LMFDB - Embedded newform 450.4.a.a.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 50)
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)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} -34.0000 q^{7} -8.00000 q^{8} +O(q^{10})$

$q-2.00000 q^{2} +4.00000 q^{4} -34.0000 q^{7} -8.00000 q^{8} -27.0000 q^{11} -28.0000 q^{13} +68.0000 q^{14} +16.0000 q^{16} -21.0000 q^{17} +35.0000 q^{19} +54.0000 q^{22} +78.0000 q^{23} +56.0000 q^{26} -136.000 q^{28} +120.000 q^{29} +182.000 q^{31} -32.0000 q^{32} +42.0000 q^{34} +146.000 q^{37} -70.0000 q^{38} -357.000 q^{41} -148.000 q^{43} -108.000 q^{44} -156.000 q^{46} +84.0000 q^{47} +813.000 q^{49} -112.000 q^{52} -702.000 q^{53} +272.000 q^{56} -240.000 q^{58} +840.000 q^{59} -238.000 q^{61} -364.000 q^{62} +64.0000 q^{64} +461.000 q^{67} -84.0000 q^{68} +708.000 q^{71} -133.000 q^{73} -292.000 q^{74} +140.000 q^{76} +918.000 q^{77} +650.000 q^{79} +714.000 q^{82} +903.000 q^{83} +296.000 q^{86} +216.000 q^{88} -735.000 q^{89} +952.000 q^{91} +312.000 q^{92} -168.000 q^{94} +1106.00 q^{97} -1626.00 q^{98} +O(q^{100})$

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
$7$	−34.0000	−1.83583	−0.917914	−	0.396780i	$-0.870128\pi$
$7$	−34.0000	−1.83583	−0.917914	+	0.396780i	$0.870128\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−27.0000	−0.740073	−0.370037	−	0.929017i	$-0.620655\pi$
$11$	−27.0000	−0.740073	−0.370037	+	0.929017i	$0.620655\pi$
$12$	0	0
$13$	−28.0000	−0.597369	−0.298685	−	0.954352i	$-0.596548\pi$
$13$	−28.0000	−0.597369	−0.298685	+	0.954352i	$0.596548\pi$
$14$	68.0000	1.29813
$15$	0	0
$16$	16.0000	0.250000
$17$	−21.0000	−0.299603	−0.149801	−	0.988716i	$-0.547863\pi$
$17$	−21.0000	−0.299603	−0.149801	+	0.988716i	$0.547863\pi$
$18$	0	0
$19$	35.0000	0.422608	0.211304	−	0.977420i	$-0.432229\pi$
$19$	35.0000	0.422608	0.211304	+	0.977420i	$0.432229\pi$
$20$	0	0
$21$	0	0
$22$	54.0000	0.523311
$23$	78.0000	0.707136	0.353568	−	0.935409i	$-0.384968\pi$
$23$	78.0000	0.707136	0.353568	+	0.935409i	$0.384968\pi$
$24$	0	0
$25$	0	0
$26$	56.0000	0.422404
$27$	0	0
$28$	−136.000	−0.917914
$29$	120.000	0.768395	0.384197	−	0.923251i	$-0.374478\pi$
$29$	120.000	0.768395	0.384197	+	0.923251i	$0.374478\pi$
$30$	0	0
$31$	182.000	1.05446	0.527228	−	0.849724i	$-0.323231\pi$
$31$	182.000	1.05446	0.527228	+	0.849724i	$0.323231\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	42.0000	0.211851
$35$	0	0
