LMFDB - Embedded newform 450.4.a.s.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:	yes
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)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} +11.0000 q^{7} +8.00000 q^{8} +O(q^{10})$

$q+2.00000 q^{2} +4.00000 q^{4} +11.0000 q^{7} +8.00000 q^{8} +36.0000 q^{11} +17.0000 q^{13} +22.0000 q^{14} +16.0000 q^{16} +12.0000 q^{17} -91.0000 q^{19} +72.0000 q^{22} -60.0000 q^{23} +34.0000 q^{26} +44.0000 q^{28} +276.000 q^{29} +191.000 q^{31} +32.0000 q^{32} +24.0000 q^{34} +254.000 q^{37} -182.000 q^{38} +60.0000 q^{41} -49.0000 q^{43} +144.000 q^{44} -120.000 q^{46} +600.000 q^{47} -222.000 q^{49} +68.0000 q^{52} -612.000 q^{53} +88.0000 q^{56} +552.000 q^{58} +744.000 q^{59} +167.000 q^{61} +382.000 q^{62} +64.0000 q^{64} -457.000 q^{67} +48.0000 q^{68} +588.000 q^{71} -970.000 q^{73} +508.000 q^{74} -364.000 q^{76} +396.000 q^{77} +164.000 q^{79} +120.000 q^{82} -696.000 q^{83} -98.0000 q^{86} +288.000 q^{88} +1248.00 q^{89} +187.000 q^{91} -240.000 q^{92} +1200.00 q^{94} -1099.00 q^{97} -444.000 q^{98} +O(q^{100})$

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
$7$	11.0000	0.593944	0.296972	−	0.954886i	$-0.404023\pi$
$7$	11.0000	0.593944	0.296972	+	0.954886i	$0.404023\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	0	0
$11$	36.0000	0.986764	0.493382	−	0.869813i	$-0.335760\pi$
$11$	36.0000	0.986764	0.493382	+	0.869813i	$0.335760\pi$
$12$	0	0
$13$	17.0000	0.362689	0.181344	−	0.983420i	$-0.441955\pi$
$13$	17.0000	0.362689	0.181344	+	0.983420i	$0.441955\pi$
$14$	22.0000	0.419982
$15$	0	0
$16$	16.0000	0.250000
$17$	12.0000	0.171202	0.0856008	−	0.996330i	$-0.472719\pi$
$17$	12.0000	0.171202	0.0856008	+	0.996330i	$0.472719\pi$
$18$	0	0
$19$	−91.0000	−1.09878	−0.549390	−	0.835566i	$-0.685140\pi$
$19$	−91.0000	−1.09878	−0.549390	+	0.835566i	$0.685140\pi$
$20$	0	0
$21$	0	0
$22$	72.0000	0.697748
$23$	−60.0000	−0.543951	−0.271975	−	0.962304i	$-0.587677\pi$
$23$	−60.0000	−0.543951	−0.271975	+	0.962304i	$0.587677\pi$
$24$	0	0
$25$	0	0
$26$	34.0000	0.256460
$27$	0	0
$28$	44.0000	0.296972
$29$	276.000	1.76731	0.883654	−	0.468141i	$-0.155076\pi$
$29$	276.000	1.76731	0.883654	+	0.468141i	$0.155076\pi$
$30$	0	0
$31$	191.000	1.10660	0.553300	−	0.832982i	$-0.313368\pi$
$31$	191.000	1.10660	0.553300	+	0.832982i	$0.313368\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	24.0000	0.121058
$35$	0	0
$36$	0	0
$37$	254.000	1.12858	0.564288	−	0.825578i	$-0.309151\pi$
$37$	254.000	1.12858	0.564288	+	0.825578i	$0.309151\pi$
$38$	−182.000	−0.776955
$39$	0	0
$40$	0	0
$41$	60.0000	0.228547	0.114273	−	0.993449i	$-0.463546\pi$
$41$	60.0000	0.228547	0.114273	+	0.993449i	$0.463546\pi$
$42$	0	0
$43$	−49.0000	−0.173777	−0.0868887	−	0.996218i	$-0.527692\pi$
$43$	−49.0000	−0.173777	−0.0868887	+	0.996218i	$0.527692\pi$
$44$	144.000	0.493382
$45$	0	0
$46$	−120.000	−0.384631
$47$	600.000	1.86211	0.931053	−	0.364884i	$-0.118891\pi$
$47$	600.000	1.86211	0.931053	+	0.364884i	$0.118891\pi$
$48$	0	0
$49$	−222.000	−0.647230
$50$	0	0
$51$	0	0
$52$	68.0000	0.181344
$53$	−612.000	−1.58613	−0.793063	−	0.609140i	$-0.791515\pi$
$53$	−612.000	−1.58613	−0.793063	+	0.609140i	$0.791515\pi$
$54$	0	0
$55$	0	0
$56$	88.0000	0.209991
$57$	0	0
$58$	552.000	1.24968
$59$	744.000	1.64170	0.820852	−	0.571141i	$-0.193499\pi$
$59$	744.000	1.64170	0.820852	+	0.571141i	$0.193499\pi$
$60$	0	0
$61$	167.000	0.350527	0.175264	−	0.984522i	$-0.443922\pi$
$61$	167.000	0.350527	0.175264	+	0.984522i	$0.443922\pi$
$62$	382.000	0.782485
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	−457.000	−0.833305	−0.416653	−	0.909066i	$-0.636797\pi$
$67$	−457.000	−0.833305	−0.416653	+	0.909066i	$0.636797\pi$
$68$	48.0000	0.0856008
$69$	0	0
$70$	0	0
$71$	588.000	0.982856	0.491428	−	0.870918i	$-0.336475\pi$
$71$	588.000	0.982856	0.491428	+	0.870918i	$0.336475\pi$
$72$	0	0
$73$	−970.000	−1.55520	−0.777602	−	0.628757i	$-0.783564\pi$
$73$	−970.000	−1.55520	−0.777602	+	0.628757i	$0.783564\pi$
$74$	508.000	0.798024
$75$	0	0
$76$	−364.000	−0.549390
$77$	396.000	0.586083
$78$	0	0
$79$	164.000	0.233563	0.116781	−	0.993158i	$-0.462742\pi$
$79$	164.000	0.233563	0.116781	+	0.993158i	$0.462742\pi$
$80$	0	0
$81$	0	0
$82$	120.000	0.161607
$83$	−696.000	−0.920433	−0.460216	−	0.887807i	$-0.652228\pi$
$83$	−696.000	−0.920433	−0.460216	+	0.887807i	$0.652228\pi$
$84$	0	0
$85$	0	0
$86$	−98.0000	−0.122879
