$822$	0	0
$823$	24407.0	1.03375	0.516874	−	0.856062i	$-0.327096\pi$
$823$	24407.0	1.03375	0.516874	+	0.856062i	$0.327096\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15048.0	−0.632733	−0.316367	−	0.948637i	$-0.602463\pi$
$827$	−15048.0	−0.632733	−0.316367	+	0.948637i	$0.602463\pi$
$828$	0	0
$829$	−28406.0	−1.19009	−0.595043	−	0.803694i	$-0.702865\pi$
$829$	−28406.0	−1.19009	−0.595043	+	0.803694i	$0.702865\pi$
$830$	0	0
$831$	−14631.0	−0.610763
$832$	0	0
$833$	−1044.00	−0.0434243
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−945.000	−0.0390251
$838$	0	0
$839$	26914.0	1.10748	0.553739	−	0.832690i	$-0.313200\pi$
$839$	26914.0	1.10748	0.553739	+	0.832690i	$0.313200\pi$
$840$	0	0
$841$	−23905.0	−0.980155
$842$	0	0
$843$	11274.0	0.460614
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−16093.0	−0.652848
$848$	0	0
$849$	2805.00	0.113389
$850$	0	0
$851$	15660.0	0.630808
$852$	0	0
$853$	21275.0	0.853977	0.426988	−	0.904257i	$-0.359575\pi$
$853$	21275.0	0.853977	0.426988	+	0.904257i	$0.359575\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39132.0	1.55977	0.779885	−	0.625922i	$-0.215277\pi$
$857$	39132.0	1.55977	0.779885	+	0.625922i	$0.215277\pi$
$858$	0	0
$859$	448.000	0.0177946	0.00889730	−	0.999960i	$-0.497168\pi$
$859$	448.000	0.0177946	0.00889730	+	0.999960i	$0.497168\pi$
$860$	0	0
$861$	26676.0	1.05588
$862$	0	0
$863$	26856.0	1.05932	0.529658	−	0.848212i	$-0.322320\pi$
$863$	26856.0	1.05932	0.529658	+	0.848212i	$0.322320\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4647.00	0.182030
$868$	0	0
$869$	30096.0	1.17484
$870$	0	0
$871$	811.000	0.0315496
$872$	0	0
$873$	−9711.00	−0.376481
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3653.00	−0.140653	−0.0703267	−	0.997524i	$-0.522404\pi$
$877$	−3653.00	−0.140653	−0.0703267	+	0.997524i	$0.522404\pi$
$878$	0	0
$879$	−18642.0	−0.715335
$880$	0	0
$881$	−6552.00	−0.250559	−0.125280	−	0.992121i	$-0.539983\pi$
$881$	−6552.00	−0.250559	−0.125280	+	0.992121i	$0.539983\pi$
$882$	0	0
$883$	−4481.00	−0.170779	−0.0853894	−	0.996348i	$-0.527213\pi$
$883$	−4481.00	−0.170779	−0.0853894	+	0.996348i	$0.527213\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14666.0	0.555170	0.277585	−	0.960701i	$-0.410466\pi$
$887$	14666.0	0.555170	0.277585	+	0.960701i	$0.410466\pi$
$888$	0	0
$889$	−9576.00	−0.361270
$890$	0	0
$891$	−1782.00	−0.0670025
$892$	0	0
$893$	12190.0	0.456800
$894$	0	0
$895$	0	0
$896$	0	0
$897$	174.000	0.00647680
$898$	0	0
$899$	770.000	0.0285661
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−24567.0	−0.905358
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−51804.0	−1.89650	−0.948249	−	0.317528i	$-0.897147\pi$
$907$	−51804.0	−1.89650	−0.948249	+	0.317528i	$0.897147\pi$
$908$	0	0
$909$	−12960.0	−0.472889
$910$	0	0
$911$	−31198.0	−1.13462	−0.567308	−	0.823505i	$-0.692015\pi$
$911$	−31198.0	−1.13462	−0.567308	+	0.823505i	$0.692015\pi$
$912$	0	0
$913$	−25036.0	−0.907525
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1368.00	−0.0492643
$918$	0	0
$919$	27001.0	0.969185	0.484592	−	0.874740i	$-0.338968\pi$
$919$	27001.0	0.969185	0.484592	+	0.874740i	$0.338968\pi$
$920$	0	0
$921$	−11715.0	−0.419134
$922$	0	0
$923$	−36.0000	−0.00128381
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1116.00	−0.0395407
$928$	0	0
$929$	−22694.0	−0.801470	−0.400735	−	0.916194i	$-0.631245\pi$
$929$	−22694.0	−0.801470	−0.400735	+	0.916194i	$0.631245\pi$
$930$	0	0
$931$	954.000	0.0335833
$932$	0	0
$933$	28986.0	1.01711
$934$	0	0
$935$	0	0
$936$	0	0
$937$	29503.0	1.02862	0.514312	−	0.857603i	$-0.328047\pi$
$937$	29503.0	1.02862	0.514312	+	0.857603i	$0.328047\pi$
$938$	0	0
$939$	−15441.0	−0.536633
$940$	0	0
$941$	14566.0	0.504610	0.252305	−	0.967648i	$-0.418811\pi$
$941$	14566.0	0.504610	0.252305	+	0.967648i	$0.418811\pi$
$942$	0	0
$943$	27144.0	0.937360
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42322.0	−1.45225	−0.726125	−	0.687563i	$-0.758681\pi$
$947$	−42322.0	−1.45225	−0.726125	+	0.687563i	$0.758681\pi$
$948$	0	0
$949$	522.000	0.0178555
$950$	0	0
$951$	27648.0	0.942742
$952$	0	0
$953$	43416.0	1.47574	0.737871	−	0.674942i	$-0.235831\pi$
$953$	43416.0	1.47574	0.737871	+	0.674942i	$0.235831\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1452.00	0.0490454
$958$	0	0
$959$	−35150.0	−1.18358
$960$	0	0
$961$	−28566.0	−0.958880
$962$	0	0
$963$	3888.00	0.130103
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−21528.0	−0.715919	−0.357960	−	0.933737i	$-0.616527\pi$
$967$	−21528.0	−0.715919	−0.357960	+	0.933737i	$0.616527\pi$
$968$	0	0
$969$	9222.00	0.305731
$970$	0	0
$971$	−27050.0	−0.894002	−0.447001	−	0.894533i	$-0.647508\pi$
$971$	−27050.0	−0.894002	−0.447001	+	0.894533i	$0.647508\pi$
$972$	0	0
$973$	−34884.0	−1.14936
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−24934.0	−0.816489	−0.408244	−	0.912873i	$-0.633859\pi$
$977$	−24934.0	−0.816489	−0.408244	+	0.912873i	$0.633859\pi$
$978$	0	0
$979$	−3168.00	−0.103422
$980$	0	0
$981$	6309.00	0.205332
$982$	0	0
$983$	−8388.00	−0.272162	−0.136081	−	0.990698i	$-0.543451\pi$
$983$	−8388.00	−0.272162	−0.136081	+	0.990698i	$0.543451\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−13110.0	−0.422792
$988$	0	0
$989$	−24998.0	−0.803731
$990$	0	0
$991$	−29033.0	−0.930639	−0.465320	−	0.885143i	$-0.654061\pi$
$991$	−29033.0	−0.930639	−0.465320	+	0.885143i	$0.654061\pi$
$992$	0	0
$993$	6588.00	0.210538
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−25326.0	−0.804496	−0.402248	−	0.915531i	$-0.631771\pi$
$997$	−25326.0	−0.804496	−0.402248	+	0.915531i	$0.631771\pi$
$998$	0	0
$999$	7290.00	0.230876