$36$	0	0
$37$	146.000	0.648710	0.324355	−	0.945936i	$-0.394853\pi$
$37$	146.000	0.648710	0.324355	+	0.945936i	$0.394853\pi$
$38$	−70.0000	−0.298829
$39$	0	0
$40$	0	0
$41$	−357.000	−1.35985	−0.679927	−	0.733280i	$-0.737989\pi$
$41$	−357.000	−1.35985	−0.679927	+	0.733280i	$0.737989\pi$
$42$	0	0
$43$	−148.000	−0.524879	−0.262439	−	0.964948i	$-0.584527\pi$
$43$	−148.000	−0.524879	−0.262439	+	0.964948i	$0.584527\pi$
$44$	−108.000	−0.370037
$45$	0	0
$46$	−156.000	−0.500021
$47$	84.0000	0.260695	0.130347	−	0.991468i	$-0.458391\pi$
$47$	84.0000	0.260695	0.130347	+	0.991468i	$0.458391\pi$
$48$	0	0
$49$	813.000	2.37026
$50$	0	0
$51$	0	0
$52$	−112.000	−0.298685
$53$	−702.000	−1.81938	−0.909690	−	0.415288i	$-0.863681\pi$
$53$	−702.000	−1.81938	−0.909690	+	0.415288i	$0.863681\pi$
$54$	0	0
$55$	0	0
$56$	272.000	0.649063
$57$	0	0
$58$	−240.000	−0.543337
$59$	840.000	1.85354	0.926769	−	0.375633i	$-0.122575\pi$
$59$	840.000	1.85354	0.926769	+	0.375633i	$0.122575\pi$
$60$	0	0
$61$	−238.000	−0.499554	−0.249777	−	0.968303i	$-0.580357\pi$
$61$	−238.000	−0.499554	−0.249777	+	0.968303i	$0.580357\pi$
$62$	−364.000	−0.745614
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	461.000	0.840599	0.420299	−	0.907386i	$-0.361925\pi$
$67$	461.000	0.840599	0.420299	+	0.907386i	$0.361925\pi$
$68$	−84.0000	−0.149801
$69$	0	0
$70$	0	0
$71$	708.000	1.18344	0.591719	−	0.806144i	$-0.298449\pi$
$71$	708.000	1.18344	0.591719	+	0.806144i	$0.298449\pi$
$72$	0	0
$73$	−133.000	−0.213239	−0.106620	−	0.994300i	$-0.534003\pi$
$73$	−133.000	−0.213239	−0.106620	+	0.994300i	$0.534003\pi$
$74$	−292.000	−0.458707
$75$	0	0
$76$	140.000	0.211304
$77$	918.000	1.35865
$78$	0	0
$79$	650.000	0.925705	0.462853	−	0.886435i	$-0.346826\pi$
$79$	650.000	0.925705	0.462853	+	0.886435i	$0.346826\pi$
$80$	0	0
$81$	0	0
$82$	714.000	0.961562
$83$	903.000	1.19418	0.597091	−	0.802173i	$-0.296323\pi$
$83$	903.000	1.19418	0.597091	+	0.802173i	$0.296323\pi$
$84$	0	0
$85$	0	0
$86$	296.000	0.371145
$87$	0	0
$88$	216.000	0.261655
$89$	−735.000	−0.875392	−0.437696	−	0.899123i	$-0.644205\pi$
$89$	−735.000	−0.875392	−0.437696	+	0.899123i	$0.644205\pi$
$90$	0	0
$91$	952.000	1.09667
$92$	312.000	0.353568
$93$	0	0
$94$	−168.000	−0.184339
$95$	0	0
$96$	0	0
$97$	1106.00	1.15770	0.578852	−	0.815433i	$-0.303501\pi$
$97$	1106.00	1.15770	0.578852	+	0.815433i	$0.303501\pi$
$98$	−1626.00	−1.67603
$99$	0	0
$100$	0	0