$87$	0	0
$88$	288.000	0.348874
$89$	1248.00	1.48638	0.743190	−	0.669081i	$-0.233312\pi$
$89$	1248.00	1.48638	0.743190	+	0.669081i	$0.233312\pi$
$90$	0	0
$91$	187.000	0.215417
$92$	−240.000	−0.271975
$93$	0	0
$94$	1200.00	1.31671
$95$	0	0
$96$	0	0
$97$	−1099.00	−1.15038	−0.575188	−	0.818021i	$-0.695071\pi$
$97$	−1099.00	−1.15038	−0.575188	+	0.818021i	$0.695071\pi$
$98$	−444.000	−0.457661
$99$	0	0
$100$	0	0
$101$	444.000	0.437422	0.218711	−	0.975790i	$-0.429815\pi$
$101$	444.000	0.437422	0.218711	+	0.975790i	$0.429815\pi$
$102$	0	0
$103$	−916.000	−0.876273	−0.438137	−	0.898908i	$-0.644361\pi$
$103$	−916.000	−0.876273	−0.438137	+	0.898908i	$0.644361\pi$
$104$	136.000	0.128230
$105$	0	0
$106$	−1224.00	−1.12156
$107$	−204.000	−0.184312	−0.0921562	−	0.995745i	$-0.529376\pi$
$107$	−204.000	−0.184312	−0.0921562	+	0.995745i	$0.529376\pi$
$108$	0	0
$109$	−967.000	−0.849741	−0.424871	−	0.905254i	$-0.639680\pi$
$109$	−967.000	−0.849741	−0.424871	+	0.905254i	$0.639680\pi$
$110$	0	0
$111$	0	0
$112$	176.000	0.148486
$113$	672.000	0.559438	0.279719	−	0.960082i	$-0.409759\pi$
$113$	672.000	0.559438	0.279719	+	0.960082i	$0.409759\pi$
$114$	0	0
$115$	0	0
$116$	1104.00	0.883654
$117$	0	0
$118$	1488.00	1.16086
$119$	132.000	0.101684
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	334.000	0.247860
$123$	0	0
$124$	764.000	0.553300
$125$	0	0
$126$	0	0
$127$	−1924.00	−1.34431	−0.672155	−	0.740410i	$-0.734631\pi$
$127$	−1924.00	−1.34431	−0.672155	+	0.740410i	$0.734631\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	0	0
$131$	−756.000	−0.504214	−0.252107	−	0.967699i	$-0.581123\pi$
$131$	−756.000	−0.504214	−0.252107	+	0.967699i	$0.581123\pi$
$132$	0	0
$133$	−1001.00	−0.652614
$134$	−914.000	−0.589236
$135$	0	0
$136$	96.0000	0.0605289
$137$	888.000	0.553773	0.276887	−	0.960903i	$-0.410697\pi$
$137$	888.000	0.553773	0.276887	+	0.960903i	$0.410697\pi$
$138$	0	0
$139$	−160.000	−0.0976333	−0.0488166	−	0.998808i	$-0.515545\pi$
$139$	−160.000	−0.0976333	−0.0488166	+	0.998808i	$0.515545\pi$
$140$	0	0
$141$	0	0
$142$	1176.00	0.694984
$143$	612.000	0.357888
$144$	0	0
$145$	0	0
$146$	−1940.00	−1.09970
$147$	0	0
$148$	1016.00	0.564288
$149$	−3096.00	−1.70224	−0.851121	−	0.524969i	$-0.824077\pi$
$149$	−3096.00	−1.70224	−0.851121	+	0.524969i	$0.824077\pi$
$150$	0	0
$151$	1049.00	0.565340	0.282670	−	0.959217i	$-0.408780\pi$
$151$	1049.00	0.565340	0.282670	+	0.959217i	$0.408780\pi$
$152$	−728.000	−0.388478
$153$	0	0
$154$	792.000	0.414423
$155$	0	0
$156$	0	0
$157$	2363.00	1.20120	0.600599	−	0.799551i	$-0.294929\pi$
$157$	2363.00	1.20120	0.600599	+	0.799551i	$0.294929\pi$
$158$	328.000	0.165154
$159$	0	0
$160$	0	0
$161$	−660.000	−0.323076
$162$	0	0
$163$	−3235.00	−1.55451	−0.777254	−	0.629187i	$-0.783388\pi$
$163$	−3235.00	−1.55451	−0.777254	+	0.629187i	$0.783388\pi$
$164$	240.000	0.114273
$165$	0	0
$166$	−1392.00	−0.650844
$167$	−2772.00	−1.28445	−0.642227	−	0.766515i	$-0.721989\pi$
$167$	−2772.00	−1.28445	−0.642227	+	0.766515i	$0.721989\pi$
$168$	0	0
$169$	−1908.00	−0.868457
$170$	0	0
$171$	0	0
$172$	−196.000	−0.0868887
$173$	−84.0000	−0.0369156	−0.0184578	−	0.999830i	$-0.505876\pi$
$173$	−84.0000	−0.0369156	−0.0184578	+	0.999830i	$0.505876\pi$
$174$	0	0
$175$	0	0
$176$	576.000	0.246691
$177$	0	0
$178$	2496.00	1.05103
$179$	−2736.00	−1.14245	−0.571224	−	0.820794i	$-0.693531\pi$
$179$	−2736.00	−1.14245	−0.571224	+	0.820794i	$0.693531\pi$
$180$	0	0
$181$	4397.00	1.80567	0.902835	−	0.429986i	$-0.141482\pi$
$181$	4397.00	1.80567	0.902835	+	0.429986i	$0.141482\pi$
$182$	374.000	0.152323
$183$	0	0
$184$	−480.000	−0.192316
$185$	0	0
$186$	0	0
$187$	432.000	0.168936
$188$	2400.00	0.931053
$189$	0	0
$190$	0	0
$191$	−3108.00	−1.17742	−0.588709	−	0.808345i	$-0.700364\pi$
$191$	−3108.00	−1.17742	−0.588709	+	0.808345i	$0.700364\pi$
$192$	0	0
$193$	2615.00	0.975294	0.487647	−	0.873041i	$-0.337855\pi$
$193$	2615.00	0.975294	0.487647	+	0.873041i	$0.337855\pi$
$194$	−2198.00	−0.813439
$195$	0	0
$196$	−888.000	−0.323615
$197$	−3624.00	−1.31066	−0.655328	−	0.755344i	$-0.727470\pi$
$197$	−3624.00	−1.31066	−0.655328	+	0.755344i	$0.727470\pi$
$198$	0	0
$199$	−1819.00	−0.647967	−0.323984	−	0.946063i	$-0.605022\pi$
$199$	−1819.00	−0.647967	−0.323984	+	0.946063i	$0.605022\pi$
$200$	0	0
$201$	0	0
$202$	888.000	0.309304
$203$	3036.00	1.04968
$204$	0	0
$205$	0	0
$206$	−1832.00	−0.619619
$207$	0	0
$208$	272.000	0.0906721
$209$	−3276.00	−1.08424
