LMFDB - Embedded newform 450.6.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("450.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.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:	$72.1727189158$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 90)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +16.0000 q^{4} -98.0000 q^{7} +64.0000 q^{8} +O(q^{10})$

$q+4.00000 q^{2} +16.0000 q^{4} -98.0000 q^{7} +64.0000 q^{8} -354.000 q^{11} -404.000 q^{13} -392.000 q^{14} +256.000 q^{16} +654.000 q^{17} +1796.00 q^{19} -1416.00 q^{22} -1080.00 q^{23} -1616.00 q^{26} -1568.00 q^{28} +5754.00 q^{29} +10196.0 q^{31} +1024.00 q^{32} +2616.00 q^{34} -5552.00 q^{37} +7184.00 q^{38} +12960.0 q^{41} +8968.00 q^{43} -5664.00 q^{44} -4320.00 q^{46} -5400.00 q^{47} -7203.00 q^{49} -6464.00 q^{52} +8214.00 q^{53} -6272.00 q^{56} +23016.0 q^{58} -3954.00 q^{59} +962.000 q^{61} +40784.0 q^{62} +4096.00 q^{64} +17956.0 q^{67} +10464.0 q^{68} +56148.0 q^{71} +85690.0 q^{73} -22208.0 q^{74} +28736.0 q^{76} +34692.0 q^{77} -26044.0 q^{79} +51840.0 q^{82} -93468.0 q^{83} +35872.0 q^{86} -22656.0 q^{88} -73428.0 q^{89} +39592.0 q^{91} -17280.0 q^{92} -21600.0 q^{94} -128978. q^{97} -28812.0 q^{98} +O(q^{100})$

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−98.0000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−98.0000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	0	0
$11$	−354.000	−0.882108	−0.441054	−	0.897481i	$-0.645395\pi$
$11$	−354.000	−0.882108	−0.441054	+	0.897481i	$0.645395\pi$
$12$	0	0
$13$	−404.000	−0.663014	−0.331507	−	0.943453i	$-0.607557\pi$
$13$	−404.000	−0.663014	−0.331507	+	0.943453i	$0.607557\pi$
$14$	−392.000	−0.534522
$15$	0	0
$16$	256.000	0.250000
$17$	654.000	0.548852	0.274426	−	0.961608i	$-0.411512\pi$
$17$	654.000	0.548852	0.274426	+	0.961608i	$0.411512\pi$
$18$	0	0
$19$	1796.00	1.14136	0.570680	−	0.821173i	$-0.306680\pi$
$19$	1796.00	1.14136	0.570680	+	0.821173i	$0.306680\pi$
$20$	0	0
$21$	0	0
$22$	−1416.00	−0.623744
$23$	−1080.00	−0.425701	−0.212850	−	0.977085i	$-0.568275\pi$
$23$	−1080.00	−0.425701	−0.212850	+	0.977085i	$0.568275\pi$
$24$	0	0
$25$	0	0
$26$	−1616.00	−0.468822
$27$	0	0
$28$	−1568.00	−0.377964
$29$	5754.00	1.27050	0.635250	−	0.772306i	$-0.280897\pi$
$29$	5754.00	1.27050	0.635250	+	0.772306i	$0.280897\pi$
$30$	0	0
$31$	10196.0	1.90557	0.952787	−	0.303641i	$-0.0982024\pi$
$31$	10196.0	1.90557	0.952787	+	0.303641i	$0.0982024\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	2616.00	0.388097
$35$	0	0
$36$	0	0
$37$	−5552.00	−0.666723	−0.333361	−	0.942799i	$-0.608183\pi$
$37$	−5552.00	−0.666723	−0.333361	+	0.942799i	$0.608183\pi$
$38$	7184.00	0.807063
$39$	0	0
$40$	0	0
$41$	12960.0	1.20405	0.602026	−	0.798476i	$-0.294360\pi$
$41$	12960.0	1.20405	0.602026	+	0.798476i	$0.294360\pi$
$42$	0	0
$43$	8968.00	0.739647	0.369823	−	0.929102i	$-0.379418\pi$
$43$	8968.00	0.739647	0.369823	+	0.929102i	$0.379418\pi$
$44$	−5664.00	−0.441054
$45$	0	0
$46$	−4320.00	−0.301016
$47$	−5400.00	−0.356574	−0.178287	−	0.983979i	$-0.557055\pi$
$47$	−5400.00	−0.356574	−0.178287	+	0.983979i	$0.557055\pi$
$48$	0	0
$49$	−7203.00	−0.428571
$50$	0	0
$51$	0	0
$52$	−6464.00	−0.331507
$53$	8214.00	0.401666	0.200833	−	0.979625i	$-0.435635\pi$
$53$	8214.00	0.401666	0.200833	+	0.979625i	$0.435635\pi$
$54$	0	0
$55$	0	0
$56$	−6272.00	−0.267261
$57$	0	0
$58$	23016.0	0.898380
$59$	−3954.00	−0.147879	−0.0739395	−	0.997263i	$-0.523557\pi$
$59$	−3954.00	−0.147879	−0.0739395	+	0.997263i	$0.523557\pi$
$60$	0	0
$61$	962.000	0.0331017	0.0165509	−	0.999863i	$-0.494731\pi$
$61$	962.000	0.0331017	0.0165509	+	0.999863i	$0.494731\pi$
$62$	40784.0	1.34744
$63$	0	0
$64$	4096.00	0.125000
$65$	0	0
$66$	0	0
$67$	17956.0	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	17956.0	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	10464.0	0.274426
$69$	0	0
$70$	0	0
$71$	56148.0	1.32187	0.660935	−	0.750444i	$-0.270160\pi$
$71$	56148.0	1.32187	0.660935	+	0.750444i	$0.270160\pi$
$72$	0	0
$73$	85690.0	1.88201	0.941007	−	0.338386i	$-0.109881\pi$
$73$	85690.0	1.88201	0.941007	+	0.338386i	$0.109881\pi$
$74$	−22208.0	−0.471444
$75$	0	0
$76$	28736.0	0.570680
$77$	34692.0	0.666811
$78$	0	0
$79$	−26044.0	−0.469505	−0.234752	−	0.972055i	$-0.575428\pi$
$79$	−26044.0	−0.469505	−0.234752	+	0.972055i	$0.575428\pi$
$80$	0	0
$81$	0	0
$82$	51840.0	0.851394
$83$	−93468.0	−1.48925	−0.744625	−	0.667483i	$-0.767372\pi$
$83$	−93468.0	−1.48925	−0.744625	+	0.667483i	$0.767372\pi$
$84$	0	0
$85$	0	0
$86$	35872.0	0.523009
$87$	0	0
