LMFDB - Embedded newform 450.4.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,4,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, 4, names="a")

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

chi := DirichletCharacter("450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$26.5508595026$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
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)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} -23.0000 q^{7} +8.00000 q^{8} +O(q^{10})$

$q+2.00000 q^{2} +4.00000 q^{4} -23.0000 q^{7} +8.00000 q^{8} +30.0000 q^{11} -29.0000 q^{13} -46.0000 q^{14} +16.0000 q^{16} +78.0000 q^{17} +149.000 q^{19} +60.0000 q^{22} +150.000 q^{23} -58.0000 q^{26} -92.0000 q^{28} +234.000 q^{29} -217.000 q^{31} +32.0000 q^{32} +156.000 q^{34} -146.000 q^{37} +298.000 q^{38} +156.000 q^{41} +433.000 q^{43} +120.000 q^{44} +300.000 q^{46} +30.0000 q^{47} +186.000 q^{49} -116.000 q^{52} -552.000 q^{53} -184.000 q^{56} +468.000 q^{58} +270.000 q^{59} +275.000 q^{61} -434.000 q^{62} +64.0000 q^{64} -803.000 q^{67} +312.000 q^{68} -660.000 q^{71} +646.000 q^{73} -292.000 q^{74} +596.000 q^{76} -690.000 q^{77} +992.000 q^{79} +312.000 q^{82} -846.000 q^{83} +866.000 q^{86} +240.000 q^{88} +1488.00 q^{89} +667.000 q^{91} +600.000 q^{92} +60.0000 q^{94} +319.000 q^{97} +372.000 q^{98} +O(q^{100})$

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−23.0000	−1.24188	−0.620942	−	0.783857i	$-0.713250\pi$
$7$	−23.0000	−1.24188	−0.620942	+	0.783857i	$0.713250\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	0	0
$11$	30.0000	0.822304	0.411152	−	0.911567i	$-0.365127\pi$
$11$	30.0000	0.822304	0.411152	+	0.911567i	$0.365127\pi$
$12$	0	0
$13$	−29.0000	−0.618704	−0.309352	−	0.950948i	$-0.600112\pi$
$13$	−29.0000	−0.618704	−0.309352	+	0.950948i	$0.600112\pi$
$14$	−46.0000	−0.878144
$15$	0	0
$16$	16.0000	0.250000
$17$	78.0000	1.11281	0.556405	−	0.830911i	$-0.312180\pi$
$17$	78.0000	1.11281	0.556405	+	0.830911i	$0.312180\pi$
$18$	0	0
$19$	149.000	1.79910	0.899551	−	0.436815i	$-0.143894\pi$
$19$	149.000	1.79910	0.899551	+	0.436815i	$0.143894\pi$
$20$	0	0
$21$	0	0
$22$	60.0000	0.581456
$23$	150.000	1.35988	0.679938	−	0.733269i	$-0.262007\pi$
$23$	150.000	1.35988	0.679938	+	0.733269i	$0.262007\pi$
$24$	0	0
$25$	0	0
$26$	−58.0000	−0.437490
$27$	0	0
$28$	−92.0000	−0.620942
$29$	234.000	1.49837	0.749185	−	0.662361i	$-0.230446\pi$
$29$	234.000	1.49837	0.749185	+	0.662361i	$0.230446\pi$
$30$	0	0
$31$	−217.000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−217.000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	156.000	0.786876
$35$	0	0
$36$	0	0
$37$	−146.000	−0.648710	−0.324355	−	0.945936i	$-0.605147\pi$
$37$	−146.000	−0.648710	−0.324355	+	0.945936i	$0.605147\pi$
$38$	298.000	1.27216
$39$	0	0
$40$	0	0
$41$	156.000	0.594222	0.297111	−	0.954843i	$-0.403977\pi$
$41$	156.000	0.594222	0.297111	+	0.954843i	$0.403977\pi$
$42$	0	0
$43$	433.000	1.53563	0.767813	−	0.640675i	$-0.221345\pi$
$43$	433.000	1.53563	0.767813	+	0.640675i	$0.221345\pi$
$44$	120.000	0.411152
$45$	0	0
$46$	300.000	0.961578
$47$	30.0000	0.0931053	0.0465527	−	0.998916i	$-0.485176\pi$
$47$	30.0000	0.0931053	0.0465527	+	0.998916i	$0.485176\pi$
$48$	0	0
$49$	186.000	0.542274
$50$	0	0
$51$	0	0
$52$	−116.000	−0.309352
$53$	−552.000	−1.43062	−0.715312	−	0.698806i	$-0.753715\pi$
$53$	−552.000	−1.43062	−0.715312	+	0.698806i	$0.753715\pi$
$54$	0	0
$55$	0	0
$56$	−184.000	−0.439072
$57$	0	0
$58$	468.000	1.05951
$59$	270.000	0.595780	0.297890	−	0.954600i	$-0.403717\pi$
$59$	270.000	0.595780	0.297890	+	0.954600i	$0.403717\pi$
$60$	0	0
$61$	275.000	0.577215	0.288608	−	0.957447i	$-0.406808\pi$
$61$	275.000	0.577215	0.288608	+	0.957447i	$0.406808\pi$
$62$	−434.000	−0.889001
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	−803.000	−1.46421	−0.732105	−	0.681192i	$-0.761462\pi$
$67$	−803.000	−1.46421	−0.732105	+	0.681192i	$0.761462\pi$
$68$	312.000	0.556405
$69$	0	0
$70$	0	0
$71$	−660.000	−1.10321	−0.551603	−	0.834107i	$-0.685984\pi$
$71$	−660.000	−1.10321	−0.551603	+	0.834107i	$0.685984\pi$
$72$	0	0
$73$	646.000	1.03573	0.517867	−	0.855461i	$-0.326726\pi$
$73$	646.000	1.03573	0.517867	+	0.855461i	$0.326726\pi$
$74$	−292.000	−0.458707
$75$	0	0
$76$	596.000	0.899551
$77$	−690.000	−1.02121
$78$	0	0
$79$	992.000	1.41277	0.706384	−	0.707829i	$-0.250325\pi$
$79$	992.000	1.41277	0.706384	+	0.707829i	$0.250325\pi$
$80$	0	0
$81$	0	0
$82$	312.000	0.420178