$210$	0	0
$211$	−1999.00	−0.652212	−0.326106	−	0.945333i	$-0.605737\pi$
$211$	−1999.00	−0.652212	−0.326106	+	0.945333i	$0.605737\pi$
$212$	−2448.00	−0.793063
$213$	0	0
$214$	−408.000	−0.130329
$215$	0	0
$216$	0	0
$217$	2101.00	0.657259
$218$	−1934.00	−0.600858
$219$	0	0
$220$	0	0
$221$	204.000	0.0620929
$222$	0	0
$223$	−493.000	−0.148044	−0.0740218	−	0.997257i	$-0.523583\pi$
$223$	−493.000	−0.148044	−0.0740218	+	0.997257i	$0.523583\pi$
$224$	352.000	0.104995
$225$	0	0
$226$	1344.00	0.395582
$227$	300.000	0.0877167	0.0438584	−	0.999038i	$-0.486035\pi$
$227$	300.000	0.0877167	0.0438584	+	0.999038i	$0.486035\pi$
$228$	0	0
$229$	−79.0000	−0.0227968	−0.0113984	−	0.999935i	$-0.503628\pi$
$229$	−79.0000	−0.0227968	−0.0113984	+	0.999935i	$0.503628\pi$
$230$	0	0
$231$	0	0
$232$	2208.00	0.624838
$233$	6108.00	1.71738	0.858688	−	0.512500i	$-0.171280\pi$
$233$	6108.00	1.71738	0.858688	+	0.512500i	$0.171280\pi$
$234$	0	0
$235$	0	0
$236$	2976.00	0.820852
$237$	0	0
$238$	264.000	0.0719016
$239$	−2868.00	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−2868.00	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	2705.00	0.723006	0.361503	−	0.932371i	$-0.382264\pi$
$241$	2705.00	0.723006	0.361503	+	0.932371i	$0.382264\pi$
$242$	−70.0000	−0.0185941
$243$	0	0
$244$	668.000	0.175264
$245$	0	0
$246$	0	0
$247$	−1547.00	−0.398515
$248$	1528.00	0.391242
$249$	0	0
$250$	0	0
$251$	4008.00	1.00790	0.503950	−	0.863733i	$-0.331880\pi$
$251$	4008.00	1.00790	0.503950	+	0.863733i	$0.331880\pi$
$252$	0	0
$253$	−2160.00	−0.536751
$254$	−3848.00	−0.950571
$255$	0	0
$256$	256.000	0.0625000
$257$	−1992.00	−0.483492	−0.241746	−	0.970340i	$-0.577720\pi$
$257$	−1992.00	−0.483492	−0.241746	+	0.970340i	$0.577720\pi$
$258$	0	0
$259$	2794.00	0.670312
$260$	0	0
$261$	0	0
$262$	−1512.00	−0.356533
$263$	−972.000	−0.227894	−0.113947	−	0.993487i	$-0.536349\pi$
$263$	−972.000	−0.227894	−0.113947	+	0.993487i	$0.536349\pi$
$264$	0	0
$265$	0	0
$266$	−2002.00	−0.461468
$267$	0	0
$268$	−1828.00	−0.416653
$269$	1812.00	0.410705	0.205352	−	0.978688i	$-0.434166\pi$
$269$	1812.00	0.410705	0.205352	+	0.978688i	$0.434166\pi$
$270$	0	0
$271$	−5092.00	−1.14139	−0.570696	−	0.821162i	$-0.693326\pi$
$271$	−5092.00	−1.14139	−0.570696	+	0.821162i	$0.693326\pi$
$272$	192.000	0.0428004
$273$	0	0
$274$	1776.00	0.391577
$275$	0	0
$276$	0	0
$277$	569.000	0.123422	0.0617110	−	0.998094i	$-0.480344\pi$
$277$	569.000	0.123422	0.0617110	+	0.998094i	$0.480344\pi$
$278$	−320.000	−0.0690371
$279$	0	0
$280$	0	0
$281$	−6468.00	−1.37313	−0.686563	−	0.727070i	$-0.740882\pi$
$281$	−6468.00	−1.37313	−0.686563	+	0.727070i	$0.740882\pi$
$282$	0	0
$283$	3557.00	0.747144	0.373572	−	0.927601i	$-0.378133\pi$
$283$	3557.00	0.747144	0.373572	+	0.927601i	$0.378133\pi$
$284$	2352.00	0.491428
$285$	0	0
$286$	1224.00	0.253065
$287$	660.000	0.135744
$288$	0	0
$289$	−4769.00	−0.970690
$290$	0	0
$291$	0	0
$292$	−3880.00	−0.777602
$293$	−5376.00	−1.07191	−0.535954	−	0.844247i	$-0.680048\pi$
$293$	−5376.00	−1.07191	−0.535954	+	0.844247i	$0.680048\pi$
$294$	0	0
$295$	0	0
$296$	2032.00	0.399012
$297$	0	0
$298$	−6192.00	−1.20367
$299$	−1020.00	−0.197285
$300$	0	0
$301$	−539.000	−0.103214
$302$	2098.00	0.399756
$303$	0	0
$304$	−1456.00	−0.274695
$305$	0	0
$306$	0	0
$307$	−9343.00	−1.73692	−0.868458	−	0.495763i	$-0.834889\pi$
$307$	−9343.00	−1.73692	−0.868458	+	0.495763i	$0.834889\pi$
$308$	1584.00	0.293041
$309$	0	0
$310$	0	0
$311$	1968.00	0.358827	0.179413	−	0.983774i	$-0.442580\pi$
$311$	1968.00	0.358827	0.179413	+	0.983774i	$0.442580\pi$
$312$	0	0
$313$	5675.00	1.02482	0.512412	−	0.858740i	$-0.328752\pi$
$313$	5675.00	1.02482	0.512412	+	0.858740i	$0.328752\pi$
$314$	4726.00	0.849375
$315$	0	0
$316$	656.000	0.116781
$317$	2616.00	0.463499	0.231750	−	0.972775i	$-0.425555\pi$
$317$	2616.00	0.463499	0.231750	+	0.972775i	$0.425555\pi$
$318$	0	0
$319$	9936.00	1.74392
$320$	0	0
$321$	0	0
$322$	−1320.00	−0.228449
$323$	−1092.00	−0.188113
$324$	0	0
$325$	0	0
$326$	−6470.00	−1.09920
$327$	0	0
$328$	480.000	0.0808036
$329$	6600.00	1.10599
$330$	0	0
$331$	−9664.00	−1.60478	−0.802389	−	0.596801i	$-0.796438\pi$
$331$	−9664.00	−1.60478	−0.802389	+	0.596801i	$0.796438\pi$
$332$	−2784.00	−0.460216
$333$	0	0
$334$	−5544.00	−0.908246
$335$	0	0
$336$	0	0
$337$	7637.00	1.23446	0.617231	−	0.786782i	$-0.288254\pi$
$337$	7637.00	1.23446	0.617231	+	0.786782i	$0.288254\pi$
$338$	−3816.00	−0.614092