$88$	−22656.0	−0.311872
$89$	−73428.0	−0.982622	−0.491311	−	0.870984i	$-0.663482\pi$
$89$	−73428.0	−0.982622	−0.491311	+	0.870984i	$0.663482\pi$
$90$	0	0
$91$	39592.0	0.501192
$92$	−17280.0	−0.212850
$93$	0	0
$94$	−21600.0	−0.252136
$95$	0	0
$96$	0	0
$97$	−128978.	−1.39183	−0.695915	−	0.718124i	$-0.745001\pi$
$97$	−128978.	−1.39183	−0.695915	+	0.718124i	$0.745001\pi$
$98$	−28812.0	−0.303046
$99$	0	0
$100$	0	0
$101$	154794.	1.50991	0.754954	−	0.655777i	$-0.227659\pi$
$101$	154794.	1.50991	0.754954	+	0.655777i	$0.227659\pi$
$102$	0	0
$103$	−27698.0	−0.257250	−0.128625	−	0.991693i	$-0.541056\pi$
$103$	−27698.0	−0.257250	−0.128625	+	0.991693i	$0.541056\pi$
$104$	−25856.0	−0.234411
$105$	0	0
$106$	32856.0	0.284021
$107$	221172.	1.86754	0.933772	−	0.357869i	$-0.116497\pi$
$107$	221172.	1.86754	0.933772	+	0.357869i	$0.116497\pi$
$108$	0	0
$109$	123122.	0.992589	0.496294	−	0.868154i	$-0.334694\pi$
$109$	123122.	0.992589	0.496294	+	0.868154i	$0.334694\pi$
$110$	0	0
$111$	0	0
$112$	−25088.0	−0.188982
$113$	220386.	1.62363	0.811817	−	0.583913i	$-0.198479\pi$
$113$	220386.	1.62363	0.811817	+	0.583913i	$0.198479\pi$
$114$	0	0
$115$	0	0
$116$	92064.0	0.635250
$117$	0	0
$118$	−15816.0	−0.104566
$119$	−64092.0	−0.414893
$120$	0	0
$121$	−35735.0	−0.221886
$122$	3848.00	0.0234064
$123$	0	0
$124$	163136.	0.952787
$125$	0	0
$126$	0	0
$127$	−181298.	−0.997433	−0.498716	−	0.866765i	$-0.666195\pi$
$127$	−181298.	−0.997433	−0.498716	+	0.866765i	$0.666195\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	0	0
$131$	254514.	1.29579	0.647893	−	0.761731i	$-0.275650\pi$
$131$	254514.	1.29579	0.647893	+	0.761731i	$0.275650\pi$
$132$	0	0
$133$	−176008.	−0.862786
$134$	71824.0	0.345547
$135$	0	0
$136$	41856.0	0.194049
$137$	415506.	1.89137	0.945684	−	0.325088i	$-0.105394\pi$
$137$	415506.	1.89137	0.945684	+	0.325088i	$0.105394\pi$
$138$	0	0
$139$	−312340.	−1.37117	−0.685584	−	0.727994i	$-0.740453\pi$
$139$	−312340.	−1.37117	−0.685584	+	0.727994i	$0.740453\pi$
$140$	0	0
$141$	0	0
$142$	224592.	0.934703
$143$	143016.	0.584850
$144$	0	0
$145$	0	0
$146$	342760.	1.33079
$147$	0	0
$148$	−88832.0	−0.333361
$149$	−300954.	−1.11054	−0.555270	−	0.831670i	$-0.687385\pi$
$149$	−300954.	−1.11054	−0.555270	+	0.831670i	$0.687385\pi$
$150$	0	0
$151$	−37576.0	−0.134112	−0.0670561	−	0.997749i	$-0.521361\pi$
$151$	−37576.0	−0.134112	−0.0670561	+	0.997749i	$0.521361\pi$
$152$	114944.	0.403531
$153$	0	0
$154$	138768.	0.471506
$155$	0	0
$156$	0	0
$157$	378316.	1.22491	0.612457	−	0.790504i	$-0.290181\pi$
$157$	378316.	1.22491	0.612457	+	0.790504i	$0.290181\pi$
$158$	−104176.	−0.331990
$159$	0	0
$160$	0	0
$161$	105840.	0.321799
$162$	0	0
$163$	−66980.0	−0.197459	−0.0987293	−	0.995114i	$-0.531478\pi$
$163$	−66980.0	−0.197459	−0.0987293	+	0.995114i	$0.531478\pi$
$164$	207360.	0.602026
$165$	0	0
$166$	−373872.	−1.05306
$167$	−83424.0	−0.231473	−0.115736	−	0.993280i	$-0.536923\pi$
$167$	−83424.0	−0.231473	−0.115736	+	0.993280i	$0.536923\pi$
$168$	0	0
$169$	−208077.	−0.560412
$170$	0	0
$171$	0	0
$172$	143488.	0.369823
$173$	42318.0	0.107500	0.0537502	−	0.998554i	$-0.482883\pi$
$173$	42318.0	0.107500	0.0537502	+	0.998554i	$0.482883\pi$
$174$	0	0
$175$	0	0
$176$	−90624.0	−0.220527
$177$	0	0
$178$	−293712.	−0.694819
$179$	167406.	0.390516	0.195258	−	0.980752i	$-0.437446\pi$
$179$	167406.	0.390516	0.195258	+	0.980752i	$0.437446\pi$
$180$	0	0
$181$	−483118.	−1.09612	−0.548058	−	0.836440i	$-0.684633\pi$
$181$	−483118.	−1.09612	−0.548058	+	0.836440i	$0.684633\pi$
$182$	158368.	0.354396
$183$	0	0
$184$	−69120.0	−0.150508
$185$	0	0
$186$	0	0
$187$	−231516.	−0.484147
$188$	−86400.0	−0.178287
$189$	0	0
$190$	0	0
$191$	772572.	1.53234	0.766171	−	0.642637i	$-0.222160\pi$
$191$	772572.	1.53234	0.766171	+	0.642637i	$0.222160\pi$
$192$	0	0
$193$	670570.	1.29584	0.647919	−	0.761709i	$-0.275639\pi$
$193$	670570.	1.29584	0.647919	+	0.761709i	$0.275639\pi$
$194$	−515912.	−0.984173
$195$	0	0
$196$	−115248.	−0.214286
$197$	851922.	1.56399	0.781996	−	0.623284i	$-0.214202\pi$
$197$	851922.	1.56399	0.781996	+	0.623284i	$0.214202\pi$
$198$	0	0
$199$	−561256.	−1.00468	−0.502341	−	0.864670i	$-0.667528\pi$
$199$	−561256.	−1.00468	−0.502341	+	0.864670i	$0.667528\pi$
$200$	0	0
$201$	0	0
$202$	619176.	1.06767
$203$	−563892.	−0.960408
$204$	0	0
$205$	0	0
$206$	−110792.	−0.181903
$207$	0	0
$208$	−103424.	−0.165754
$209$	−635784.	−1.00680
$210$	0	0
$211$	−578764.	−0.894943	−0.447471	−	0.894298i	$-0.647675\pi$
$211$	−578764.	−0.894943	−0.447471	+	0.894298i	$0.647675\pi$
$212$	131424.	0.200833
$213$	0	0