$83$	−846.000	−1.11880	−0.559401	−	0.828897i	$-0.688969\pi$
$83$	−846.000	−1.11880	−0.559401	+	0.828897i	$0.688969\pi$
$84$	0	0
$85$	0	0
$86$	866.000	1.08585
$87$	0	0
$88$	240.000	0.290728
$89$	1488.00	1.77222	0.886111	−	0.463474i	$-0.153397\pi$
$89$	1488.00	1.77222	0.886111	+	0.463474i	$0.153397\pi$
$90$	0	0
$91$	667.000	0.768358
$92$	600.000	0.679938
$93$	0	0
$94$	60.0000	0.0658354
$95$	0	0
$96$	0	0
$97$	319.000	0.333913	0.166956	−	0.985964i	$-0.446606\pi$
$97$	319.000	0.333913	0.166956	+	0.985964i	$0.446606\pi$
$98$	372.000	0.383446
$99$	0	0
$100$	0	0
$101$	792.000	0.780267	0.390133	−	0.920758i	$-0.372429\pi$
$101$	792.000	0.780267	0.390133	+	0.920758i	$0.372429\pi$
$102$	0	0
$103$	−812.000	−0.776784	−0.388392	−	0.921494i	$-0.626969\pi$
$103$	−812.000	−0.776784	−0.388392	+	0.921494i	$0.626969\pi$
$104$	−232.000	−0.218745
$105$	0	0
$106$	−1104.00	−1.01160
$107$	−1416.00	−1.27934	−0.639672	−	0.768648i	$-0.720930\pi$
$107$	−1416.00	−1.27934	−0.639672	+	0.768648i	$0.720930\pi$
$108$	0	0
$109$	−55.0000	−0.0483307	−0.0241653	−	0.999708i	$-0.507693\pi$
$109$	−55.0000	−0.0483307	−0.0241653	+	0.999708i	$0.507693\pi$
$110$	0	0
$111$	0	0
$112$	−368.000	−0.310471
$113$	1404.00	1.16882	0.584412	−	0.811457i	$-0.301325\pi$
$113$	1404.00	1.16882	0.584412	+	0.811457i	$0.301325\pi$
$114$	0	0
$115$	0	0
$116$	936.000	0.749185
$117$	0	0
$118$	540.000	0.421280
$119$	−1794.00	−1.38198
$120$	0	0
$121$	−431.000	−0.323817
$122$	550.000	0.408153
$123$	0	0
$124$	−868.000	−0.628619
$125$	0	0
$126$	0	0
$127$	−1280.00	−0.894344	−0.447172	−	0.894448i	$-0.647569\pi$
$127$	−1280.00	−0.894344	−0.447172	+	0.894448i	$0.647569\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	0	0
$131$	−480.000	−0.320136	−0.160068	−	0.987106i	$-0.551171\pi$
$131$	−480.000	−0.320136	−0.160068	+	0.987106i	$0.551171\pi$
$132$	0	0
$133$	−3427.00	−2.23428
$134$	−1606.00	−1.03535
$135$	0	0
$136$	624.000	0.393438
$137$	−282.000	−0.175860	−0.0879302	−	0.996127i	$-0.528025\pi$
$137$	−282.000	−0.175860	−0.0879302	+	0.996127i	$0.528025\pi$
$138$	0	0
$139$	1604.00	0.978773	0.489387	−	0.872067i	$-0.337221\pi$
$139$	1604.00	0.978773	0.489387	+	0.872067i	$0.337221\pi$
$140$	0	0
$141$	0	0
$142$	−1320.00	−0.780084
$143$	−870.000	−0.508763
$144$	0	0
$145$	0	0
$146$	1292.00	0.732375
$147$	0	0
$148$	−584.000	−0.324355
$149$	774.000	0.425561	0.212780	−	0.977100i	$-0.431748\pi$
$149$	774.000	0.425561	0.212780	+	0.977100i	$0.431748\pi$
$150$	0	0
$151$	293.000	0.157907	0.0789536	−	0.996878i	$-0.474842\pi$
$151$	293.000	0.157907	0.0789536	+	0.996878i	$0.474842\pi$
$152$	1192.00	0.636079
$153$	0	0
$154$	−1380.00	−0.722101
$155$	0	0
$156$	0	0
$157$	1729.00	0.878912	0.439456	−	0.898264i	$-0.355171\pi$
$157$	1729.00	0.878912	0.439456	+	0.898264i	$0.355171\pi$
$158$	1984.00	0.998978
$159$	0	0
$160$	0	0
$161$	−3450.00	−1.68881
$162$	0	0
$163$	1123.00	0.539633	0.269816	−	0.962912i	$-0.413037\pi$
$163$	1123.00	0.539633	0.269816	+	0.962912i	$0.413037\pi$
$164$	624.000	0.297111
$165$	0	0
$166$	−1692.00	−0.791112
$167$	1200.00	0.556041	0.278020	−	0.960575i	$-0.410322\pi$
$167$	1200.00	0.556041	0.278020	+	0.960575i	$0.410322\pi$
$168$	0	0
$169$	−1356.00	−0.617205
$170$	0	0
$171$	0	0
$172$	1732.00	0.767813
$173$	−1734.00	−0.762044	−0.381022	−	0.924566i	$-0.624428\pi$
$173$	−1734.00	−0.762044	−0.381022	+	0.924566i	$0.624428\pi$
$174$	0	0
$175$	0	0
$176$	480.000	0.205576
$177$	0	0
$178$	2976.00	1.25315
$179$	−2586.00	−1.07981	−0.539907	−	0.841725i	$-0.681541\pi$
$179$	−2586.00	−1.07981	−0.539907	+	0.841725i	$0.681541\pi$
$180$	0	0
$181$	−3931.00	−1.61430	−0.807152	−	0.590344i	$-0.798992\pi$
$181$	−3931.00	−1.61430	−0.807152	+	0.590344i	$0.798992\pi$
$182$	1334.00	0.543311
$183$	0	0
$184$	1200.00	0.480789
$185$	0	0
$186$	0	0
$187$	2340.00	0.915068
$188$	120.000	0.0465527
$189$	0	0
$190$	0	0
$191$	−1566.00	−0.593255	−0.296628	−	0.954993i	$-0.595862\pi$
$191$	−1566.00	−0.593255	−0.296628	+	0.954993i	$0.595862\pi$
$192$	0	0
$193$	−2291.00	−0.854455	−0.427227	−	0.904144i	$-0.640510\pi$
$193$	−2291.00	−0.854455	−0.427227	+	0.904144i	$0.640510\pi$
$194$	638.000	0.236112
$195$	0	0
$196$	744.000	0.271137
$197$	2142.00	0.774676	0.387338	−	0.921938i	$-0.373395\pi$
$197$	2142.00	0.774676	0.387338	+	0.921938i	$0.373395\pi$
$198$	0	0
$199$	−4903.00	−1.74656	−0.873278	−	0.487223i	$-0.838010\pi$
$199$	−4903.00	−1.74656	−0.873278	+	0.487223i	$0.838010\pi$
$200$	0	0
$201$	0	0
$202$	1584.00	0.551732
$203$	−5382.00	−1.86080