$101$	−462.000	−0.455156	−0.227578	−	0.973760i	$-0.573081\pi$
$101$	−462.000	−0.455156	−0.227578	+	0.973760i	$0.573081\pi$
$102$	0	0
$103$	812.000	0.776784	0.388392	−	0.921494i	$-0.373031\pi$
$103$	812.000	0.776784	0.388392	+	0.921494i	$0.373031\pi$
$104$	224.000	0.211202
$105$	0	0
$106$	1404.00	1.28650
$107$	789.000	0.712855	0.356428	−	0.934323i	$-0.383995\pi$
$107$	789.000	0.712855	0.356428	+	0.934323i	$0.383995\pi$
$108$	0	0
$109$	230.000	0.202110	0.101055	−	0.994881i	$-0.467778\pi$
$109$	230.000	0.202110	0.101055	+	0.994881i	$0.467778\pi$
$110$	0	0
$111$	0	0
$112$	−544.000	−0.458957
$113$	2073.00	1.72576	0.862882	−	0.505405i	$-0.168657\pi$
$113$	2073.00	1.72576	0.862882	+	0.505405i	$0.168657\pi$
$114$	0	0
$115$	0	0
$116$	480.000	0.384197
$117$	0	0
$118$	−1680.00	−1.31065
$119$	714.000	0.550019
$120$	0	0
$121$	−602.000	−0.452292
$122$	476.000	0.353238
$123$	0	0
$124$	728.000	0.527228
$125$	0	0
$126$	0	0
$127$	−1114.00	−0.778358	−0.389179	−	0.921162i	$-0.627241\pi$
$127$	−1114.00	−0.778358	−0.389179	+	0.921162i	$0.627241\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−252.000	−0.168071	−0.0840357	−	0.996463i	$-0.526781\pi$
$131$	−252.000	−0.168071	−0.0840357	+	0.996463i	$0.526781\pi$
$132$	0	0
$133$	−1190.00	−0.775835
$134$	−922.000	−0.594393
$135$	0	0
$136$	168.000	0.105926
$137$	−1941.00	−1.21044	−0.605222	−	0.796057i	$-0.706916\pi$
$137$	−1941.00	−1.21044	−0.605222	+	0.796057i	$0.706916\pi$
$138$	0	0
$139$	−1645.00	−1.00379	−0.501896	−	0.864928i	$-0.667364\pi$
$139$	−1645.00	−1.00379	−0.501896	+	0.864928i	$0.667364\pi$
$140$	0	0
$141$	0	0
$142$	−1416.00	−0.836817
$143$	756.000	0.442097
$144$	0	0
$145$	0	0
$146$	266.000	0.150783
$147$	0	0
$148$	584.000	0.324355
$149$	1800.00	0.989676	0.494838	−	0.868985i	$-0.335227\pi$
$149$	1800.00	0.989676	0.494838	+	0.868985i	$0.335227\pi$
$150$	0	0
$151$	−3298.00	−1.77740	−0.888700	−	0.458489i	$-0.848391\pi$
$151$	−3298.00	−1.77740	−0.888700	+	0.458489i	$0.848391\pi$
$152$	−280.000	−0.149414
$153$	0	0
$154$	−1836.00	−0.960708
$155$	0	0
$156$	0	0
$157$	266.000	0.135217	0.0676086	−	0.997712i	$-0.478463\pi$
$157$	266.000	0.135217	0.0676086	+	0.997712i	$0.478463\pi$
$158$	−1300.00	−0.654572
$159$	0	0
$160$	0	0
$161$	−2652.00	−1.29818
$162$	0	0
$163$	1157.00	0.555971	0.277985	−	0.960585i	$-0.410333\pi$
$163$	1157.00	0.555971	0.277985	+	0.960585i	$0.410333\pi$
$164$	−1428.00	−0.679927
$165$	0	0
$166$	−1806.00	−0.844414