$214$	884688.	1.32055
$215$	0	0
$216$	0	0
$217$	−999208.	−1.44048
$218$	492488.	0.701866
$219$	0	0
$220$	0	0
$221$	−264216.	−0.363897
$222$	0	0
$223$	−863114.	−1.16227	−0.581134	−	0.813808i	$-0.697391\pi$
$223$	−863114.	−1.16227	−0.581134	+	0.813808i	$0.697391\pi$
$224$	−100352.	−0.133631
$225$	0	0
$226$	881544.	1.14808
$227$	−712800.	−0.918128	−0.459064	−	0.888403i	$-0.651815\pi$
$227$	−712800.	−0.918128	−0.459064	+	0.888403i	$0.651815\pi$
$228$	0	0
$229$	−343666.	−0.433060	−0.216530	−	0.976276i	$-0.569474\pi$
$229$	−343666.	−0.433060	−0.216530	+	0.976276i	$0.569474\pi$
$230$	0	0
$231$	0	0
$232$	368256.	0.449190
$233$	218814.	0.264049	0.132025	−	0.991246i	$-0.457852\pi$
$233$	218814.	0.264049	0.132025	+	0.991246i	$0.457852\pi$
$234$	0	0
$235$	0	0
$236$	−63264.0	−0.0739395
$237$	0	0
$238$	−256368.	−0.293374
$239$	1.02095e6	1.15614	0.578068	−	0.815989i	$-0.303807\pi$
$239$	1.02095e6	1.15614	0.578068	+	0.815989i	$0.303807\pi$
$240$	0	0
$241$	1.37183e6	1.52145	0.760725	−	0.649074i	$-0.224844\pi$
$241$	1.37183e6	1.52145	0.760725	+	0.649074i	$0.224844\pi$
$242$	−142940.	−0.156897
$243$	0	0
$244$	15392.0	0.0165509
$245$	0	0
$246$	0	0
$247$	−725584.	−0.756738
$248$	652544.	0.673722
$249$	0	0
$250$	0	0
$251$	932058.	0.933810	0.466905	−	0.884307i	$-0.345369\pi$
$251$	932058.	0.933810	0.466905	+	0.884307i	$0.345369\pi$
$252$	0	0
$253$	382320.	0.375514
$254$	−725192.	−0.705292
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.26569e6	−1.19535	−0.597676	−	0.801738i	$-0.703909\pi$
$257$	−1.26569e6	−1.19535	−0.597676	+	0.801738i	$0.703909\pi$
$258$	0	0
$259$	544096.	0.503995
$260$	0	0
$261$	0	0
$262$	1.01806e6	0.916259
$263$	−699936.	−0.623978	−0.311989	−	0.950086i	$-0.600995\pi$
$263$	−699936.	−0.623978	−0.311989	+	0.950086i	$0.600995\pi$
$264$	0	0
$265$	0	0
$266$	−704032.	−0.610082
$267$	0	0
$268$	287296.	0.244339
$269$	488178.	0.411337	0.205668	−	0.978622i	$-0.434063\pi$
$269$	488178.	0.411337	0.205668	+	0.978622i	$0.434063\pi$
$270$	0	0
$271$	525968.	0.435047	0.217523	−	0.976055i	$-0.430202\pi$
$271$	525968.	0.435047	0.217523	+	0.976055i	$0.430202\pi$
$272$	167424.	0.137213
$273$	0	0
$274$	1.66202e6	1.33740
$275$	0	0
$276$	0	0
$277$	−2.42931e6	−1.90232	−0.951161	−	0.308696i	$-0.900107\pi$
$277$	−2.42931e6	−1.90232	−0.951161	+	0.308696i	$0.900107\pi$
$278$	−1.24936e6	−0.969562
$279$	0	0
$280$	0	0
$281$	−2.29031e6	−1.73033	−0.865163	−	0.501490i	$-0.832785\pi$
$281$	−2.29031e6	−1.73033	−0.865163	+	0.501490i	$0.832785\pi$
$282$	0	0
$283$	2.09156e6	1.55240	0.776200	−	0.630487i	$-0.217145\pi$
$283$	2.09156e6	1.55240	0.776200	+	0.630487i	$0.217145\pi$
$284$	898368.	0.660935
$285$	0	0
$286$	572064.	0.413551
$287$	−1.27008e6	−0.910178
$288$	0	0
$289$	−992141.	−0.698761
$290$	0	0
$291$	0	0
$292$	1.37104e6	0.941007
$293$	−107118.	−0.0728943	−0.0364471	−	0.999336i	$-0.511604\pi$
$293$	−107118.	−0.0728943	−0.0364471	+	0.999336i	$0.511604\pi$
$294$	0	0
$295$	0	0
$296$	−355328.	−0.235722
$297$	0	0
$298$	−1.20382e6	−0.785271
$299$	436320.	0.282246
$300$	0	0
$301$	−878864.	−0.559121
$302$	−150304.	−0.0948316
$303$	0	0
$304$	459776.	0.285340
$305$	0	0
$306$	0	0
$307$	−1.12938e6	−0.683900	−0.341950	−	0.939718i	$-0.611087\pi$
$307$	−1.12938e6	−0.683900	−0.341950	+	0.939718i	$0.611087\pi$
$308$	555072.	0.333405
$309$	0	0
$310$	0	0
$311$	−3.06145e6	−1.79484	−0.897422	−	0.441174i	$-0.854562\pi$
$311$	−3.06145e6	−1.79484	−0.897422	+	0.441174i	$0.854562\pi$
$312$	0	0
$313$	1.23025e6	0.709794	0.354897	−	0.934905i	$-0.384516\pi$
$313$	1.23025e6	0.709794	0.354897	+	0.934905i	$0.384516\pi$
$314$	1.51326e6	0.866145
$315$	0	0
$316$	−416704.	−0.234752
$317$	679122.	0.379577	0.189788	−	0.981825i	$-0.439220\pi$
$317$	679122.	0.379577	0.189788	+	0.981825i	$0.439220\pi$
$318$	0	0
$319$	−2.03692e6	−1.12072
$320$	0	0
$321$	0	0
$322$	423360.	0.227546
$323$	1.17458e6	0.626438
$324$	0	0
$325$	0	0
$326$	−267920.	−0.139624
$327$	0	0
$328$	829440.	0.425697
$329$	529200.	0.269544
$330$	0	0
$331$	98516.0	0.0494239	0.0247119	−	0.999695i	$-0.492133\pi$
$331$	98516.0	0.0494239	0.0247119	+	0.999695i	$0.492133\pi$
$332$	−1.49549e6	−0.744625
$333$	0	0
$334$	−333696.	−0.163676
$335$	0	0
$336$	0	0
$337$	1.99779e6	0.958244	0.479122	−	0.877748i	$-0.340955\pi$
$337$	1.99779e6	0.958244	0.479122	+	0.877748i	$0.340955\pi$
$338$	−832308.	−0.396271
$339$	0	0
$340$	0	0
$341$	−3.60938e6	−1.68092
$342$	0	0
$343$	2.35298e6	1.07990
$344$	573952.	0.261505
$345$	0	0
$346$	169272.	0.0760142
$347$	−2.65032e6	−1.18161	−0.590806	−	0.806814i	$-0.701190\pi$
$347$	−2.65032e6	−1.18161	−0.590806	+	0.806814i	$0.701190\pi$