$204$	0	0
$205$	0	0
$206$	−1624.00	−0.549269
$207$	0	0
$208$	−464.000	−0.154676
$209$	4470.00	1.47941
$210$	0	0
$211$	605.000	0.197393	0.0986965	−	0.995118i	$-0.468533\pi$
$211$	605.000	0.197393	0.0986965	+	0.995118i	$0.468533\pi$
$212$	−2208.00	−0.715312
$213$	0	0
$214$	−2832.00	−0.904633
$215$	0	0
$216$	0	0
$217$	4991.00	1.56134
$218$	−110.000	−0.0341750
$219$	0	0
$220$	0	0
$221$	−2262.00	−0.688500
$222$	0	0
$223$	145.000	0.0435422	0.0217711	−	0.999763i	$-0.493069\pi$
$223$	145.000	0.0435422	0.0217711	+	0.999763i	$0.493069\pi$
$224$	−736.000	−0.219536
$225$	0	0
$226$	2808.00	0.826484
$227$	2964.00	0.866641	0.433321	−	0.901240i	$-0.357342\pi$
$227$	2964.00	0.866641	0.433321	+	0.901240i	$0.357342\pi$
$228$	0	0
$229$	−5635.00	−1.62608	−0.813038	−	0.582211i	$-0.802188\pi$
$229$	−5635.00	−1.62608	−0.813038	+	0.582211i	$0.802188\pi$
$230$	0	0
$231$	0	0
$232$	1872.00	0.529754
$233$	4164.00	1.17078	0.585392	−	0.810750i	$-0.300941\pi$
$233$	4164.00	1.17078	0.585392	+	0.810750i	$0.300941\pi$
$234$	0	0
$235$	0	0
$236$	1080.00	0.297890
$237$	0	0
$238$	−3588.00	−0.977208
$239$	1944.00	0.526138	0.263069	−	0.964777i	$-0.415265\pi$
$239$	1944.00	0.526138	0.263069	+	0.964777i	$0.415265\pi$
$240$	0	0
$241$	857.000	0.229063	0.114532	−	0.993420i	$-0.463463\pi$
$241$	857.000	0.229063	0.114532	+	0.993420i	$0.463463\pi$
$242$	−862.000	−0.228973
$243$	0	0
$244$	1100.00	0.288608
$245$	0	0
$246$	0	0
$247$	−4321.00	−1.11311
$248$	−1736.00	−0.444500
$249$	0	0
$250$	0	0
$251$	3924.00	0.986776	0.493388	−	0.869809i	$-0.335758\pi$
$251$	3924.00	0.986776	0.493388	+	0.869809i	$0.335758\pi$
$252$	0	0
$253$	4500.00	1.11823
$254$	−2560.00	−0.632396
$255$	0	0
$256$	256.000	0.0625000
$257$	−2844.00	−0.690287	−0.345144	−	0.938550i	$-0.612170\pi$
$257$	−2844.00	−0.690287	−0.345144	+	0.938550i	$0.612170\pi$
$258$	0	0
$259$	3358.00	0.805621
$260$	0	0
$261$	0	0
$262$	−960.000	−0.226370
$263$	−6060.00	−1.42082	−0.710410	−	0.703788i	$-0.751490\pi$
$263$	−6060.00	−1.42082	−0.710410	+	0.703788i	$0.751490\pi$
$264$	0	0
$265$	0	0
$266$	−6854.00	−1.57987
$267$	0	0
$268$	−3212.00	−0.732105
$269$	−3906.00	−0.885327	−0.442664	−	0.896688i	$-0.645966\pi$
$269$	−3906.00	−0.885327	−0.442664	+	0.896688i	$0.645966\pi$
$270$	0	0
$271$	2144.00	0.480586	0.240293	−	0.970700i	$-0.422757\pi$
$271$	2144.00	0.480586	0.240293	+	0.970700i	$0.422757\pi$
$272$	1248.00	0.278203
$273$	0	0
$274$	−564.000	−0.124352
$275$	0	0
$276$	0	0
$277$	−2321.00	−0.503449	−0.251725	−	0.967799i	$-0.580998\pi$
$277$	−2321.00	−0.503449	−0.251725	+	0.967799i	$0.580998\pi$
$278$	3208.00	0.692097
$279$	0	0
$280$	0	0
$281$	6822.00	1.44828	0.724140	−	0.689654i	$-0.242237\pi$
$281$	6822.00	1.44828	0.724140	+	0.689654i	$0.242237\pi$
$282$	0	0
$283$	−4049.00	−0.850488	−0.425244	−	0.905079i	$-0.639812\pi$
$283$	−4049.00	−0.850488	−0.425244	+	0.905079i	$0.639812\pi$
$284$	−2640.00	−0.551603
$285$	0	0
$286$	−1740.00	−0.359749
$287$	−3588.00	−0.737955
$288$	0	0
$289$	1171.00	0.238347
$290$	0	0
$291$	0	0
$292$	2584.00	0.517867
$293$	2238.00	0.446230	0.223115	−	0.974792i	$-0.428377\pi$
$293$	2238.00	0.446230	0.223115	+	0.974792i	$0.428377\pi$
$294$	0	0
$295$	0	0
$296$	−1168.00	−0.229353
$297$	0	0
$298$	1548.00	0.300917
$299$	−4350.00	−0.841361
$300$	0	0
$301$	−9959.00	−1.90707
$302$	586.000	0.111657
$303$	0	0
$304$	2384.00	0.449776
$305$	0	0
$306$	0	0
$307$	−1385.00	−0.257479	−0.128740	−	0.991678i	$-0.541093\pi$
$307$	−1385.00	−0.257479	−0.128740	+	0.991678i	$0.541093\pi$
$308$	−2760.00	−0.510603
$309$	0	0
$310$	0	0
$311$	5670.00	1.03381	0.516907	−	0.856042i	$-0.327083\pi$
$311$	5670.00	1.03381	0.516907	+	0.856042i	$0.327083\pi$
$312$	0	0
$313$	421.000	0.0760266	0.0380133	−	0.999277i	$-0.487897\pi$
$313$	421.000	0.0760266	0.0380133	+	0.999277i	$0.487897\pi$
$314$	3458.00	0.621485
$315$	0	0
$316$	3968.00	0.706384
$317$	9984.00	1.76895	0.884475	−	0.466587i	$-0.154517\pi$
$317$	9984.00	1.76895	0.884475	+	0.466587i	$0.154517\pi$
$318$	0	0
$319$	7020.00	1.23211
$320$	0	0
$321$	0	0
$322$	−6900.00	−1.19417
$323$	11622.0	2.00206
$324$	0	0
$325$	0	0
$326$	2246.00	0.381578
$327$	0	0
$328$	1248.00	0.210089
$329$	−690.000	−0.115626
$330$	0	0
$331$	−4228.00	−0.702090	−0.351045	−	0.936359i	$-0.614174\pi$
$331$	−4228.00	−0.702090	−0.351045	+	0.936359i	$0.614174\pi$
$332$	−3384.00	−0.559401
$333$	0	0
$334$	2400.00	0.393180
$335$	0	0
$336$	0	0
$337$	−5393.00	−0.871737	−0.435869	−	0.900010i	$-0.643559\pi$
$337$	−5393.00	−0.871737	−0.435869	+	0.900010i	$0.643559\pi$