$339$	0	0
$340$	0	0
$341$	6876.00	1.09195
$342$	0	0
$343$	−6215.00	−0.978363
$344$	−392.000	−0.0614396
$345$	0	0
$346$	−168.000	−0.0261033
$347$	5640.00	0.872539	0.436270	−	0.899816i	$-0.356299\pi$
$347$	5640.00	0.872539	0.436270	+	0.899816i	$0.356299\pi$
$348$	0	0
$349$	7382.00	1.13223	0.566117	−	0.824325i	$-0.308445\pi$
$349$	7382.00	1.13223	0.566117	+	0.824325i	$0.308445\pi$
$350$	0	0
$351$	0	0
$352$	1152.00	0.174437
$353$	−10740.0	−1.61936	−0.809678	−	0.586875i	$-0.800358\pi$
$353$	−10740.0	−1.61936	−0.809678	+	0.586875i	$0.800358\pi$
$354$	0	0
$355$	0	0
$356$	4992.00	0.743190
$357$	0	0
$358$	−5472.00	−0.807833
$359$	9276.00	1.36370	0.681850	−	0.731492i	$-0.261176\pi$
$359$	9276.00	1.36370	0.681850	+	0.731492i	$0.261176\pi$
$360$	0	0
$361$	1422.00	0.207319
$362$	8794.00	1.27680
$363$	0	0
$364$	748.000	0.107708
$365$	0	0
$366$	0	0
$367$	−7387.00	−1.05068	−0.525338	−	0.850893i	$-0.676061\pi$
$367$	−7387.00	−1.05068	−0.525338	+	0.850893i	$0.676061\pi$
$368$	−960.000	−0.135988
$369$	0	0
$370$	0	0
$371$	−6732.00	−0.942070
$372$	0	0
$373$	10259.0	1.42410	0.712052	−	0.702127i	$-0.247766\pi$
$373$	10259.0	1.42410	0.712052	+	0.702127i	$0.247766\pi$
$374$	864.000	0.119456
$375$	0	0
$376$	4800.00	0.658354
$377$	4692.00	0.640982
$378$	0	0
$379$	7655.00	1.03750	0.518748	−	0.854927i	$-0.326398\pi$
$379$	7655.00	1.03750	0.518748	+	0.854927i	$0.326398\pi$
$380$	0	0
$381$	0	0
$382$	−6216.00	−0.832561
$383$	−600.000	−0.0800485	−0.0400242	−	0.999199i	$-0.512744\pi$
$383$	−600.000	−0.0800485	−0.0400242	+	0.999199i	$0.512744\pi$
$384$	0	0
$385$	0	0
$386$	5230.00	0.689637
$387$	0	0
$388$	−4396.00	−0.575188
$389$	4068.00	0.530221	0.265110	−	0.964218i	$-0.414592\pi$
$389$	4068.00	0.530221	0.265110	+	0.964218i	$0.414592\pi$
$390$	0	0
$391$	−720.000	−0.0931252
$392$	−1776.00	−0.228830
$393$	0	0
$394$	−7248.00	−0.926774
$395$	0	0
$396$	0	0
$397$	12647.0	1.59883	0.799414	−	0.600781i	$-0.205143\pi$
$397$	12647.0	1.59883	0.799414	+	0.600781i	$0.205143\pi$
$398$	−3638.00	−0.458182
$399$	0	0
$400$	0	0
$401$	9924.00	1.23586	0.617931	−	0.786232i	$-0.287971\pi$
$401$	9924.00	1.23586	0.617931	+	0.786232i	$0.287971\pi$
$402$	0	0
$403$	3247.00	0.401351
$404$	1776.00	0.218711
$405$	0	0
$406$	6072.00	0.742237
$407$	9144.00	1.11364
$408$	0	0
$409$	−10903.0	−1.31814	−0.659069	−	0.752082i	$-0.729050\pi$
$409$	−10903.0	−1.31814	−0.659069	+	0.752082i	$0.729050\pi$
$410$	0	0
$411$	0	0
$412$	−3664.00	−0.438137
$413$	8184.00	0.975081
$414$	0	0
$415$	0	0
$416$	544.000	0.0641149
$417$	0	0
$418$	−6552.00	−0.766672
$419$	14796.0	1.72514	0.862568	−	0.505941i	$-0.168855\pi$
$419$	14796.0	1.72514	0.862568	+	0.505941i	$0.168855\pi$
$420$	0	0
$421$	−4606.00	−0.533213	−0.266607	−	0.963805i	$-0.585902\pi$
$421$	−4606.00	−0.533213	−0.266607	+	0.963805i	$0.585902\pi$
$422$	−3998.00	−0.461184
$423$	0	0
$424$	−4896.00	−0.560780
$425$	0	0
$426$	0	0
$427$	1837.00	0.208194
$428$	−816.000	−0.0921562
$429$	0	0
$430$	0	0
$431$	13860.0	1.54899	0.774493	−	0.632583i	$-0.218005\pi$
$431$	13860.0	1.54899	0.774493	+	0.632583i	$0.218005\pi$
$432$	0	0
$433$	8051.00	0.893548	0.446774	−	0.894647i	$-0.352573\pi$
$433$	8051.00	0.893548	0.446774	+	0.894647i	$0.352573\pi$
$434$	4202.00	0.464752
$435$	0	0
$436$	−3868.00	−0.424871
$437$	5460.00	0.597682
$438$	0	0
$439$	2087.00	0.226895	0.113448	−	0.993544i	$-0.463811\pi$
$439$	2087.00	0.226895	0.113448	+	0.993544i	$0.463811\pi$
$440$	0	0
$441$	0	0
$442$	408.000	0.0439063
$443$	−13368.0	−1.43371	−0.716854	−	0.697223i	$-0.754419\pi$
$443$	−13368.0	−1.43371	−0.716854	+	0.697223i	$0.754419\pi$
$444$	0	0
$445$	0	0
$446$	−986.000	−0.104683
$447$	0	0
$448$	704.000	0.0742430
$449$	5352.00	0.562531	0.281266	−	0.959630i	$-0.409246\pi$
$449$	5352.00	0.562531	0.281266	+	0.959630i	$0.409246\pi$
$450$	0	0
$451$	2160.00	0.225522
$452$	2688.00	0.279719
$453$	0	0
$454$	600.000	0.0620251
$455$	0	0
$456$	0	0
$457$	13718.0	1.40416	0.702080	−	0.712098i	$-0.252255\pi$
$457$	13718.0	1.40416	0.702080	+	0.712098i	$0.252255\pi$
$458$	−158.000	−0.0161198
$459$	0	0
$460$	0	0
$461$	−11868.0	−1.19902	−0.599510	−	0.800368i	$-0.704638\pi$
$461$	−11868.0	−1.19902	−0.599510	+	0.800368i	$0.704638\pi$
$462$	0	0
$463$	6572.00	0.659669	0.329834	−	0.944039i	$-0.393007\pi$
$463$	6572.00	0.659669	0.329834	+	0.944039i	$0.393007\pi$
$464$	4416.00	0.441827
$465$	0	0
$466$	12216.0	1.21437
$467$	−18636.0	−1.84662	−0.923310	−	0.384056i	$-0.874527\pi$