$348$	0	0
$349$	−2.03378e6	−0.893801	−0.446900	−	0.894584i	$-0.647472\pi$
$349$	−2.03378e6	−0.893801	−0.446900	+	0.894584i	$0.647472\pi$
$350$	0	0
$351$	0	0
$352$	−362496.	−0.155936
$353$	948630.	0.405191	0.202596	−	0.979262i	$-0.435062\pi$
$353$	948630.	0.405191	0.202596	+	0.979262i	$0.435062\pi$
$354$	0	0
$355$	0	0
$356$	−1.17485e6	−0.491311
$357$	0	0
$358$	669624.	0.276136
$359$	−494616.	−0.202550	−0.101275	−	0.994858i	$-0.532292\pi$
$359$	−494616.	−0.202550	−0.101275	+	0.994858i	$0.532292\pi$
$360$	0	0
$361$	749517.	0.302701
$362$	−1.93247e6	−0.775072
$363$	0	0
$364$	633472.	0.250596
$365$	0	0
$366$	0	0
$367$	75046.0	0.0290846	0.0145423	−	0.999894i	$-0.495371\pi$
$367$	75046.0	0.0290846	0.0145423	+	0.999894i	$0.495371\pi$
$368$	−276480.	−0.106425
$369$	0	0
$370$	0	0
$371$	−804972.	−0.303631
$372$	0	0
$373$	−533468.	−0.198535	−0.0992673	−	0.995061i	$-0.531650\pi$
$373$	−533468.	−0.198535	−0.0992673	+	0.995061i	$0.531650\pi$
$374$	−926064.	−0.342343
$375$	0	0
$376$	−345600.	−0.126068
$377$	−2.32462e6	−0.842360
$378$	0	0
$379$	−199780.	−0.0714421	−0.0357210	−	0.999362i	$-0.511373\pi$
$379$	−199780.	−0.0714421	−0.0357210	+	0.999362i	$0.511373\pi$
$380$	0	0
$381$	0	0
$382$	3.09029e6	1.08353
$383$	−869400.	−0.302847	−0.151423	−	0.988469i	$-0.548386\pi$
$383$	−869400.	−0.302847	−0.151423	+	0.988469i	$0.548386\pi$
$384$	0	0
$385$	0	0
$386$	2.68228e6	0.916296
$387$	0	0
$388$	−2.06365e6	−0.695915
$389$	1.05190e6	0.352453	0.176227	−	0.984350i	$-0.443611\pi$
$389$	1.05190e6	0.352453	0.176227	+	0.984350i	$0.443611\pi$
$390$	0	0
$391$	−706320.	−0.233647
$392$	−460992.	−0.151523
$393$	0	0
$394$	3.40769e6	1.10591
$395$	0	0
$396$	0	0
$397$	−4.19262e6	−1.33508	−0.667542	−	0.744572i	$-0.732654\pi$
$397$	−4.19262e6	−1.33508	−0.667542	+	0.744572i	$0.732654\pi$
$398$	−2.24502e6	−0.710417
$399$	0	0
$400$	0	0
$401$	1.90292e6	0.590963	0.295482	−	0.955348i	$-0.404520\pi$
$401$	1.90292e6	0.590963	0.295482	+	0.955348i	$0.404520\pi$
$402$	0	0
$403$	−4.11918e6	−1.26342
$404$	2.47670e6	0.754954
$405$	0	0
$406$	−2.25557e6	−0.679111
$407$	1.96541e6	0.588121
$408$	0	0
$409$	1.62910e6	0.481547	0.240774	−	0.970581i	$-0.422599\pi$
$409$	1.62910e6	0.481547	0.240774	+	0.970581i	$0.422599\pi$
$410$	0	0
$411$	0	0
$412$	−443168.	−0.128625
$413$	387492.	0.111786
$414$	0	0
$415$	0	0
$416$	−413696.	−0.117206
$417$	0	0
$418$	−2.54314e6	−0.711916
$419$	2.33027e6	0.648443	0.324222	−	0.945981i	$-0.394898\pi$
$419$	2.33027e6	0.648443	0.324222	+	0.945981i	$0.394898\pi$
$420$	0	0
$421$	612974.	0.168553	0.0842766	−	0.996442i	$-0.473142\pi$
$421$	612974.	0.168553	0.0842766	+	0.996442i	$0.473142\pi$
$422$	−2.31506e6	−0.632820
$423$	0	0
$424$	525696.	0.142010
$425$	0	0
$426$	0	0
$427$	−94276.0	−0.0250225
$428$	3.53875e6	0.933772
$429$	0	0
$430$	0	0
$431$	−6.69384e6	−1.73573	−0.867865	−	0.496800i	$-0.834508\pi$
$431$	−6.69384e6	−1.73573	−0.867865	+	0.496800i	$0.834508\pi$
$432$	0	0
$433$	−2.20194e6	−0.564399	−0.282199	−	0.959356i	$-0.591064\pi$
$433$	−2.20194e6	−0.564399	−0.282199	+	0.959356i	$0.591064\pi$
$434$	−3.99683e6	−1.01857
$435$	0	0
$436$	1.96995e6	0.496294
$437$	−1.93968e6	−0.485877
$438$	0	0
$439$	1.10437e6	0.273497	0.136748	−	0.990606i	$-0.456335\pi$
$439$	1.10437e6	0.273497	0.136748	+	0.990606i	$0.456335\pi$
$440$	0	0
$441$	0	0
$442$	−1.05686e6	−0.257314
$443$	−3.82202e6	−0.925303	−0.462652	−	0.886540i	$-0.653102\pi$
$443$	−3.82202e6	−0.925303	−0.462652	+	0.886540i	$0.653102\pi$
$444$	0	0
$445$	0	0
$446$	−3.45246e6	−0.821847
$447$	0	0
$448$	−401408.	−0.0944911
$449$	−3.93985e6	−0.922283	−0.461141	−	0.887327i	$-0.652560\pi$
$449$	−3.93985e6	−0.922283	−0.461141	+	0.887327i	$0.652560\pi$
$450$	0	0
$451$	−4.58784e6	−1.06210
$452$	3.52618e6	0.811817
$453$	0	0
$454$	−2.85120e6	−0.649214
$455$	0	0
$456$	0	0
$457$	1.01753e6	0.227906	0.113953	−	0.993486i	$-0.463649\pi$
$457$	1.01753e6	0.227906	0.113953	+	0.993486i	$0.463649\pi$
$458$	−1.37466e6	−0.306220
$459$	0	0
$460$	0	0
$461$	6.80830e6	1.49206	0.746030	−	0.665912i	$-0.231957\pi$
$461$	6.80830e6	1.49206	0.746030	+	0.665912i	$0.231957\pi$
$462$	0	0
$463$	−6.74281e6	−1.46180	−0.730901	−	0.682483i	$-0.760900\pi$
$463$	−6.74281e6	−1.46180	−0.730901	+	0.682483i	$0.760900\pi$
$464$	1.47302e6	0.317625
$465$	0	0
$466$	875256.	0.186711
$467$	3.22619e6	0.684538	0.342269	−	0.939602i	$-0.388805\pi$
$467$	3.22619e6	0.684538	0.342269	+	0.939602i	$0.388805\pi$
$468$	0	0
$469$	−1.75969e6	−0.369406
$470$	0	0
$471$	0	0
$472$	−253056.	−0.0522831
$473$	−3.17467e6	−0.652448
$474$	0	0
$475$	0	0