$338$	−2712.00	−0.436430
$339$	0	0
$340$	0	0
$341$	−6510.00	−1.03383
$342$	0	0
$343$	3611.00	0.568442
$344$	3464.00	0.542925
$345$	0	0
$346$	−3468.00	−0.538846
$347$	−7914.00	−1.22434	−0.612170	−	0.790726i	$-0.709703\pi$
$347$	−7914.00	−1.22434	−0.612170	+	0.790726i	$0.709703\pi$
$348$	0	0
$349$	1010.00	0.154911	0.0774557	−	0.996996i	$-0.475320\pi$
$349$	1010.00	0.154911	0.0774557	+	0.996996i	$0.475320\pi$
$350$	0	0
$351$	0	0
$352$	960.000	0.145364
$353$	4722.00	0.711974	0.355987	−	0.934491i	$-0.384145\pi$
$353$	4722.00	0.711974	0.355987	+	0.934491i	$0.384145\pi$
$354$	0	0
$355$	0	0
$356$	5952.00	0.886111
$357$	0	0
$358$	−5172.00	−0.763544
$359$	−6204.00	−0.912074	−0.456037	−	0.889961i	$-0.650732\pi$
$359$	−6204.00	−0.912074	−0.456037	+	0.889961i	$0.650732\pi$
$360$	0	0
$361$	15342.0	2.23677
$362$	−7862.00	−1.14148
$363$	0	0
$364$	2668.00	0.384179
$365$	0	0
$366$	0	0
$367$	−1361.00	−0.193579	−0.0967897	−	0.995305i	$-0.530857\pi$
$367$	−1361.00	−0.193579	−0.0967897	+	0.995305i	$0.530857\pi$
$368$	2400.00	0.339969
$369$	0	0
$370$	0	0
$371$	12696.0	1.77667
$372$	0	0
$373$	913.000	0.126738	0.0633691	−	0.997990i	$-0.479815\pi$
$373$	913.000	0.126738	0.0633691	+	0.997990i	$0.479815\pi$
$374$	4680.00	0.647051
$375$	0	0
$376$	240.000	0.0329177
$377$	−6786.00	−0.927047
$378$	0	0
$379$	−8881.00	−1.20366	−0.601829	−	0.798625i	$-0.705561\pi$
$379$	−8881.00	−1.20366	−0.601829	+	0.798625i	$0.705561\pi$
$380$	0	0
$381$	0	0
$382$	−3132.00	−0.419495
$383$	5460.00	0.728441	0.364221	−	0.931313i	$-0.381335\pi$
$383$	5460.00	0.728441	0.364221	+	0.931313i	$0.381335\pi$
$384$	0	0
$385$	0	0
$386$	−4582.00	−0.604191
$387$	0	0
$388$	1276.00	0.166956
$389$	13884.0	1.80963	0.904816	−	0.425803i	$-0.140008\pi$
$389$	13884.0	1.80963	0.904816	+	0.425803i	$0.140008\pi$
$390$	0	0
$391$	11700.0	1.51328
$392$	1488.00	0.191723
$393$	0	0
$394$	4284.00	0.547779
$395$	0	0
$396$	0	0
$397$	3781.00	0.477992	0.238996	−	0.971021i	$-0.423182\pi$
$397$	3781.00	0.477992	0.238996	+	0.971021i	$0.423182\pi$
$398$	−9806.00	−1.23500
$399$	0	0
$400$	0	0
$401$	−9024.00	−1.12378	−0.561892	−	0.827211i	$-0.689926\pi$
$401$	−9024.00	−1.12378	−0.561892	+	0.827211i	$0.689926\pi$
$402$	0	0
$403$	6293.00	0.777858
$404$	3168.00	0.390133
$405$	0	0
$406$	−10764.0	−1.31578
$407$	−4380.00	−0.533436
$408$	0	0
$409$	14789.0	1.78794	0.893972	−	0.448123i	$-0.147907\pi$
$409$	14789.0	1.78794	0.893972	+	0.448123i	$0.147907\pi$
$410$	0	0
$411$	0	0
$412$	−3248.00	−0.388392
$413$	−6210.00	−0.739889
$414$	0	0
$415$	0	0
$416$	−928.000	−0.109372
$417$	0	0
$418$	8940.00	1.04610
$419$	−9840.00	−1.14729	−0.573646	−	0.819103i	$-0.694472\pi$
$419$	−9840.00	−1.14729	−0.573646	+	0.819103i	$0.694472\pi$
$420$	0	0
$421$	5510.00	0.637865	0.318932	−	0.947778i	$-0.396676\pi$
$421$	5510.00	0.637865	0.318932	+	0.947778i	$0.396676\pi$
$422$	1210.00	0.139578
$423$	0	0
$424$	−4416.00	−0.505802
$425$	0	0
$426$	0	0
$427$	−6325.00	−0.716834
$428$	−5664.00	−0.639672
$429$	0	0
$430$	0	0
$431$	−11070.0	−1.23718	−0.618588	−	0.785715i	$-0.712295\pi$
$431$	−11070.0	−1.23718	−0.618588	+	0.785715i	$0.712295\pi$
$432$	0	0
$433$	12133.0	1.34659	0.673297	−	0.739373i	$-0.264878\pi$
$433$	12133.0	1.34659	0.673297	+	0.739373i	$0.264878\pi$
$434$	9982.00	1.10404
$435$	0	0
$436$	−220.000	−0.0241653
$437$	22350.0	2.44656
$438$	0	0
$439$	−1873.00	−0.203630	−0.101815	−	0.994803i	$-0.532465\pi$
$439$	−1873.00	−0.203630	−0.101815	+	0.994803i	$0.532465\pi$
$440$	0	0
$441$	0	0
$442$	−4524.00	−0.486843
$443$	−576.000	−0.0617756	−0.0308878	−	0.999523i	$-0.509833\pi$
$443$	−576.000	−0.0617756	−0.0308878	+	0.999523i	$0.509833\pi$
$444$	0	0
$445$	0	0
$446$	290.000	0.0307890
$447$	0	0
$448$	−1472.00	−0.155235
$449$	4884.00	0.513341	0.256671	−	0.966499i	$-0.417374\pi$
$449$	4884.00	0.513341	0.256671	+	0.966499i	$0.417374\pi$
$450$	0	0
$451$	4680.00	0.488631
$452$	5616.00	0.584412
$453$	0	0
$454$	5928.00	0.612808
$455$	0	0
$456$	0	0
$457$	15802.0	1.61748	0.808738	−	0.588169i	$-0.200151\pi$
$457$	15802.0	1.61748	0.808738	+	0.588169i	$0.200151\pi$
$458$	−11270.0	−1.14981
$459$	0	0
$460$	0	0
$461$	15360.0	1.55181	0.775907	−	0.630847i	$-0.217292\pi$
$461$	15360.0	1.55181	0.775907	+	0.630847i	$0.217292\pi$
$462$	0	0
$463$	−1712.00	−0.171843	−0.0859216	−	0.996302i	$-0.527383\pi$
$463$	−1712.00	−0.171843	−0.0859216	+	0.996302i	$0.527383\pi$
$464$	3744.00	0.374592
$465$	0	0
$466$	8328.00	0.827869
$467$	−16278.0	−1.61297	−0.806484	−	0.591256i	$-0.798632\pi$