$167$	1764.00	0.817380	0.408690	−	0.912673i	$-0.365986\pi$
$167$	1764.00	0.817380	0.408690	+	0.912673i	$0.365986\pi$
$168$	0	0
$169$	−1413.00	−0.643150
$170$	0	0
$171$	0	0
$172$	−592.000	−0.262439
$173$	1848.00	0.812144	0.406072	−	0.913841i	$-0.366898\pi$
$173$	1848.00	0.812144	0.406072	+	0.913841i	$0.366898\pi$
$174$	0	0
$175$	0	0
$176$	−432.000	−0.185018
$177$	0	0
$178$	1470.00	0.618995
$179$	−135.000	−0.0563708	−0.0281854	−	0.999603i	$-0.508973\pi$
$179$	−135.000	−0.0563708	−0.0281854	+	0.999603i	$0.508973\pi$
$180$	0	0
$181$	2282.00	0.937126	0.468563	−	0.883430i	$-0.344772\pi$
$181$	2282.00	0.937126	0.468563	+	0.883430i	$0.344772\pi$
$182$	−1904.00	−0.775461
$183$	0	0
$184$	−624.000	−0.250010
$185$	0	0
$186$	0	0
$187$	567.000	0.221728
$188$	336.000	0.130347
$189$	0	0
$190$	0	0
$191$	1398.00	0.529611	0.264806	−	0.964302i	$-0.414692\pi$
$191$	1398.00	0.529611	0.264806	+	0.964302i	$0.414692\pi$
$192$	0	0
$193$	3317.00	1.23711	0.618557	−	0.785740i	$-0.287718\pi$
$193$	3317.00	1.23711	0.618557	+	0.785740i	$0.287718\pi$
$194$	−2212.00	−0.818620
$195$	0	0
$196$	3252.00	1.18513
$197$	−1686.00	−0.609759	−0.304880	−	0.952391i	$-0.598616\pi$
$197$	−1686.00	−0.609759	−0.304880	+	0.952391i	$0.598616\pi$
$198$	0	0
$199$	−1540.00	−0.548581	−0.274291	−	0.961647i	$-0.588443\pi$
$199$	−1540.00	−0.548581	−0.274291	+	0.961647i	$0.588443\pi$
$200$	0	0
$201$	0	0
$202$	924.000	0.321844
$203$	−4080.00	−1.41064
$204$	0	0
$205$	0	0
$206$	−1624.00	−0.549269
$207$	0	0
$208$	−448.000	−0.149342
$209$	−945.000	−0.312761
$210$	0	0
$211$	−3043.00	−0.992838	−0.496419	−	0.868083i	$-0.665352\pi$
$211$	−3043.00	−0.992838	−0.496419	+	0.868083i	$0.665352\pi$
$212$	−2808.00	−0.909690
$213$	0	0
$214$	−1578.00	−0.504065
$215$	0	0
$216$	0	0
$217$	−6188.00	−1.93580
$218$	−460.000	−0.142913
$219$	0	0
$220$	0	0
$221$	588.000	0.178974
$222$	0	0
$223$	3332.00	1.00057	0.500285	−	0.865861i	$-0.333228\pi$
$223$	3332.00	1.00057	0.500285	+	0.865861i	$0.333228\pi$
$224$	1088.00	0.324532
$225$	0	0
$226$	−4146.00	−1.22030
$227$	−1596.00	−0.466653	−0.233327	−	0.972398i	$-0.574961\pi$
$227$	−1596.00	−0.466653	−0.233327	+	0.972398i	$0.574961\pi$
$228$	0	0
$229$	4340.00	1.25238	0.626191	−	0.779670i	$-0.284613\pi$
$229$	4340.00	1.25238	0.626191	+	0.779670i	$0.284613\pi$
$230$	0	0
$231$	0	0
$232$	−960.000	−0.271668
$233$	3018.00	0.848565	0.424283	−	0.905530i	$-0.360526\pi$
$233$	3018.00	0.848565	0.424283	+	0.905530i	$0.360526\pi$