$476$	−1.02547e6	−0.207447
$477$	0	0
$478$	4.08379e6	0.817512
$479$	−6.21212e6	−1.23709	−0.618545	−	0.785749i	$-0.712278\pi$
$479$	−6.21212e6	−1.23709	−0.618545	+	0.785749i	$0.712278\pi$
$480$	0	0
$481$	2.24301e6	0.442047
$482$	5.48732e6	1.07583
$483$	0	0
$484$	−571760.	−0.110943
$485$	0	0
$486$	0	0
$487$	1.51326e6	0.289128	0.144564	−	0.989495i	$-0.453822\pi$
$487$	1.51326e6	0.289128	0.144564	+	0.989495i	$0.453822\pi$
$488$	61568.0	0.0117032
$489$	0	0
$490$	0	0
$491$	3.43835e6	0.643646	0.321823	−	0.946800i	$-0.395704\pi$
$491$	3.43835e6	0.643646	0.321823	+	0.946800i	$0.395704\pi$
$492$	0	0
$493$	3.76312e6	0.697317
$494$	−2.90234e6	−0.535094
$495$	0	0
$496$	2.61018e6	0.476393
$497$	−5.50250e6	−0.999239
$498$	0	0
$499$	5.27340e6	0.948067	0.474033	−	0.880507i	$-0.342798\pi$
$499$	5.27340e6	0.948067	0.474033	+	0.880507i	$0.342798\pi$
$500$	0	0
$501$	0	0
$502$	3.72823e6	0.660304
$503$	8.81172e6	1.55289	0.776445	−	0.630185i	$-0.217021\pi$
$503$	8.81172e6	1.55289	0.776445	+	0.630185i	$0.217021\pi$
$504$	0	0
$505$	0	0
$506$	1.52928e6	0.265528
$507$	0	0
$508$	−2.90077e6	−0.498716
$509$	1.03788e7	1.77563	0.887817	−	0.460197i	$-0.152221\pi$
$509$	1.03788e7	1.77563	0.887817	+	0.460197i	$0.152221\pi$
$510$	0	0
$511$	−8.39762e6	−1.42267
$512$	262144.	0.0441942
$513$	0	0
$514$	−5.06278e6	−0.845242
$515$	0	0
$516$	0	0
$517$	1.91160e6	0.314536
$518$	2.17638e6	0.356378
$519$	0	0
$520$	0	0
$521$	4.72177e6	0.762098	0.381049	−	0.924555i	$-0.375563\pi$
$521$	4.72177e6	0.762098	0.381049	+	0.924555i	$0.375563\pi$
$522$	0	0
$523$	4.24340e6	0.678359	0.339179	−	0.940722i	$-0.389851\pi$
$523$	4.24340e6	0.678359	0.339179	+	0.940722i	$0.389851\pi$
$524$	4.07222e6	0.647893
$525$	0	0
$526$	−2.79974e6	−0.441219
$527$	6.66818e6	1.04588
$528$	0	0
$529$	−5.26994e6	−0.818779
$530$	0	0
$531$	0	0
$532$	−2.81613e6	−0.431393
$533$	−5.23584e6	−0.798304
$534$	0	0
$535$	0	0
$536$	1.14918e6	0.172774
$537$	0	0
$538$	1.95271e6	0.290859
$539$	2.54986e6	0.378046
$540$	0	0
$541$	−1.30682e6	−0.191966	−0.0959828	−	0.995383i	$-0.530599\pi$
$541$	−1.30682e6	−0.191966	−0.0959828	+	0.995383i	$0.530599\pi$
$542$	2.10387e6	0.307625
$543$	0	0
$544$	669696.	0.0970243
$545$	0	0
$546$	0	0
$547$	9.69388e6	1.38525	0.692627	−	0.721296i	$-0.256453\pi$
$547$	9.69388e6	1.38525	0.692627	+	0.721296i	$0.256453\pi$
$548$	6.64810e6	0.945684
$549$	0	0
$550$	0	0
$551$	1.03342e7	1.45010
$552$	0	0
$553$	2.55231e6	0.354912
$554$	−9.71725e6	−1.34514
$555$	0	0
$556$	−4.99744e6	−0.685584
$557$	8.47666e6	1.15768	0.578838	−	0.815443i	$-0.303506\pi$
$557$	8.47666e6	1.15768	0.578838	+	0.815443i	$0.303506\pi$
$558$	0	0
$559$	−3.62307e6	−0.490397
$560$	0	0
$561$	0	0
$562$	−9.16123e6	−1.22353
$563$	1.15102e6	0.153042	0.0765210	−	0.997068i	$-0.475619\pi$
$563$	1.15102e6	0.153042	0.0765210	+	0.997068i	$0.475619\pi$
$564$	0	0
$565$	0	0
$566$	8.36622e6	1.09771
$567$	0	0
$568$	3.59347e6	0.467351
$569$	45396.0	0.00587810	0.00293905	−	0.999996i	$-0.499064\pi$
$569$	45396.0	0.00587810	0.00293905	+	0.999996i	$0.499064\pi$
$570$	0	0
$571$	6.43696e6	0.826211	0.413105	−	0.910683i	$-0.364444\pi$
$571$	6.43696e6	0.826211	0.413105	+	0.910683i	$0.364444\pi$
$572$	2.28826e6	0.292425
$573$	0	0
$574$	−5.08032e6	−0.643593
$575$	0	0
$576$	0	0
$577$	578758.	0.0723698	0.0361849	−	0.999345i	$-0.488479\pi$
$577$	578758.	0.0723698	0.0361849	+	0.999345i	$0.488479\pi$
$578$	−3.96856e6	−0.494099
$579$	0	0
$580$	0	0
$581$	9.15986e6	1.12577
$582$	0	0
$583$	−2.90776e6	−0.354313
$584$	5.48416e6	0.665393
$585$	0	0
$586$	−428472.	−0.0515440
$587$	960804.	0.115091	0.0575453	−	0.998343i	$-0.481673\pi$
$587$	960804.	0.115091	0.0575453	+	0.998343i	$0.481673\pi$
$588$	0	0
$589$	1.83120e7	2.17494
$590$	0	0
$591$	0	0
$592$	−1.42131e6	−0.166681
$593$	5.70220e6	0.665895	0.332948	−	0.942945i	$-0.391957\pi$
$593$	5.70220e6	0.665895	0.332948	+	0.942945i	$0.391957\pi$
$594$	0	0
$595$	0	0
$596$	−4.81526e6	−0.555270
$597$	0	0
$598$	1.74528e6	0.199578
$599$	−1.53569e7	−1.74879	−0.874393	−	0.485219i	$-0.838740\pi$
$599$	−1.53569e7	−1.74879	−0.874393	+	0.485219i	$0.838740\pi$
$600$	0	0
$601$	1.21690e7	1.37425	0.687127	−	0.726537i	$-0.258872\pi$
$601$	1.21690e7	1.37425	0.687127	+	0.726537i	$0.258872\pi$
$602$	−3.51546e6	−0.395358
$603$	0	0
$604$	−601216.	−0.0670561
$605$	0	0
$606$	0	0
$607$	2.94630e6	0.324567	0.162284	−	0.986744i	$-0.448114\pi$
$607$	2.94630e6	0.324567	0.162284	+	0.986744i	$0.448114\pi$
$608$	1.83910e6	0.201766
$609$	0	0
$610$	0	0
$611$	2.18160e6	0.236413
$612$	0	0
$613$	1.06180e7	1.14128	0.570642	−	0.821199i	$-0.306695\pi$