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	1200.4.a.n.1.1		1
4.3	odd	2		600.4.a.j.1.1	yes	1
5.2	odd	4		1200.4.f.f.49.2		2
5.3	odd	4		1200.4.f.f.49.1		2
5.4	even	2		1200.4.a.x.1.1		1
12.11	even	2		1800.4.a.g.1.1		1
20.3	even	4		600.4.f.h.49.2		2
20.7	even	4		600.4.f.h.49.1		2
20.19	odd	2		600.4.a.g.1.1	✓	1
60.23	odd	4		1800.4.f.g.649.2		2
60.47	odd	4		1800.4.f.g.649.1		2
60.59	even	2		1800.4.a.bf.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.4.a.g.1.1	✓	1	20.19	odd	2
600.4.a.j.1.1	yes	1	4.3	odd	2
600.4.f.h.49.1		2	20.7	even	4
600.4.f.h.49.2		2	20.3	even	4
1200.4.a.n.1.1		1	1.1	even	1	trivial
1200.4.a.x.1.1		1	5.4	even	2
1200.4.f.f.49.1		2	5.3	odd	4
1200.4.f.f.49.2		2	5.2	odd	4
1800.4.a.g.1.1		1	12.11	even	2
1800.4.a.bf.1.1		1	60.59	even	2
1800.4.f.g.649.1		2	60.47	odd	4
1800.4.f.g.649.2		2	60.23	odd	4

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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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	1200.4.a.n.1.1

Level	$1200$
Weight	$4$
Character	1200.1
Self dual	yes
Analytic conductor	$70.802$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$