$467$	−18636.0	−1.84662	−0.923310	+	0.384056i	$0.874527\pi$
$468$	0	0
$469$	−5027.00	−0.494937
$470$	0	0
$471$	0	0
$472$	5952.00	0.580430
$473$	−1764.00	−0.171477
$474$	0	0
$475$	0	0
$476$	528.000	0.0508421
$477$	0	0
$478$	−5736.00	−0.548867
$479$	−5256.00	−0.501363	−0.250681	−	0.968070i	$-0.580655\pi$
$479$	−5256.00	−0.501363	−0.250681	+	0.968070i	$0.580655\pi$
$480$	0	0
$481$	4318.00	0.409322
$482$	5410.00	0.511242
$483$	0	0
$484$	−140.000	−0.0131480
$485$	0	0
$486$	0	0
$487$	−9991.00	−0.929642	−0.464821	−	0.885405i	$-0.653881\pi$
$487$	−9991.00	−0.929642	−0.464821	+	0.885405i	$0.653881\pi$
$488$	1336.00	0.123930
$489$	0	0
$490$	0	0
$491$	−1056.00	−0.0970603	−0.0485302	−	0.998822i	$-0.515454\pi$
$491$	−1056.00	−0.0970603	−0.0485302	+	0.998822i	$0.515454\pi$
$492$	0	0
$493$	3312.00	0.302566
$494$	−3094.00	−0.281793
$495$	0	0
$496$	3056.00	0.276650
$497$	6468.00	0.583761
$498$	0	0
$499$	12329.0	1.10606	0.553028	−	0.833163i	$-0.313472\pi$
$499$	12329.0	1.10606	0.553028	+	0.833163i	$0.313472\pi$
$500$	0	0
$501$	0	0
$502$	8016.00	0.712692
$503$	−2460.00	−0.218064	−0.109032	−	0.994038i	$-0.534775\pi$
$503$	−2460.00	−0.218064	−0.109032	+	0.994038i	$0.534775\pi$
$504$	0	0
$505$	0	0
$506$	−4320.00	−0.379540
$507$	0	0
$508$	−7696.00	−0.672155
$509$	−1848.00	−0.160926	−0.0804628	−	0.996758i	$-0.525640\pi$
$509$	−1848.00	−0.160926	−0.0804628	+	0.996758i	$0.525640\pi$
$510$	0	0
$511$	−10670.0	−0.923705
$512$	512.000	0.0441942
$513$	0	0
$514$	−3984.00	−0.341881
$515$	0	0
$516$	0	0
$517$	21600.0	1.83746
$518$	5588.00	0.473982
$519$	0	0
$520$	0	0
$521$	12732.0	1.07063	0.535316	−	0.844652i	$-0.320193\pi$
$521$	12732.0	1.07063	0.535316	+	0.844652i	$0.320193\pi$
$522$	0	0
$523$	9977.00	0.834156	0.417078	−	0.908871i	$-0.363054\pi$
$523$	9977.00	0.834156	0.417078	+	0.908871i	$0.363054\pi$
$524$	−3024.00	−0.252107
$525$	0	0
$526$	−1944.00	−0.161145
$527$	2292.00	0.189452
$528$	0	0
$529$	−8567.00	−0.704118
$530$	0	0
$531$	0	0
$532$	−4004.00	−0.326307
$533$	1020.00	0.0828914
$534$	0	0
$535$	0	0
$536$	−3656.00	−0.294618
$537$	0	0
$538$	3624.00	0.290412
$539$	−7992.00	−0.638664
$540$	0	0
$541$	16733.0	1.32977	0.664887	−	0.746944i	$-0.268480\pi$
$541$	16733.0	1.32977	0.664887	+	0.746944i	$0.268480\pi$
$542$	−10184.0	−0.807085
$543$	0	0
$544$	384.000	0.0302645
$545$	0	0
$546$	0	0
$547$	−16360.0	−1.27880	−0.639400	−	0.768875i	$-0.720817\pi$
$547$	−16360.0	−1.27880	−0.639400	+	0.768875i	$0.720817\pi$
$548$	3552.00	0.276887
$549$	0	0
$550$	0	0
$551$	−25116.0	−1.94188
$552$	0	0
$553$	1804.00	0.138723
$554$	1138.00	0.0872725
$555$	0	0
$556$	−640.000	−0.0488166
$557$	−3984.00	−0.303066	−0.151533	−	0.988452i	$-0.548421\pi$
$557$	−3984.00	−0.303066	−0.151533	+	0.988452i	$0.548421\pi$
$558$	0	0
$559$	−833.000	−0.0630271
$560$	0	0
$561$	0	0
$562$	−12936.0	−0.970947
$563$	16332.0	1.22258	0.611289	−	0.791407i	$-0.290651\pi$
$563$	16332.0	1.22258	0.611289	+	0.791407i	$0.290651\pi$
$564$	0	0
$565$	0	0
$566$	7114.00	0.528310
$567$	0	0
$568$	4704.00	0.347492
$569$	−25716.0	−1.89468	−0.947338	−	0.320235i	$-0.896238\pi$
$569$	−25716.0	−1.89468	−0.947338	+	0.320235i	$0.896238\pi$
$570$	0	0
$571$	−15091.0	−1.10602	−0.553011	−	0.833174i	$-0.686521\pi$
$571$	−15091.0	−1.10602	−0.553011	+	0.833174i	$0.686521\pi$
$572$	2448.00	0.178944
$573$	0	0
$574$	1320.00	0.0959856
$575$	0	0
$576$	0	0
$577$	2579.00	0.186075	0.0930374	−	0.995663i	$-0.470342\pi$
$577$	2579.00	0.186075	0.0930374	+	0.995663i	$0.470342\pi$
$578$	−9538.00	−0.686381
$579$	0	0
$580$	0	0
$581$	−7656.00	−0.546686
$582$	0	0
$583$	−22032.0	−1.56513
$584$	−7760.00	−0.549848
$585$	0	0
$586$	−10752.0	−0.757954
$587$	11892.0	0.836176	0.418088	−	0.908407i	$-0.362700\pi$
$587$	11892.0	0.836176	0.418088	+	0.908407i	$0.362700\pi$
$588$	0	0
$589$	−17381.0	−1.21591
$590$	0	0
$591$	0	0
$592$	4064.00	0.282144
$593$	6564.00	0.454555	0.227278	−	0.973830i	$-0.427018\pi$
$593$	6564.00	0.454555	0.227278	+	0.973830i	$0.427018\pi$
$594$	0	0
$595$	0	0
$596$	−12384.0	−0.851121
$597$	0	0
$598$	−2040.00	−0.139501
$599$	−9096.00	−0.620455	−0.310227	−	0.950662i	$-0.600405\pi$
$599$	−9096.00	−0.620455	−0.310227	+	0.950662i	$0.600405\pi$
$600$	0	0
$601$	6575.00	0.446256	0.223128	−	0.974789i	$-0.428373\pi$
$601$	6575.00	0.446256	0.223128	+	0.974789i	$0.428373\pi$
$602$	−1078.00	−0.0729834
$603$	0	0
$604$	4196.00	0.282670
$605$	0	0
$606$	0	0
$607$	−12436.0	−0.831568	−0.415784	−	0.909463i	$-0.636493\pi$