$613$	1.06180e7	1.14128	0.570642	+	0.821199i	$0.306695\pi$
$614$	−4.51750e6	−0.483590
$615$	0	0
$616$	2.22029e6	0.235753
$617$	1.14174e7	1.20741	0.603706	−	0.797207i	$-0.293690\pi$
$617$	1.14174e7	1.20741	0.603706	+	0.797207i	$0.293690\pi$
$618$	0	0
$619$	−1.28242e6	−0.134525	−0.0672626	−	0.997735i	$-0.521427\pi$
$619$	−1.28242e6	−0.134525	−0.0672626	+	0.997735i	$0.521427\pi$
$620$	0	0
$621$	0	0
$622$	−1.22458e7	−1.26915
$623$	7.19594e6	0.742793
$624$	0	0
$625$	0	0
$626$	4.92100e6	0.501900
$627$	0	0
$628$	6.05306e6	0.612457
$629$	−3.63101e6	−0.365932
$630$	0	0
$631$	7.37894e6	0.737769	0.368885	−	0.929475i	$-0.379740\pi$
$631$	7.37894e6	0.737769	0.368885	+	0.929475i	$0.379740\pi$
$632$	−1.66682e6	−0.165995
$633$	0	0
$634$	2.71649e6	0.268401
$635$	0	0
$636$	0	0
$637$	2.91001e6	0.284149
$638$	−8.14766e6	−0.792467
$639$	0	0
$640$	0	0
$641$	1.63802e6	0.157462	0.0787309	−	0.996896i	$-0.474913\pi$
$641$	1.63802e6	0.157462	0.0787309	+	0.996896i	$0.474913\pi$
$642$	0	0
$643$	−2.03847e7	−1.94436	−0.972180	−	0.234236i	$-0.924741\pi$
$643$	−2.03847e7	−1.94436	−0.972180	+	0.234236i	$0.924741\pi$
$644$	1.69344e6	0.160900
$645$	0	0
$646$	4.69834e6	0.442958
$647$	2.10318e7	1.97522	0.987608	−	0.156939i	$-0.0501625\pi$
$647$	2.10318e7	1.97522	0.987608	+	0.156939i	$0.0501625\pi$
$648$	0	0
$649$	1.39972e6	0.130445
$650$	0	0
$651$	0	0
$652$	−1.07168e6	−0.0987293
$653$	−7.85504e6	−0.720884	−0.360442	−	0.932782i	$-0.617374\pi$
$653$	−7.85504e6	−0.720884	−0.360442	+	0.932782i	$0.617374\pi$
$654$	0	0
$655$	0	0
$656$	3.31776e6	0.301013
$657$	0	0
$658$	2.11680e6	0.190597
$659$	−9.32727e6	−0.836645	−0.418322	−	0.908299i	$-0.637382\pi$
$659$	−9.32727e6	−0.836645	−0.418322	+	0.908299i	$0.637382\pi$
$660$	0	0
$661$	1.57325e7	1.40053	0.700267	−	0.713881i	$-0.253064\pi$
$661$	1.57325e7	1.40053	0.700267	+	0.713881i	$0.253064\pi$
$662$	394064.	0.0349480
$663$	0	0
$664$	−5.98195e6	−0.526530
$665$	0	0
$666$	0	0
$667$	−6.21432e6	−0.540853
$668$	−1.33478e6	−0.115736
$669$	0	0
$670$	0	0
$671$	−340548.	−0.0291993
$672$	0	0
$673$	1.26562e7	1.07712	0.538562	−	0.842586i	$-0.318968\pi$
$673$	1.26562e7	1.07712	0.538562	+	0.842586i	$0.318968\pi$
$674$	7.99118e6	0.677581
$675$	0	0
$676$	−3.32923e6	−0.280206
$677$	−2.65061e6	−0.222267	−0.111133	−	0.993805i	$-0.535448\pi$
$677$	−2.65061e6	−0.222267	−0.111133	+	0.993805i	$0.535448\pi$
$678$	0	0
$679$	1.26398e7	1.05212
$680$	0	0
$681$	0	0
$682$	−1.44375e7	−1.18859
$683$	−5.74475e6	−0.471215	−0.235608	−	0.971848i	$-0.575708\pi$
$683$	−5.74475e6	−0.471215	−0.235608	+	0.971848i	$0.575708\pi$
$684$	0	0
$685$	0	0
$686$	9.41192e6	0.763604
$687$	0	0
$688$	2.29581e6	0.184912
$689$	−3.31846e6	−0.266310
$690$	0	0
$691$	−1.43749e7	−1.14527	−0.572636	−	0.819809i	$-0.694079\pi$
$691$	−1.43749e7	−1.14527	−0.572636	+	0.819809i	$0.694079\pi$
$692$	677088.	0.0537502
$693$	0	0
$694$	−1.06013e7	−0.835525
$695$	0	0
$696$	0	0
$697$	8.47584e6	0.660847
$698$	−8.13513e6	−0.632013
$699$	0	0
$700$	0	0
$701$	−4.26164e6	−0.327553	−0.163776	−	0.986497i	$-0.552368\pi$
$701$	−4.26164e6	−0.327553	−0.163776	+	0.986497i	$0.552368\pi$
$702$	0	0
$703$	−9.97139e6	−0.760970
$704$	−1.44998e6	−0.110263
$705$	0	0
$706$	3.79452e6	0.286514
$707$	−1.51698e7	−1.14138
$708$	0	0
$709$	1.97088e7	1.47246	0.736231	−	0.676731i	$-0.236604\pi$
$709$	1.97088e7	1.47246	0.736231	+	0.676731i	$0.236604\pi$
$710$	0	0
$711$	0	0
$712$	−4.69939e6	−0.347409
$713$	−1.10117e7	−0.811203
$714$	0	0
$715$	0	0
$716$	2.67850e6	0.195258
$717$	0	0
$718$	−1.97846e6	−0.143224
$719$	1.02211e7	0.737353	0.368676	−	0.929558i	$-0.379811\pi$
$719$	1.02211e7	0.737353	0.368676	+	0.929558i	$0.379811\pi$
$720$	0	0
$721$	2.71440e6	0.194463
$722$	2.99807e6	0.214042
$723$	0	0
$724$	−7.72989e6	−0.548058
$725$	0	0
$726$	0	0
$727$	1.32103e7	0.926996	0.463498	−	0.886098i	$-0.346594\pi$
$727$	1.32103e7	0.926996	0.463498	+	0.886098i	$0.346594\pi$
$728$	2.53389e6	0.177198
$729$	0	0
$730$	0	0
$731$	5.86507e6	0.405957
$732$	0	0
$733$	1.30586e7	0.897713	0.448857	−	0.893604i	$-0.351831\pi$
$733$	1.30586e7	0.897713	0.448857	+	0.893604i	$0.351831\pi$
$734$	300184.	0.0205659
$735$	0	0
$736$	−1.10592e6	−0.0752539
$737$	−6.35642e6	−0.431066
$738$	0	0
$739$	−1.53767e7	−1.03574	−0.517870	−	0.855459i	$-0.673275\pi$
$739$	−1.53767e7	−1.03574	−0.517870	+	0.855459i	$0.673275\pi$
$740$	0	0
$741$	0	0
$742$	−3.21989e6	−0.214699
$743$	2.62537e7	1.74469	0.872346	−	0.488889i	$-0.162598\pi$
$743$	2.62537e7	1.74469	0.872346	+	0.488889i	$0.162598\pi$
$744$	0	0
$745$	0	0
$746$	−2.13387e6	−0.140385
$747$	0	0
$748$	−3.70426e6	−0.242073