$467$	−16278.0	−1.61297	−0.806484	+	0.591256i	$0.798632\pi$
$468$	0	0
$469$	18469.0	1.81838
$470$	0	0
$471$	0	0
$472$	2160.00	0.210640
$473$	12990.0	1.26275
$474$	0	0
$475$	0	0
$476$	−7176.00	−0.690990
$477$	0	0
$478$	3888.00	0.372036
$479$	14766.0	1.40851	0.704254	−	0.709948i	$-0.251281\pi$
$479$	14766.0	1.40851	0.704254	+	0.709948i	$0.251281\pi$
$480$	0	0
$481$	4234.00	0.401359
$482$	1714.00	0.161972
$483$	0	0
$484$	−1724.00	−0.161908
$485$	0	0
$486$	0	0
$487$	3319.00	0.308826	0.154413	−	0.988006i	$-0.450651\pi$
$487$	3319.00	0.308826	0.154413	+	0.988006i	$0.450651\pi$
$488$	2200.00	0.204076
$489$	0	0
$490$	0	0
$491$	−11064.0	−1.01693	−0.508464	−	0.861083i	$-0.669786\pi$
$491$	−11064.0	−1.01693	−0.508464	+	0.861083i	$0.669786\pi$
$492$	0	0
$493$	18252.0	1.66740
$494$	−8642.00	−0.787089
$495$	0	0
$496$	−3472.00	−0.314309
$497$	15180.0	1.37005
$498$	0	0
$499$	−14131.0	−1.26772	−0.633858	−	0.773449i	$-0.718530\pi$
$499$	−14131.0	−1.26772	−0.633858	+	0.773449i	$0.718530\pi$
$500$	0	0
$501$	0	0
$502$	7848.00	0.697756
$503$	11988.0	1.06266	0.531331	−	0.847165i	$-0.321692\pi$
$503$	11988.0	1.06266	0.531331	+	0.847165i	$0.321692\pi$
$504$	0	0
$505$	0	0
$506$	9000.00	0.790709
$507$	0	0
$508$	−5120.00	−0.447172
$509$	−10806.0	−0.940997	−0.470499	−	0.882401i	$-0.655926\pi$
$509$	−10806.0	−0.940997	−0.470499	+	0.882401i	$0.655926\pi$
$510$	0	0
$511$	−14858.0	−1.28626
$512$	512.000	0.0441942
$513$	0	0
$514$	−5688.00	−0.488107
$515$	0	0
$516$	0	0
$517$	900.000	0.0765608
$518$	6716.00	0.569660
$519$	0	0
$520$	0	0
$521$	−22578.0	−1.89858	−0.949290	−	0.314402i	$-0.898196\pi$
$521$	−22578.0	−1.89858	−0.949290	+	0.314402i	$0.898196\pi$
$522$	0	0
$523$	−12065.0	−1.00873	−0.504365	−	0.863491i	$-0.668273\pi$
$523$	−12065.0	−1.00873	−0.504365	+	0.863491i	$0.668273\pi$
$524$	−1920.00	−0.160068
$525$	0	0
$526$	−12120.0	−1.00467
$527$	−16926.0	−1.39907
$528$	0	0
$529$	10333.0	0.849264
$530$	0	0
$531$	0	0
$532$	−13708.0	−1.11714
$533$	−4524.00	−0.367648
$534$	0	0
$535$	0	0
$536$	−6424.00	−0.517676
$537$	0	0
$538$	−7812.00	−0.626021
$539$	5580.00	0.445914
$540$	0	0
$541$	−12055.0	−0.958013	−0.479006	−	0.877811i	$-0.659003\pi$
$541$	−12055.0	−0.958013	−0.479006	+	0.877811i	$0.659003\pi$
$542$	4288.00	0.339825
$543$	0	0
$544$	2496.00	0.196719
$545$	0	0
$546$	0	0
$547$	−6176.00	−0.482754	−0.241377	−	0.970431i	$-0.577599\pi$
$547$	−6176.00	−0.482754	−0.241377	+	0.970431i	$0.577599\pi$
$548$	−1128.00	−0.0879302
$549$	0	0
$550$	0	0
$551$	34866.0	2.69572
$552$	0	0
$553$	−22816.0	−1.75449
$554$	−4642.00	−0.355992
$555$	0	0
$556$	6416.00	0.489387
$557$	8274.00	0.629409	0.314704	−	0.949190i	$-0.398095\pi$
$557$	8274.00	0.629409	0.314704	+	0.949190i	$0.398095\pi$
$558$	0	0
$559$	−12557.0	−0.950098
$560$	0	0
$561$	0	0
$562$	13644.0	1.02409
$563$	966.000	0.0723127	0.0361563	−	0.999346i	$-0.488489\pi$
$563$	966.000	0.0723127	0.0361563	+	0.999346i	$0.488489\pi$
$564$	0	0
$565$	0	0
$566$	−8098.00	−0.601386
$567$	0	0
$568$	−5280.00	−0.390042
$569$	−19002.0	−1.40001	−0.700005	−	0.714138i	$-0.746819\pi$
$569$	−19002.0	−1.40001	−0.700005	+	0.714138i	$0.746819\pi$
$570$	0	0
$571$	8645.00	0.633594	0.316797	−	0.948493i	$-0.397393\pi$
$571$	8645.00	0.633594	0.316797	+	0.948493i	$0.397393\pi$
$572$	−3480.00	−0.254381
$573$	0	0
$574$	−7176.00	−0.521813
$575$	0	0
$576$	0	0
$577$	−10931.0	−0.788672	−0.394336	−	0.918966i	$-0.629025\pi$
$577$	−10931.0	−0.788672	−0.394336	+	0.918966i	$0.629025\pi$
$578$	2342.00	0.168537
$579$	0	0
$580$	0	0
$581$	19458.0	1.38942
$582$	0	0
$583$	−16560.0	−1.17641
$584$	5168.00	0.366187
$585$	0	0
$586$	4476.00	0.315532
$587$	8904.00	0.626077	0.313039	−	0.949740i	$-0.398653\pi$
$587$	8904.00	0.626077	0.313039	+	0.949740i	$0.398653\pi$
$588$	0	0
$589$	−32333.0	−2.26190
$590$	0	0
$591$	0	0
$592$	−2336.00	−0.162177
$593$	8820.00	0.610782	0.305391	−	0.952227i	$-0.401213\pi$
$593$	8820.00	0.610782	0.305391	+	0.952227i	$0.401213\pi$
$594$	0	0
$595$	0	0
$596$	3096.00	0.212780
$597$	0	0
$598$	−8700.00	−0.594932
$599$	−9804.00	−0.668749	−0.334374	−	0.942440i	$-0.608525\pi$
$599$	−9804.00	−0.668749	−0.334374	+	0.942440i	$0.608525\pi$
$600$	0	0
$601$	−23437.0	−1.59071	−0.795354	−	0.606146i	$-0.792715\pi$
$601$	−23437.0	−1.59071	−0.795354	+	0.606146i	$0.792715\pi$
$602$	−19918.0	−1.34850
$603$	0	0
$604$	1172.00	0.0789536
$605$	0	0
$606$	0	0
$607$	−2648.00	−0.177066	−0.0885330	−	0.996073i	$-0.528218\pi$
$607$	−2648.00	−0.177066	−0.0885330	+	0.996073i	$0.528218\pi$