$607$	−12436.0	−0.831568	−0.415784	+	0.909463i	$0.636493\pi$
$608$	−2912.00	−0.194239
$609$	0	0
$610$	0	0
$611$	10200.0	0.675365
$612$	0	0
$613$	−4354.00	−0.286878	−0.143439	−	0.989659i	$-0.545816\pi$
$613$	−4354.00	−0.286878	−0.143439	+	0.989659i	$0.545816\pi$
$614$	−18686.0	−1.22818
$615$	0	0
$616$	3168.00	0.207212
$617$	−30060.0	−1.96138	−0.980689	−	0.195575i	$-0.937343\pi$
$617$	−30060.0	−1.96138	−0.980689	+	0.195575i	$0.937343\pi$
$618$	0	0
$619$	24845.0	1.61326	0.806628	−	0.591060i	$-0.201290\pi$
$619$	24845.0	1.61326	0.806628	+	0.591060i	$0.201290\pi$
$620$	0	0
$621$	0	0
$622$	3936.00	0.253729
$623$	13728.0	0.882826
$624$	0	0
$625$	0	0
$626$	11350.0	0.724660
$627$	0	0
$628$	9452.00	0.600599
$629$	3048.00	0.193214
$630$	0	0
$631$	−10885.0	−0.686727	−0.343364	−	0.939203i	$-0.611566\pi$
$631$	−10885.0	−0.686727	−0.343364	+	0.939203i	$0.611566\pi$
$632$	1312.00	0.0825768
$633$	0	0
$634$	5232.00	0.327743
$635$	0	0
$636$	0	0
$637$	−3774.00	−0.234743
$638$	19872.0	1.23313
$639$	0	0
$640$	0	0
$641$	−20136.0	−1.24076	−0.620378	−	0.784303i	$-0.713021\pi$
$641$	−20136.0	−1.24076	−0.620378	+	0.784303i	$0.713021\pi$
$642$	0	0
$643$	−10888.0	−0.667777	−0.333889	−	0.942613i	$-0.608361\pi$
$643$	−10888.0	−0.667777	−0.333889	+	0.942613i	$0.608361\pi$
$644$	−2640.00	−0.161538
$645$	0	0
$646$	−2184.00	−0.133016
$647$	−30384.0	−1.84624	−0.923121	−	0.384510i	$-0.874370\pi$
$647$	−30384.0	−1.84624	−0.923121	+	0.384510i	$0.874370\pi$
$648$	0	0
$649$	26784.0	1.61998
$650$	0	0
$651$	0	0
$652$	−12940.0	−0.777254
$653$	7104.00	0.425729	0.212864	−	0.977082i	$-0.431721\pi$
$653$	7104.00	0.425729	0.212864	+	0.977082i	$0.431721\pi$
$654$	0	0
$655$	0	0
$656$	960.000	0.0571367
$657$	0	0
$658$	13200.0	0.782051
$659$	−24180.0	−1.42932	−0.714658	−	0.699474i	$-0.753418\pi$
$659$	−24180.0	−1.42932	−0.714658	+	0.699474i	$0.753418\pi$
$660$	0	0
$661$	15662.0	0.921605	0.460803	−	0.887503i	$-0.347562\pi$
$661$	15662.0	0.921605	0.460803	+	0.887503i	$0.347562\pi$
$662$	−19328.0	−1.13475
$663$	0	0
$664$	−5568.00	−0.325422
$665$	0	0
$666$	0	0
$667$	−16560.0	−0.961328
$668$	−11088.0	−0.642227
$669$	0	0
$670$	0	0
$671$	6012.00	0.345888
$672$	0	0
$673$	−6622.00	−0.379286	−0.189643	−	0.981853i	$-0.560733\pi$
$673$	−6622.00	−0.379286	−0.189643	+	0.981853i	$0.560733\pi$
$674$	15274.0	0.872897
$675$	0	0
$676$	−7632.00	−0.434228
$677$	3468.00	0.196877	0.0984387	−	0.995143i	$-0.468615\pi$
$677$	3468.00	0.196877	0.0984387	+	0.995143i	$0.468615\pi$
$678$	0	0
$679$	−12089.0	−0.683260
$680$	0	0
$681$	0	0
$682$	13752.0	0.772128
$683$	19344.0	1.08372	0.541858	−	0.840470i	$-0.317721\pi$
$683$	19344.0	1.08372	0.541858	+	0.840470i	$0.317721\pi$
$684$	0	0
$685$	0	0
$686$	−12430.0	−0.691807
$687$	0	0
$688$	−784.000	−0.0434444
$689$	−10404.0	−0.575270
$690$	0	0
$691$	−9736.00	−0.535998	−0.267999	−	0.963419i	$-0.586362\pi$
$691$	−9736.00	−0.535998	−0.267999	+	0.963419i	$0.586362\pi$
$692$	−336.000	−0.0184578
$693$	0	0
$694$	11280.0	0.616978
$695$	0	0
$696$	0	0
$697$	720.000	0.0391276
$698$	14764.0	0.800610
$699$	0	0
$700$	0	0
$701$	3012.00	0.162285	0.0811424	−	0.996703i	$-0.474143\pi$
$701$	3012.00	0.162285	0.0811424	+	0.996703i	$0.474143\pi$
$702$	0	0
$703$	−23114.0	−1.24006
$704$	2304.00	0.123346
$705$	0	0
$706$	−21480.0	−1.14506
$707$	4884.00	0.259804
$708$	0	0
$709$	−23851.0	−1.26339	−0.631695	−	0.775217i	$-0.717640\pi$
$709$	−23851.0	−1.26339	−0.631695	+	0.775217i	$0.717640\pi$
$710$	0	0
$711$	0	0
$712$	9984.00	0.525514
$713$	−11460.0	−0.601936
$714$	0	0
$715$	0	0
$716$	−10944.0	−0.571224
$717$	0	0
$718$	18552.0	0.964282
$719$	−5916.00	−0.306856	−0.153428	−	0.988160i	$-0.549031\pi$
$719$	−5916.00	−0.306856	−0.153428	+	0.988160i	$0.549031\pi$
$720$	0	0
$721$	−10076.0	−0.520457
$722$	2844.00	0.146597
$723$	0	0
$724$	17588.0	0.902835
$725$	0	0
$726$	0	0
$727$	−27685.0	−1.41235	−0.706176	−	0.708036i	$-0.749581\pi$
$727$	−27685.0	−1.41235	−0.706176	+	0.708036i	$0.749581\pi$
$728$	1496.00	0.0761613
$729$	0	0
$730$	0	0
$731$	−588.000	−0.0297510
$732$	0	0
$733$	10154.0	0.511660	0.255830	−	0.966722i	$-0.417651\pi$
$733$	10154.0	0.511660	0.255830	+	0.966722i	$0.417651\pi$
$734$	−14774.0	−0.742940
$735$	0	0
$736$	−1920.00	−0.0961578
$737$	−16452.0	−0.822276
$738$	0	0
$739$	−13840.0	−0.688921	−0.344461	−	0.938801i	$-0.611938\pi$
$739$	−13840.0	−0.688921	−0.344461	+	0.938801i	$0.611938\pi$
$740$	0	0
$741$	0	0