$234$	0	0
$235$	0	0
$236$	3360.00	0.926769
$237$	0	0
$238$	−1428.00	−0.388922
$239$	−4440.00	−1.20167	−0.600836	−	0.799372i	$-0.705166\pi$
$239$	−4440.00	−1.20167	−0.600836	+	0.799372i	$0.705166\pi$
$240$	0	0
$241$	−3703.00	−0.989756	−0.494878	−	0.868962i	$-0.664787\pi$
$241$	−3703.00	−0.989756	−0.494878	+	0.868962i	$0.664787\pi$
$242$	1204.00	0.319818
$243$	0	0
$244$	−952.000	−0.249777
$245$	0	0
$246$	0	0
$247$	−980.000	−0.252453
$248$	−1456.00	−0.372807
$249$	0	0
$250$	0	0
$251$	−7077.00	−1.77967	−0.889833	−	0.456286i	$-0.849179\pi$
$251$	−7077.00	−1.77967	−0.889833	+	0.456286i	$0.849179\pi$
$252$	0	0
$253$	−2106.00	−0.523332
$254$	2228.00	0.550382
$255$	0	0
$256$	256.000	0.0625000
$257$	−6846.00	−1.66164	−0.830821	−	0.556540i	$-0.812128\pi$
$257$	−6846.00	−1.66164	−0.830821	+	0.556540i	$0.812128\pi$
$258$	0	0
$259$	−4964.00	−1.19092
$260$	0	0
$261$	0	0
$262$	504.000	0.118844
$263$	3438.00	0.806069	0.403035	−	0.915185i	$-0.367955\pi$
$263$	3438.00	0.806069	0.403035	+	0.915185i	$0.367955\pi$
$264$	0	0
$265$	0	0
$266$	2380.00	0.548598
$267$	0	0
$268$	1844.00	0.420299
$269$	1680.00	0.380786	0.190393	−	0.981708i	$-0.439024\pi$
$269$	1680.00	0.380786	0.190393	+	0.981708i	$0.439024\pi$
$270$	0	0
$271$	5222.00	1.17053	0.585266	−	0.810842i	$-0.300990\pi$
$271$	5222.00	1.17053	0.585266	+	0.810842i	$0.300990\pi$
$272$	−336.000	−0.0749007
$273$	0	0
$274$	3882.00	0.855913
$275$	0	0
$276$	0	0
$277$	−1384.00	−0.300204	−0.150102	−	0.988671i	$-0.547960\pi$
$277$	−1384.00	−0.300204	−0.150102	+	0.988671i	$0.547960\pi$
$278$	3290.00	0.709788
$279$	0	0
$280$	0	0
$281$	3858.00	0.819036	0.409518	−	0.912302i	$-0.365697\pi$
$281$	3858.00	0.819036	0.409518	+	0.912302i	$0.365697\pi$
$282$	0	0
$283$	4277.00	0.898379	0.449190	−	0.893437i	$-0.351713\pi$
$283$	4277.00	0.898379	0.449190	+	0.893437i	$0.351713\pi$
$284$	2832.00	0.591719
$285$	0	0
$286$	−1512.00	−0.312610
$287$	12138.0	2.49646
$288$	0	0
$289$	−4472.00	−0.910238
$290$	0	0
$291$	0	0
$292$	−532.000	−0.106620
$293$	−6342.00	−1.26452	−0.632259	−	0.774757i	$-0.717872\pi$
$293$	−6342.00	−1.26452	−0.632259	+	0.774757i	$0.717872\pi$
$294$	0	0
$295$	0	0
$296$	−1168.00	−0.229353
$297$	0	0
$298$	−3600.00	−0.699807
$299$	−2184.00	−0.422421
$300$	0	0
$301$	5032.00	0.963587
$302$	6596.00	1.25681
$303$	0	0
$304$	560.000	0.105652
$305$	0	0
$306$	0	0
$307$	5831.00	1.08402	0.542008	−	0.840373i	$-0.317664\pi$