$608$	4768.00	0.318039
$609$	0	0
$610$	0	0
$611$	−870.000	−0.0576046
$612$	0	0
$613$	−794.000	−0.0523154	−0.0261577	−	0.999658i	$-0.508327\pi$
$613$	−794.000	−0.0523154	−0.0261577	+	0.999658i	$0.508327\pi$
$614$	−2770.00	−0.182065
$615$	0	0
$616$	−5520.00	−0.361051
$617$	18720.0	1.22146	0.610728	−	0.791840i	$-0.290877\pi$
$617$	18720.0	1.22146	0.610728	+	0.791840i	$0.290877\pi$
$618$	0	0
$619$	−8959.00	−0.581733	−0.290866	−	0.956764i	$-0.593944\pi$
$619$	−8959.00	−0.581733	−0.290866	+	0.956764i	$0.593944\pi$
$620$	0	0
$621$	0	0
$622$	11340.0	0.731017
$623$	−34224.0	−2.20089
$624$	0	0
$625$	0	0
$626$	842.000	0.0537589
$627$	0	0
$628$	6916.00	0.439456
$629$	−11388.0	−0.721891
$630$	0	0
$631$	−12373.0	−0.780604	−0.390302	−	0.920687i	$-0.627629\pi$
$631$	−12373.0	−0.780604	−0.390302	+	0.920687i	$0.627629\pi$
$632$	7936.00	0.499489
$633$	0	0
$634$	19968.0	1.25084
$635$	0	0
$636$	0	0
$637$	−5394.00	−0.335507
$638$	14040.0	0.871237
$639$	0	0
$640$	0	0
$641$	−24900.0	−1.53431	−0.767154	−	0.641463i	$-0.778328\pi$
$641$	−24900.0	−1.53431	−0.767154	+	0.641463i	$0.778328\pi$
$642$	0	0
$643$	14668.0	0.899610	0.449805	−	0.893127i	$-0.351493\pi$
$643$	14668.0	0.899610	0.449805	+	0.893127i	$0.351493\pi$
$644$	−13800.0	−0.844404
$645$	0	0
$646$	23244.0	1.41567
$647$	−10788.0	−0.655518	−0.327759	−	0.944761i	$-0.606293\pi$
$647$	−10788.0	−0.655518	−0.327759	+	0.944761i	$0.606293\pi$
$648$	0	0
$649$	8100.00	0.489912
$650$	0	0
$651$	0	0
$652$	4492.00	0.269816
$653$	−14214.0	−0.851817	−0.425909	−	0.904766i	$-0.640046\pi$
$653$	−14214.0	−0.851817	−0.425909	+	0.904766i	$0.640046\pi$
$654$	0	0
$655$	0	0
$656$	2496.00	0.148556
$657$	0	0
$658$	−1380.00	−0.0817599
$659$	588.000	0.0347576	0.0173788	−	0.999849i	$-0.494468\pi$
$659$	588.000	0.0347576	0.0173788	+	0.999849i	$0.494468\pi$
$660$	0	0
$661$	−3166.00	−0.186298	−0.0931491	−	0.995652i	$-0.529693\pi$
$661$	−3166.00	−0.186298	−0.0931491	+	0.995652i	$0.529693\pi$
$662$	−8456.00	−0.496453
$663$	0	0
$664$	−6768.00	−0.395556
$665$	0	0
$666$	0	0
$667$	35100.0	2.03760
$668$	4800.00	0.278020
$669$	0	0
$670$	0	0
$671$	8250.00	0.474646
$672$	0	0
$673$	−9182.00	−0.525914	−0.262957	−	0.964808i	$-0.584698\pi$
$673$	−9182.00	−0.525914	−0.262957	+	0.964808i	$0.584698\pi$
$674$	−10786.0	−0.616411
$675$	0	0
$676$	−5424.00	−0.308603
$677$	11742.0	0.666590	0.333295	−	0.942823i	$-0.391839\pi$
$677$	11742.0	0.666590	0.333295	+	0.942823i	$0.391839\pi$
$678$	0	0
$679$	−7337.00	−0.414681
$680$	0	0
$681$	0	0
$682$	−13020.0	−0.731029
$683$	−6024.00	−0.337485	−0.168742	−	0.985660i	$-0.553971\pi$
$683$	−6024.00	−0.337485	−0.168742	+	0.985660i	$0.553971\pi$
$684$	0	0
$685$	0	0
$686$	7222.00	0.401949
$687$	0	0
$688$	6928.00	0.383906
$689$	16008.0	0.885132
$690$	0	0
$691$	9344.00	0.514418	0.257209	−	0.966356i	$-0.417197\pi$
$691$	9344.00	0.514418	0.257209	+	0.966356i	$0.417197\pi$
$692$	−6936.00	−0.381022
$693$	0	0
$694$	−15828.0	−0.865739
$695$	0	0
$696$	0	0
$697$	12168.0	0.661257
$698$	2020.00	0.109539
$699$	0	0
$700$	0	0
$701$	−21234.0	−1.14408	−0.572038	−	0.820227i	$-0.693847\pi$
$701$	−21234.0	−1.14408	−0.572038	+	0.820227i	$0.693847\pi$
$702$	0	0
$703$	−21754.0	−1.16709
$704$	1920.00	0.102788
$705$	0	0
$706$	9444.00	0.503441
$707$	−18216.0	−0.969000
$708$	0	0
$709$	−1723.00	−0.0912675	−0.0456337	−	0.998958i	$-0.514531\pi$
$709$	−1723.00	−0.0912675	−0.0456337	+	0.998958i	$0.514531\pi$
$710$	0	0
$711$	0	0
$712$	11904.0	0.626575
$713$	−32550.0	−1.70969
$714$	0	0
$715$	0	0
$716$	−10344.0	−0.539907
$717$	0	0
$718$	−12408.0	−0.644934
$719$	−18510.0	−0.960093	−0.480046	−	0.877243i	$-0.659380\pi$
$719$	−18510.0	−0.960093	−0.480046	+	0.877243i	$0.659380\pi$
$720$	0	0
$721$	18676.0	0.964675
$722$	30684.0	1.58163
$723$	0	0
$724$	−15724.0	−0.807152
$725$	0	0
$726$	0	0
$727$	1009.00	0.0514742	0.0257371	−	0.999669i	$-0.491807\pi$
$727$	1009.00	0.0514742	0.0257371	+	0.999669i	$0.491807\pi$
$728$	5336.00	0.271656
$729$	0	0
$730$	0	0
$731$	33774.0	1.70886
$732$	0	0
$733$	21994.0	1.10828	0.554138	−	0.832425i	$-0.313048\pi$
$733$	21994.0	1.10828	0.554138	+	0.832425i	$0.313048\pi$
$734$	−2722.00	−0.136881
$735$	0	0
$736$	4800.00	0.240394
$737$	−24090.0	−1.20403
$738$	0	0
$739$	−13948.0	−0.694297	−0.347148	−	0.937810i	$-0.612850\pi$
$739$	−13948.0	−0.694297	−0.347148	+	0.937810i	$0.612850\pi$
$740$	0	0
$741$	0	0
$742$	25392.0	1.25629
$743$	−26508.0	−1.30886	−0.654431	−	0.756122i	$-0.727092\pi$
$743$	−26508.0	−1.30886	−0.654431	+	0.756122i	$0.727092\pi$