$742$	−13464.0	−0.666144
$743$	−14160.0	−0.699166	−0.349583	−	0.936905i	$-0.613677\pi$
$743$	−14160.0	−0.699166	−0.349583	+	0.936905i	$0.613677\pi$
$744$	0	0
$745$	0	0
$746$	20518.0	1.00699
$747$	0	0
$748$	1728.00	0.0844678
$749$	−2244.00	−0.109471
$750$	0	0
$751$	36740.0	1.78517	0.892584	−	0.450881i	$-0.148890\pi$
$751$	36740.0	1.78517	0.892584	+	0.450881i	$0.148890\pi$
$752$	9600.00	0.465527
$753$	0	0
$754$	9384.00	0.453243
$755$	0	0
$756$	0	0
$757$	8285.00	0.397785	0.198893	−	0.980021i	$-0.436265\pi$
$757$	8285.00	0.397785	0.198893	+	0.980021i	$0.436265\pi$
$758$	15310.0	0.733620
$759$	0	0
$760$	0	0
$761$	4656.00	0.221787	0.110893	−	0.993832i	$-0.464629\pi$
$761$	4656.00	0.221787	0.110893	+	0.993832i	$0.464629\pi$
$762$	0	0
$763$	−10637.0	−0.504699
$764$	−12432.0	−0.588709
$765$	0	0
$766$	−1200.00	−0.0566028
$767$	12648.0	0.595427
$768$	0	0
$769$	−21169.0	−0.992684	−0.496342	−	0.868127i	$-0.665324\pi$
$769$	−21169.0	−0.992684	−0.496342	+	0.868127i	$0.665324\pi$
$770$	0	0
$771$	0	0
$772$	10460.0	0.487647
$773$	31572.0	1.46904	0.734519	−	0.678588i	$-0.237408\pi$
$773$	31572.0	1.46904	0.734519	+	0.678588i	$0.237408\pi$
$774$	0	0
$775$	0	0
$776$	−8792.00	−0.406720
$777$	0	0
$778$	8136.00	0.374923
$779$	−5460.00	−0.251123
$780$	0	0
$781$	21168.0	0.969847
$782$	−1440.00	−0.0658495
$783$	0	0
$784$	−3552.00	−0.161808
$785$	0	0
$786$	0	0
$787$	16781.0	0.760074	0.380037	−	0.924971i	$-0.375911\pi$
$787$	16781.0	0.760074	0.380037	+	0.924971i	$0.375911\pi$
$788$	−14496.0	−0.655328
$789$	0	0
$790$	0	0
$791$	7392.00	0.332275
$792$	0	0
$793$	2839.00	0.127132
$794$	25294.0	1.13054
$795$	0	0
$796$	−7276.00	−0.323984
$797$	−216.000	−0.00959989	−0.00479995	−	0.999988i	$-0.501528\pi$
$797$	−216.000	−0.00959989	−0.00479995	+	0.999988i	$0.501528\pi$
$798$	0	0
$799$	7200.00	0.318796
$800$	0	0
$801$	0	0
$802$	19848.0	0.873887
$803$	−34920.0	−1.53462
$804$	0	0
$805$	0	0
$806$	6494.00	0.283798
$807$	0	0
$808$	3552.00	0.154652
$809$	17484.0	0.759833	0.379916	−	0.925021i	$-0.375953\pi$
$809$	17484.0	0.759833	0.379916	+	0.925021i	$0.375953\pi$
$810$	0	0
$811$	−28447.0	−1.23170	−0.615850	−	0.787863i	$-0.711187\pi$
$811$	−28447.0	−1.23170	−0.615850	+	0.787863i	$0.711187\pi$
$812$	12144.0	0.524841
$813$	0	0
$814$	18288.0	0.787462
$815$	0	0
$816$	0	0
$817$	4459.00	0.190943
$818$	−21806.0	−0.932065
$819$	0	0
$820$	0	0
$821$	−15120.0	−0.642743	−0.321371	−	0.946953i	$-0.604144\pi$
$821$	−15120.0	−0.642743	−0.321371	+	0.946953i	$0.604144\pi$
$822$	0	0
$823$	−43957.0	−1.86178	−0.930890	−	0.365300i	$-0.880966\pi$
$823$	−43957.0	−1.86178	−0.930890	+	0.365300i	$0.880966\pi$
$824$	−7328.00	−0.309809
$825$	0	0
$826$	16368.0	0.689486
$827$	23892.0	1.00460	0.502301	−	0.864693i	$-0.332487\pi$
$827$	23892.0	1.00460	0.502301	+	0.864693i	$0.332487\pi$
$828$	0	0
$829$	−12166.0	−0.509702	−0.254851	−	0.966980i	$-0.582026\pi$
$829$	−12166.0	−0.509702	−0.254851	+	0.966980i	$0.582026\pi$
$830$	0	0
$831$	0	0
$832$	1088.00	0.0453361
$833$	−2664.00	−0.110807
$834$	0	0
$835$	0	0
$836$	−13104.0	−0.542119
$837$	0	0
$838$	29592.0	1.21986
$839$	25812.0	1.06213	0.531066	−	0.847330i	$-0.321792\pi$
$839$	25812.0	1.06213	0.531066	+	0.847330i	$0.321792\pi$
$840$	0	0
$841$	51787.0	2.12338
$842$	−9212.00	−0.377039
$843$	0	0
$844$	−7996.00	−0.326106
$845$	0	0
$846$	0	0
$847$	−385.000	−0.0156184
$848$	−9792.00	−0.396531
$849$	0	0
$850$	0	0
$851$	−15240.0	−0.613890
$852$	0	0
$853$	−2689.00	−0.107936	−0.0539681	−	0.998543i	$-0.517187\pi$
$853$	−2689.00	−0.107936	−0.0539681	+	0.998543i	$0.517187\pi$
$854$	3674.00	0.147215
$855$	0	0
$856$	−1632.00	−0.0651643
$857$	48456.0	1.93142	0.965709	−	0.259626i	$-0.0835994\pi$
$857$	48456.0	1.93142	0.965709	+	0.259626i	$0.0835994\pi$
$858$	0	0
$859$	−12760.0	−0.506828	−0.253414	−	0.967358i	$-0.581554\pi$
$859$	−12760.0	−0.506828	−0.253414	+	0.967358i	$0.581554\pi$
$860$	0	0
$861$	0	0
$862$	27720.0	1.09530
$863$	10008.0	0.394758	0.197379	−	0.980327i	$-0.436757\pi$
$863$	10008.0	0.394758	0.197379	+	0.980327i	$0.436757\pi$
$864$	0	0
$865$	0	0
$866$	16102.0	0.631834
$867$	0	0
$868$	8404.00	0.328629
$869$	5904.00	0.230471
$870$	0	0
$871$	−7769.00	−0.302230
$872$	−7736.00	−0.300429
$873$	0	0
$874$	10920.0	0.422625
$875$	0	0
$876$	0	0
$877$	12251.0	0.471707	0.235853	−	0.971789i	$-0.424211\pi$
$877$	12251.0	0.471707	0.235853	+	0.971789i	$0.424211\pi$
$878$	4174.00	0.160439
$879$	0	0
$880$	0	0