$749$	−2.16749e7	−1.41173
$750$	0	0
$751$	−1.36151e7	−0.880886	−0.440443	−	0.897781i	$-0.645179\pi$
$751$	−1.36151e7	−0.880886	−0.440443	+	0.897781i	$0.645179\pi$
$752$	−1.38240e6	−0.0891434
$753$	0	0
$754$	−9.29846e6	−0.595639
$755$	0	0
$756$	0	0
$757$	−8.41208e6	−0.533536	−0.266768	−	0.963761i	$-0.585956\pi$
$757$	−8.41208e6	−0.533536	−0.266768	+	0.963761i	$0.585956\pi$
$758$	−799120.	−0.0505172
$759$	0	0
$760$	0	0
$761$	−414684.	−0.0259571	−0.0129785	−	0.999916i	$-0.504131\pi$
$761$	−414684.	−0.0259571	−0.0129785	+	0.999916i	$0.504131\pi$
$762$	0	0
$763$	−1.20660e7	−0.750327
$764$	1.23612e7	0.766171
$765$	0	0
$766$	−3.47760e6	−0.214145
$767$	1.59742e6	0.0980459
$768$	0	0
$769$	6.71521e6	0.409491	0.204745	−	0.978815i	$-0.434363\pi$
$769$	6.71521e6	0.409491	0.204745	+	0.978815i	$0.434363\pi$
$770$	0	0
$771$	0	0
$772$	1.07291e7	0.647919
$773$	2.37237e7	1.42802	0.714009	−	0.700136i	$-0.246877\pi$
$773$	2.37237e7	1.42802	0.714009	+	0.700136i	$0.246877\pi$
$774$	0	0
$775$	0	0
$776$	−8.25459e6	−0.492086
$777$	0	0
$778$	4.20761e6	0.249222
$779$	2.32762e7	1.37426
$780$	0	0
$781$	−1.98764e7	−1.16603
$782$	−2.82528e6	−0.165213
$783$	0	0
$784$	−1.84397e6	−0.107143
$785$	0	0
$786$	0	0
$787$	3.17798e7	1.82900	0.914501	−	0.404585i	$-0.132584\pi$
$787$	3.17798e7	1.82900	0.914501	+	0.404585i	$0.132584\pi$
$788$	1.36308e7	0.781996
$789$	0	0
$790$	0	0
$791$	−2.15978e7	−1.22735
$792$	0	0
$793$	−388648.	−0.0219469
$794$	−1.67705e7	−0.944047
$795$	0	0
$796$	−8.98010e6	−0.502341
$797$	−7.26280e6	−0.405003	−0.202502	−	0.979282i	$-0.564907\pi$
$797$	−7.26280e6	−0.405003	−0.202502	+	0.979282i	$0.564907\pi$
$798$	0	0
$799$	−3.53160e6	−0.195706
$800$	0	0
$801$	0	0
$802$	7.61170e6	0.417874
$803$	−3.03343e7	−1.66014
$804$	0	0
$805$	0	0
$806$	−1.64767e7	−0.893375
$807$	0	0
$808$	9.90682e6	0.533833
$809$	3.22769e7	1.73388	0.866942	−	0.498409i	$-0.166082\pi$
$809$	3.22769e7	1.73388	0.866942	+	0.498409i	$0.166082\pi$
$810$	0	0
$811$	−1.00689e7	−0.537563	−0.268782	−	0.963201i	$-0.586621\pi$
$811$	−1.00689e7	−0.537563	−0.268782	+	0.963201i	$0.586621\pi$
$812$	−9.02227e6	−0.480204
$813$	0	0
$814$	7.86163e6	0.415864
$815$	0	0
$816$	0	0
$817$	1.61065e7	0.844203
$818$	6.51639e6	0.340505
$819$	0	0
$820$	0	0
$821$	−2.08671e7	−1.08045	−0.540224	−	0.841521i	$-0.681660\pi$
$821$	−2.08671e7	−1.08045	−0.540224	+	0.841521i	$0.681660\pi$
$822$	0	0
$823$	−1.40683e7	−0.724005	−0.362003	−	0.932177i	$-0.617907\pi$
$823$	−1.40683e7	−0.724005	−0.362003	+	0.932177i	$0.617907\pi$
$824$	−1.77267e6	−0.0909516
$825$	0	0
$826$	1.54997e6	0.0790447
$827$	−2.48210e7	−1.26199	−0.630995	−	0.775787i	$-0.717353\pi$
$827$	−2.48210e7	−1.26199	−0.630995	+	0.775787i	$0.717353\pi$
$828$	0	0
$829$	−1.40954e7	−0.712347	−0.356173	−	0.934420i	$-0.615919\pi$
$829$	−1.40954e7	−0.712347	−0.356173	+	0.934420i	$0.615919\pi$
$830$	0	0
$831$	0	0
$832$	−1.65478e6	−0.0828768
$833$	−4.71076e6	−0.235222
$834$	0	0
$835$	0	0
$836$	−1.01725e7	−0.503401
$837$	0	0
$838$	9.32110e6	0.458519
$839$	−6.47203e6	−0.317421	−0.158711	−	0.987325i	$-0.550734\pi$
$839$	−6.47203e6	−0.317421	−0.158711	+	0.987325i	$0.550734\pi$
$840$	0	0
$841$	1.25974e7	0.614172
$842$	2.45190e6	0.119185
$843$	0	0
$844$	−9.26022e6	−0.447471
$845$	0	0
$846$	0	0
$847$	3.50203e6	0.167730
$848$	2.10278e6	0.100416
$849$	0	0
$850$	0	0
$851$	5.99616e6	0.283824
$852$	0	0
$853$	−1.71706e7	−0.808002	−0.404001	−	0.914758i	$-0.632381\pi$
$853$	−1.71706e7	−0.808002	−0.404001	+	0.914758i	$0.632381\pi$
$854$	−377104.	−0.0176936
$855$	0	0
$856$	1.41550e7	0.660276
$857$	2.45494e6	0.114180	0.0570899	−	0.998369i	$-0.481818\pi$
$857$	2.45494e6	0.114180	0.0570899	+	0.998369i	$0.481818\pi$
$858$	0	0
$859$	1.13126e6	0.0523094	0.0261547	−	0.999658i	$-0.491674\pi$
$859$	1.13126e6	0.0523094	0.0261547	+	0.999658i	$0.491674\pi$
$860$	0	0
$861$	0	0
$862$	−2.67754e7	−1.22735
$863$	−1.84069e7	−0.841305	−0.420653	−	0.907222i	$-0.638199\pi$
$863$	−1.84069e7	−0.841305	−0.420653	+	0.907222i	$0.638199\pi$
$864$	0	0
$865$	0	0
$866$	−8.80777e6	−0.399090
$867$	0	0
$868$	−1.59873e7	−0.720239
$869$	9.21958e6	0.414154
$870$	0	0
$871$	−7.25422e6	−0.324000
$872$	7.87981e6	0.350933
$873$	0	0
$874$	−7.75872e6	−0.343567
$875$	0	0
$876$	0	0
$877$	−2.61539e7	−1.14825	−0.574127	−	0.818766i	$-0.694659\pi$
$877$	−2.61539e7	−1.14825	−0.574127	+	0.818766i	$0.694659\pi$
$878$	4.41747e6	0.193392
$879$	0	0
$880$	0	0
$881$	−1.97640e7	−0.857898	−0.428949	−	0.903329i	$-0.641116\pi$
$881$	−1.97640e7	−0.857898	−0.428949	+	0.903329i	$0.641116\pi$
$882$	0	0
$883$	−3.43551e7	−1.48282	−0.741411	−	0.671051i	$-0.765843\pi$