$744$	0	0
$745$	0	0
$746$	1826.00	0.0896174
$747$	0	0
$748$	9360.00	0.457534
$749$	32568.0	1.58880
$750$	0	0
$751$	−1600.00	−0.0777428	−0.0388714	−	0.999244i	$-0.512376\pi$
$751$	−1600.00	−0.0777428	−0.0388714	+	0.999244i	$0.512376\pi$
$752$	480.000	0.0232763
$753$	0	0
$754$	−13572.0	−0.655521
$755$	0	0
$756$	0	0
$757$	−30101.0	−1.44523	−0.722615	−	0.691250i	$-0.757060\pi$
$757$	−30101.0	−1.44523	−0.722615	+	0.691250i	$0.757060\pi$
$758$	−17762.0	−0.851115
$759$	0	0
$760$	0	0
$761$	−35628.0	−1.69713	−0.848564	−	0.529093i	$-0.822532\pi$
$761$	−35628.0	−1.69713	−0.848564	+	0.529093i	$0.822532\pi$
$762$	0	0
$763$	1265.00	0.0600211
$764$	−6264.00	−0.296628
$765$	0	0
$766$	10920.0	0.515086
$767$	−7830.00	−0.368611
$768$	0	0
$769$	−12517.0	−0.586963	−0.293482	−	0.955965i	$-0.594814\pi$
$769$	−12517.0	−0.586963	−0.293482	+	0.955965i	$0.594814\pi$
$770$	0	0
$771$	0	0
$772$	−9164.00	−0.427227
$773$	14124.0	0.657186	0.328593	−	0.944472i	$-0.393426\pi$
$773$	14124.0	0.657186	0.328593	+	0.944472i	$0.393426\pi$
$774$	0	0
$775$	0	0
$776$	2552.00	0.118056
$777$	0	0
$778$	27768.0	1.27960
$779$	23244.0	1.06907
$780$	0	0
$781$	−19800.0	−0.907170
$782$	23400.0	1.07005
$783$	0	0
$784$	2976.00	0.135569
$785$	0	0
$786$	0	0
$787$	−40433.0	−1.83136	−0.915680	−	0.401907i	$-0.868347\pi$
$787$	−40433.0	−1.83136	−0.915680	+	0.401907i	$0.868347\pi$
$788$	8568.00	0.387338
$789$	0	0
$790$	0	0
$791$	−32292.0	−1.45154
$792$	0	0
$793$	−7975.00	−0.357126
$794$	7562.00	0.337992
$795$	0	0
$796$	−19612.0	−0.873278
$797$	−27300.0	−1.21332	−0.606660	−	0.794962i	$-0.707491\pi$
$797$	−27300.0	−1.21332	−0.606660	+	0.794962i	$0.707491\pi$
$798$	0	0
$799$	2340.00	0.103609
$800$	0	0
$801$	0	0
$802$	−18048.0	−0.794635
$803$	19380.0	0.851688
$804$	0	0
$805$	0	0
$806$	12586.0	0.550028
$807$	0	0
$808$	6336.00	0.275866
$809$	2856.00	0.124118	0.0620591	−	0.998072i	$-0.480233\pi$
$809$	2856.00	0.124118	0.0620591	+	0.998072i	$0.480233\pi$
$810$	0	0
$811$	−12619.0	−0.546379	−0.273189	−	0.961960i	$-0.588079\pi$
$811$	−12619.0	−0.546379	−0.273189	+	0.961960i	$0.588079\pi$
$812$	−21528.0	−0.930400
$813$	0	0
$814$	−8760.00	−0.377196
$815$	0	0
$816$	0	0
$817$	64517.0	2.76275
$818$	29578.0	1.26427
$819$	0	0
$820$	0	0
$821$	29082.0	1.23626	0.618130	−	0.786076i	$-0.287891\pi$
$821$	29082.0	1.23626	0.618130	+	0.786076i	$0.287891\pi$
$822$	0	0
$823$	−10235.0	−0.433499	−0.216749	−	0.976227i	$-0.569545\pi$
$823$	−10235.0	−0.433499	−0.216749	+	0.976227i	$0.569545\pi$
$824$	−6496.00	−0.274635
$825$	0	0
$826$	−12420.0	−0.523180
$827$	26976.0	1.13428	0.567139	−	0.823622i	$-0.308050\pi$
$827$	26976.0	1.13428	0.567139	+	0.823622i	$0.308050\pi$
$828$	0	0
$829$	37802.0	1.58374	0.791868	−	0.610692i	$-0.209109\pi$
$829$	37802.0	1.58374	0.791868	+	0.610692i	$0.209109\pi$
$830$	0	0
$831$	0	0
$832$	−1856.00	−0.0773380
$833$	14508.0	0.603448
$834$	0	0
$835$	0	0
$836$	17880.0	0.739704
$837$	0	0
$838$	−19680.0	−0.811258
$839$	16974.0	0.698460	0.349230	−	0.937037i	$-0.386443\pi$
$839$	16974.0	0.698460	0.349230	+	0.937037i	$0.386443\pi$
$840$	0	0
$841$	30367.0	1.24511
$842$	11020.0	0.451038
$843$	0	0
$844$	2420.00	0.0986965
$845$	0	0
$846$	0	0
$847$	9913.00	0.402143
$848$	−8832.00	−0.357656
$849$	0	0
$850$	0	0
$851$	−21900.0	−0.882165
$852$	0	0
$853$	24937.0	1.00097	0.500485	−	0.865745i	$-0.333155\pi$
$853$	24937.0	1.00097	0.500485	+	0.865745i	$0.333155\pi$
$854$	−12650.0	−0.506878
$855$	0	0
$856$	−11328.0	−0.452317
$857$	15756.0	0.628022	0.314011	−	0.949419i	$-0.398327\pi$
$857$	15756.0	0.628022	0.314011	+	0.949419i	$0.398327\pi$
$858$	0	0
$859$	38144.0	1.51508	0.757542	−	0.652787i	$-0.226400\pi$
$859$	38144.0	1.51508	0.757542	+	0.652787i	$0.226400\pi$
$860$	0	0
$861$	0	0
$862$	−22140.0	−0.874816
$863$	5448.00	0.214892	0.107446	−	0.994211i	$-0.465733\pi$
$863$	5448.00	0.214892	0.107446	+	0.994211i	$0.465733\pi$
$864$	0	0
$865$	0	0
$866$	24266.0	0.952185
$867$	0	0
$868$	19964.0	0.780671
$869$	29760.0	1.16172
$870$	0	0
$871$	23287.0	0.905913
$872$	−440.000	−0.0170875
$873$	0	0
$874$	44700.0	1.72998
$875$	0	0
$876$	0	0
$877$	−21191.0	−0.815928	−0.407964	−	0.912998i	$-0.633761\pi$
$877$	−21191.0	−0.815928	−0.407964	+	0.912998i	$0.633761\pi$
$878$	−3746.00	−0.143988
$879$	0	0
$880$	0	0
$881$	−18216.0	−0.696609	−0.348305	−	0.937381i	$-0.613242\pi$
$881$	−18216.0	−0.696609	−0.348305	+	0.937381i	$0.613242\pi$
$882$	0	0
$883$	−12767.0	−0.486573	−0.243286	−	0.969955i	$-0.578225\pi$