$881$	16344.0	0.625021	0.312510	−	0.949914i	$-0.398830\pi$
$881$	16344.0	0.625021	0.312510	+	0.949914i	$0.398830\pi$
$882$	0	0
$883$	42227.0	1.60935	0.804673	−	0.593719i	$-0.202341\pi$
$883$	42227.0	1.60935	0.804673	+	0.593719i	$0.202341\pi$
$884$	816.000	0.0310464
$885$	0	0
$886$	−26736.0	−1.01378
$887$	−13896.0	−0.526023	−0.263011	−	0.964793i	$-0.584716\pi$
$887$	−13896.0	−0.526023	−0.263011	+	0.964793i	$0.584716\pi$
$888$	0	0
$889$	−21164.0	−0.798445
$890$	0	0
$891$	0	0
$892$	−1972.00	−0.0740218
$893$	−54600.0	−2.04605
$894$	0	0
$895$	0	0
$896$	1408.00	0.0524977
$897$	0	0
$898$	10704.0	0.397770
$899$	52716.0	1.95570
$900$	0	0
$901$	−7344.00	−0.271547
$902$	4320.00	0.159468
$903$	0	0
$904$	5376.00	0.197791
$905$	0	0
$906$	0	0
$907$	−1240.00	−0.0453953	−0.0226976	−	0.999742i	$-0.507226\pi$
$907$	−1240.00	−0.0453953	−0.0226976	+	0.999742i	$0.507226\pi$
$908$	1200.00	0.0438584
$909$	0	0
$910$	0	0
$911$	−6360.00	−0.231302	−0.115651	−	0.993290i	$-0.536895\pi$
$911$	−6360.00	−0.231302	−0.115651	+	0.993290i	$0.536895\pi$
$912$	0	0
$913$	−25056.0	−0.908250
$914$	27436.0	0.992891
$915$	0	0
$916$	−316.000	−0.0113984
$917$	−8316.00	−0.299475
$918$	0	0
$919$	−2143.00	−0.0769217	−0.0384609	−	0.999260i	$-0.512245\pi$
$919$	−2143.00	−0.0769217	−0.0384609	+	0.999260i	$0.512245\pi$
$920$	0	0
$921$	0	0
$922$	−23736.0	−0.847835
$923$	9996.00	0.356471
$924$	0	0
$925$	0	0
$926$	13144.0	0.466456
$927$	0	0
$928$	8832.00	0.312419
$929$	29124.0	1.02855	0.514277	−	0.857624i	$-0.328060\pi$
$929$	29124.0	1.02855	0.514277	+	0.857624i	$0.328060\pi$
$930$	0	0
$931$	20202.0	0.711164
$932$	24432.0	0.858688
$933$	0	0
$934$	−37272.0	−1.30576
$935$	0	0
$936$	0	0
$937$	24059.0	0.838819	0.419409	−	0.907797i	$-0.362237\pi$
$937$	24059.0	0.838819	0.419409	+	0.907797i	$0.362237\pi$
$938$	−10054.0	−0.349973
$939$	0	0
$940$	0	0
$941$	47088.0	1.63127	0.815635	−	0.578567i	$-0.196388\pi$
$941$	47088.0	1.63127	0.815635	+	0.578567i	$0.196388\pi$
$942$	0	0
$943$	−3600.00	−0.124318
$944$	11904.0	0.410426
$945$	0	0
$946$	−3528.00	−0.121253
$947$	−23148.0	−0.794307	−0.397154	−	0.917752i	$-0.630002\pi$
$947$	−23148.0	−0.794307	−0.397154	+	0.917752i	$0.630002\pi$
$948$	0	0
$949$	−16490.0	−0.564055
$950$	0	0
$951$	0	0
$952$	1056.00	0.0359508
$953$	−27600.0	−0.938144	−0.469072	−	0.883160i	$-0.655412\pi$
$953$	−27600.0	−0.938144	−0.469072	+	0.883160i	$0.655412\pi$
$954$	0	0
$955$	0	0
$956$	−11472.0	−0.388108
$957$	0	0
$958$	−10512.0	−0.354517
$959$	9768.00	0.328911
$960$	0	0
$961$	6690.00	0.224564
$962$	8636.00	0.289434
$963$	0	0
$964$	10820.0	0.361503
$965$	0	0
$966$	0	0
$967$	7436.00	0.247286	0.123643	−	0.992327i	$-0.460542\pi$
$967$	7436.00	0.247286	0.123643	+	0.992327i	$0.460542\pi$
$968$	−280.000	−0.00929705
$969$	0	0
$970$	0	0
$971$	27264.0	0.901075	0.450537	−	0.892758i	$-0.351232\pi$
$971$	27264.0	0.901075	0.450537	+	0.892758i	$0.351232\pi$
$972$	0	0
$973$	−1760.00	−0.0579887
$974$	−19982.0	−0.657356
$975$	0	0
$976$	2672.00	0.0876318
$977$	46632.0	1.52701	0.763506	−	0.645801i	$-0.223477\pi$
$977$	46632.0	1.52701	0.763506	+	0.645801i	$0.223477\pi$
$978$	0	0
$979$	44928.0	1.46671
$980$	0	0
$981$	0	0
$982$	−2112.00	−0.0686320
$983$	−15216.0	−0.493708	−0.246854	−	0.969053i	$-0.579397\pi$
$983$	−15216.0	−0.493708	−0.246854	+	0.969053i	$0.579397\pi$
$984$	0	0
$985$	0	0
$986$	6624.00	0.213946
$987$	0	0
$988$	−6188.00	−0.199258
$989$	2940.00	0.0945264
$990$	0	0
$991$	39047.0	1.25163	0.625817	−	0.779970i	$-0.284766\pi$
$991$	39047.0	1.25163	0.625817	+	0.779970i	$0.284766\pi$
$992$	6112.00	0.195621
$993$	0	0
$994$	12936.0	0.412782
$995$	0	0
$996$	0	0
$997$	−46258.0	−1.46941	−0.734707	−	0.678385i	$-0.762680\pi$
$997$	−46258.0	−1.46941	−0.734707	+	0.678385i	$0.762680\pi$
$998$	24658.0	0.782100
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.4.a.s.1.1	yes	1
3.2	odd	2		450.4.a.g.1.1	yes	1
5.2	odd	4		450.4.c.i.199.2		2
5.3	odd	4		450.4.c.i.199.1		2
5.4	even	2		450.4.a.d.1.1	✓	1
15.2	even	4		450.4.c.b.199.1		2
15.8	even	4		450.4.c.b.199.2		2
15.14	odd	2		450.4.a.n.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.4.a.d.1.1	✓	1	5.4	even	2
450.4.a.g.1.1	yes	1	3.2	odd	2
450.4.a.n.1.1	yes	1	15.14	odd	2
450.4.a.s.1.1	yes	1	1.1	even	1	trivial
450.4.c.b.199.1		2	15.2	even	4
450.4.c.b.199.2		2	15.8	even	4
450.4.c.i.199.1		2	5.3	odd	4
450.4.c.i.199.2		2	5.2	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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