$883$	−3.43551e7	−1.48282	−0.741411	+	0.671051i	$0.765843\pi$
$884$	−4.22746e6	−0.181948
$885$	0	0
$886$	−1.52881e7	−0.654288
$887$	−2.11254e7	−0.901561	−0.450780	−	0.892635i	$-0.648854\pi$
$887$	−2.11254e7	−0.901561	−0.450780	+	0.892635i	$0.648854\pi$
$888$	0	0
$889$	1.77672e7	0.753988
$890$	0	0
$891$	0	0
$892$	−1.38098e7	−0.581134
$893$	−9.69840e6	−0.406978
$894$	0	0
$895$	0	0
$896$	−1.60563e6	−0.0668153
$897$	0	0
$898$	−1.57594e7	−0.652152
$899$	5.86678e7	2.42103
$900$	0	0
$901$	5.37196e6	0.220455
$902$	−1.83514e7	−0.751021
$903$	0	0
$904$	1.41047e7	0.574041
$905$	0	0
$906$	0	0
$907$	−2.61308e6	−0.105471	−0.0527357	−	0.998609i	$-0.516794\pi$
$907$	−2.61308e6	−0.105471	−0.0527357	+	0.998609i	$0.516794\pi$
$908$	−1.14048e7	−0.459064
$909$	0	0
$910$	0	0
$911$	−3.27046e7	−1.30561	−0.652803	−	0.757527i	$-0.726407\pi$
$911$	−3.27046e7	−1.30561	−0.652803	+	0.757527i	$0.726407\pi$
$912$	0	0
$913$	3.30877e7	1.31368
$914$	4.07010e6	0.161154
$915$	0	0
$916$	−5.49866e6	−0.216530
$917$	−2.49424e7	−0.979522
$918$	0	0
$919$	2.06463e7	0.806406	0.403203	−	0.915111i	$-0.367897\pi$
$919$	2.06463e7	0.806406	0.403203	+	0.915111i	$0.367897\pi$
$920$	0	0
$921$	0	0
$922$	2.72332e7	1.05505
$923$	−2.26838e7	−0.876418
$924$	0	0
$925$	0	0
$926$	−2.69713e7	−1.03365
$927$	0	0
$928$	5.89210e6	0.224595
$929$	−2.54318e7	−0.966803	−0.483401	−	0.875399i	$-0.660599\pi$
$929$	−2.54318e7	−0.966803	−0.483401	+	0.875399i	$0.660599\pi$
$930$	0	0
$931$	−1.29366e7	−0.489154
$932$	3.50102e6	0.132025
$933$	0	0
$934$	1.29048e7	0.484041
$935$	0	0
$936$	0	0
$937$	−2.52800e7	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$937$	−2.52800e7	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$938$	−7.03875e6	−0.261209
$939$	0	0
$940$	0	0
$941$	−1.80180e7	−0.663336	−0.331668	−	0.943396i	$-0.607611\pi$
$941$	−1.80180e7	−0.663336	−0.331668	+	0.943396i	$0.607611\pi$
$942$	0	0
$943$	−1.39968e7	−0.512566
$944$	−1.01222e6	−0.0369698
$945$	0	0
$946$	−1.26987e7	−0.461351
$947$	−5.10485e7	−1.84973	−0.924864	−	0.380299i	$-0.875821\pi$
$947$	−5.10485e7	−1.84973	−0.924864	+	0.380299i	$0.875821\pi$
$948$	0	0
$949$	−3.46188e7	−1.24780
$950$	0	0
$951$	0	0
$952$	−4.10189e6	−0.146687
$953$	4.53610e7	1.61790	0.808948	−	0.587880i	$-0.200037\pi$
$953$	4.53610e7	1.61790	0.808948	+	0.587880i	$0.200037\pi$
$954$	0	0
$955$	0	0
$956$	1.63352e7	0.578068
$957$	0	0
$958$	−2.48485e7	−0.874755
$959$	−4.07196e7	−1.42974
$960$	0	0
$961$	7.53293e7	2.63121
$962$	8.97203e6	0.312574
$963$	0	0
$964$	2.19493e7	0.760725
$965$	0	0
$966$	0	0
$967$	1.24701e7	0.428847	0.214424	−	0.976741i	$-0.431213\pi$
$967$	1.24701e7	0.428847	0.214424	+	0.976741i	$0.431213\pi$
$968$	−2.28704e6	−0.0784486
$969$	0	0
$970$	0	0
$971$	2.13786e7	0.727664	0.363832	−	0.931465i	$-0.381468\pi$
$971$	2.13786e7	0.727664	0.363832	+	0.931465i	$0.381468\pi$
$972$	0	0
$973$	3.06093e7	1.03651
$974$	6.05303e6	0.204445
$975$	0	0
$976$	246272.	0.00827543
$977$	1.31824e7	0.441833	0.220917	−	0.975293i	$-0.429095\pi$
$977$	1.31824e7	0.441833	0.220917	+	0.975293i	$0.429095\pi$
$978$	0	0
$979$	2.59935e7	0.866779
$980$	0	0
$981$	0	0
$982$	1.37534e7	0.455126
$983$	−2.98823e7	−0.986349	−0.493175	−	0.869930i	$-0.664164\pi$
$983$	−2.98823e7	−0.986349	−0.493175	+	0.869930i	$0.664164\pi$
$984$	0	0
$985$	0	0
$986$	1.50525e7	0.493078
$987$	0	0
$988$	−1.16093e7	−0.378369
$989$	−9.68544e6	−0.314868
$990$	0	0
$991$	−3.76534e7	−1.21793	−0.608963	−	0.793199i	$-0.708414\pi$
$991$	−3.76534e7	−1.21793	−0.608963	+	0.793199i	$0.708414\pi$
$992$	1.04407e7	0.336861
$993$	0	0
$994$	−2.20100e7	−0.706569
$995$	0	0
$996$	0	0
$997$	−1.53443e7	−0.488887	−0.244444	−	0.969664i	$-0.578605\pi$
$997$	−1.53443e7	−0.488887	−0.244444	+	0.969664i	$0.578605\pi$
$998$	2.10936e7	0.670385
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.a.o.1.1		1
3.2	odd	2		450.6.a.d.1.1		1
5.2	odd	4		450.6.c.d.199.2		2
5.3	odd	4		450.6.c.d.199.1		2
5.4	even	2		90.6.a.c.1.1	✓	1
15.2	even	4		450.6.c.l.199.1		2
15.8	even	4		450.6.c.l.199.2		2
15.14	odd	2		90.6.a.e.1.1	yes	1
20.19	odd	2		720.6.a.o.1.1		1
60.59	even	2		720.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.6.a.c.1.1	✓	1	5.4	even	2
90.6.a.e.1.1	yes	1	15.14	odd	2
450.6.a.d.1.1		1	3.2	odd	2
450.6.a.o.1.1		1	1.1	even	1	trivial
450.6.c.d.199.1		2	5.3	odd	4
450.6.c.d.199.2		2	5.2	odd	4
450.6.c.l.199.1		2	15.2	even	4
450.6.c.l.199.2		2	15.8	even	4
720.6.a.c.1.1		1	60.59	even	2
720.6.a.o.1.1		1	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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