$883$	−12767.0	−0.486573	−0.243286	+	0.969955i	$0.578225\pi$
$884$	−9048.00	−0.344250
$885$	0	0
$886$	−1152.00	−0.0436819
$887$	11010.0	0.416775	0.208388	−	0.978046i	$-0.433178\pi$
$887$	11010.0	0.416775	0.208388	+	0.978046i	$0.433178\pi$
$888$	0	0
$889$	29440.0	1.11067
$890$	0	0
$891$	0	0
$892$	580.000	0.0217711
$893$	4470.00	0.167506
$894$	0	0
$895$	0	0
$896$	−2944.00	−0.109768
$897$	0	0
$898$	9768.00	0.362987
$899$	−50778.0	−1.88381
$900$	0	0
$901$	−43056.0	−1.59201
$902$	9360.00	0.345514
$903$	0	0
$904$	11232.0	0.413242
$905$	0	0
$906$	0	0
$907$	−22772.0	−0.833662	−0.416831	−	0.908984i	$-0.636859\pi$
$907$	−22772.0	−0.833662	−0.416831	+	0.908984i	$0.636859\pi$
$908$	11856.0	0.433321
$909$	0	0
$910$	0	0
$911$	−29802.0	−1.08385	−0.541923	−	0.840428i	$-0.682304\pi$
$911$	−29802.0	−1.08385	−0.541923	+	0.840428i	$0.682304\pi$
$912$	0	0
$913$	−25380.0	−0.919995
$914$	31604.0	1.14373
$915$	0	0
$916$	−22540.0	−0.813038
$917$	11040.0	0.397571
$918$	0	0
$919$	48941.0	1.75671	0.878354	−	0.478011i	$-0.158642\pi$
$919$	48941.0	1.75671	0.878354	+	0.478011i	$0.158642\pi$
$920$	0	0
$921$	0	0
$922$	30720.0	1.09730
$923$	19140.0	0.682558
$924$	0	0
$925$	0	0
$926$	−3424.00	−0.121511
$927$	0	0
$928$	7488.00	0.264877
$929$	−31026.0	−1.09573	−0.547863	−	0.836568i	$-0.684559\pi$
$929$	−31026.0	−1.09573	−0.547863	+	0.836568i	$0.684559\pi$
$930$	0	0
$931$	27714.0	0.975607
$932$	16656.0	0.585392
$933$	0	0
$934$	−32556.0	−1.14054
$935$	0	0
$936$	0	0
$937$	−11183.0	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$937$	−11183.0	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$938$	36938.0	1.28579
$939$	0	0
$940$	0	0
$941$	2562.00	0.0887554	0.0443777	−	0.999015i	$-0.485870\pi$
$941$	2562.00	0.0887554	0.0443777	+	0.999015i	$0.485870\pi$
$942$	0	0
$943$	23400.0	0.808069
$944$	4320.00	0.148945
$945$	0	0
$946$	25980.0	0.892899
$947$	7638.00	0.262093	0.131046	−	0.991376i	$-0.458166\pi$
$947$	7638.00	0.262093	0.131046	+	0.991376i	$0.458166\pi$
$948$	0	0
$949$	−18734.0	−0.640813
$950$	0	0
$951$	0	0
$952$	−14352.0	−0.488604
$953$	51432.0	1.74821	0.874106	−	0.485735i	$-0.161448\pi$
$953$	51432.0	1.74821	0.874106	+	0.485735i	$0.161448\pi$
$954$	0	0
$955$	0	0
$956$	7776.00	0.263069
$957$	0	0
$958$	29532.0	0.995966
$959$	6486.00	0.218398
$960$	0	0
$961$	17298.0	0.580645
$962$	8468.00	0.283804
$963$	0	0
$964$	3428.00	0.114532
$965$	0	0
$966$	0	0
$967$	−39728.0	−1.32116	−0.660582	−	0.750754i	$-0.729691\pi$
$967$	−39728.0	−1.32116	−0.660582	+	0.750754i	$0.729691\pi$
$968$	−3448.00	−0.114486
$969$	0	0
$970$	0	0
$971$	47946.0	1.58461	0.792307	−	0.610123i	$-0.208880\pi$
$971$	47946.0	1.58461	0.792307	+	0.610123i	$0.208880\pi$
$972$	0	0
$973$	−36892.0	−1.21552
$974$	6638.00	0.218373
$975$	0	0
$976$	4400.00	0.144304
$977$	−22326.0	−0.731087	−0.365544	−	0.930794i	$-0.619117\pi$
$977$	−22326.0	−0.731087	−0.365544	+	0.930794i	$0.619117\pi$
$978$	0	0
$979$	44640.0	1.45730
$980$	0	0
$981$	0	0
$982$	−22128.0	−0.719076
$983$	−48468.0	−1.57262	−0.786312	−	0.617830i	$-0.788012\pi$
$983$	−48468.0	−1.57262	−0.786312	+	0.617830i	$0.788012\pi$
$984$	0	0
$985$	0	0
$986$	36504.0	1.17903
$987$	0	0
$988$	−17284.0	−0.556556
$989$	64950.0	2.08826
$990$	0	0
$991$	−25141.0	−0.805883	−0.402942	−	0.915226i	$-0.632012\pi$
$991$	−25141.0	−0.805883	−0.402942	+	0.915226i	$0.632012\pi$
$992$	−6944.00	−0.222250
$993$	0	0
$994$	30360.0	0.968773
$995$	0	0
$996$	0	0
$997$	35422.0	1.12520	0.562601	−	0.826729i	$-0.309801\pi$
$997$	35422.0	1.12520	0.562601	+	0.826729i	$0.309801\pi$
$998$	−28262.0	−0.896411
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.4.a.l.1.1		1
3.2	odd	2		150.4.a.c.1.1	✓	1
5.2	odd	4		450.4.c.h.199.2		2
5.3	odd	4		450.4.c.h.199.1		2
5.4	even	2		450.4.a.i.1.1		1
12.11	even	2		1200.4.a.r.1.1		1
15.2	even	4		150.4.c.b.49.1		2
15.8	even	4		150.4.c.b.49.2		2
15.14	odd	2		150.4.a.g.1.1	yes	1
60.23	odd	4		1200.4.f.q.49.1		2
60.47	odd	4		1200.4.f.q.49.2		2
60.59	even	2		1200.4.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.4.a.c.1.1	✓	1	3.2	odd	2
150.4.a.g.1.1	yes	1	15.14	odd	2
150.4.c.b.49.1		2	15.2	even	4
150.4.c.b.49.2		2	15.8	even	4
450.4.a.i.1.1		1	5.4	even	2
450.4.a.l.1.1		1	1.1	even	1	trivial
450.4.c.h.199.1		2	5.3	odd	4
450.4.c.h.199.2		2	5.2	odd	4
1200.4.a.r.1.1		1	12.11	even	2
1200.4.a.v.1.1		1	60.59	even	2
1200.4.f.q.49.1		2	60.23	odd	4
1200.4.f.q.49.2		2	60.47	odd	4

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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