$307$	5831.00	1.08402	0.542008	+	0.840373i	$0.317664\pi$
$308$	3672.00	0.679323
$309$	0	0
$310$	0	0
$311$	2478.00	0.451815	0.225908	−	0.974149i	$-0.427465\pi$
$311$	2478.00	0.451815	0.225908	+	0.974149i	$0.427465\pi$
$312$	0	0
$313$	−2758.00	−0.498056	−0.249028	−	0.968496i	$-0.580111\pi$
$313$	−2758.00	−0.498056	−0.249028	+	0.968496i	$0.580111\pi$
$314$	−532.000	−0.0956130
$315$	0	0
$316$	2600.00	0.462853
$317$	−636.000	−0.112686	−0.0563428	−	0.998411i	$-0.517944\pi$
$317$	−636.000	−0.112686	−0.0563428	+	0.998411i	$0.517944\pi$
$318$	0	0
$319$	−3240.00	−0.568668
$320$	0	0
$321$	0	0
$322$	5304.00	0.917951
$323$	−735.000	−0.126615
$324$	0	0
$325$	0	0
$326$	−2314.00	−0.393131
$327$	0	0
$328$	2856.00	0.480781
$329$	−2856.00	−0.478591
$330$	0	0
$331$	6887.00	1.14364	0.571818	−	0.820380i	$-0.306238\pi$
$331$	6887.00	1.14364	0.571818	+	0.820380i	$0.306238\pi$
$332$	3612.00	0.597091
$333$	0	0
$334$	−3528.00	−0.577975
$335$	0	0
$336$	0	0
$337$	7331.00	1.18500	0.592500	−	0.805570i	$-0.298141\pi$
$337$	7331.00	1.18500	0.592500	+	0.805570i	$0.298141\pi$
$338$	2826.00	0.454776
$339$	0	0
$340$	0	0
$341$	−4914.00	−0.780375
$342$	0	0
$343$	−15980.0	−2.51557
$344$	1184.00	0.185573
$345$	0	0
$346$	−3696.00	−0.574272
$347$	5349.00	0.827520	0.413760	−	0.910386i	$-0.364215\pi$
$347$	5349.00	0.827520	0.413760	+	0.910386i	$0.364215\pi$
$348$	0	0
$349$	11270.0	1.72857	0.864283	−	0.503007i	$-0.167773\pi$
$349$	11270.0	1.72857	0.864283	+	0.503007i	$0.167773\pi$
$350$	0	0
$351$	0	0
$352$	864.000	0.130828
$353$	−1302.00	−0.196313	−0.0981565	−	0.995171i	$-0.531295\pi$
$353$	−1302.00	−0.196313	−0.0981565	+	0.995171i	$0.531295\pi$
$354$	0	0
$355$	0	0
$356$	−2940.00	−0.437696
$357$	0	0
$358$	270.000	0.0398602
$359$	750.000	0.110260	0.0551302	−	0.998479i	$-0.482443\pi$
$359$	750.000	0.110260	0.0551302	+	0.998479i	$0.482443\pi$
$360$	0	0
$361$	−5634.00	−0.821403
$362$	−4564.00	−0.662648
$363$	0	0
$364$	3808.00	0.548334
$365$	0	0
$366$	0	0
$367$	−1204.00	−0.171249	−0.0856244	−	0.996327i	$-0.527288\pi$
$367$	−1204.00	−0.171249	−0.0856244	+	0.996327i	$0.527288\pi$
$368$	1248.00	0.176784
$369$	0	0
$370$	0	0
$371$	23868.0	3.34007
$372$	0	0
$373$	−1198.00	−0.166301	−0.0831503	−	0.996537i	$-0.526498\pi$
$373$	−1198.00	−0.166301	−0.0831503	+	0.996537i	$0.526498\pi$
$374$	−1134.00	−0.156785
$375$	0	0
$376$	−672.000	−0.0921696
$377$	−3360.00	−0.459015
$378$	0	0
$379$	8105.00	1.09849	0.549243	−	0.835663i	$-0.314916\pi$
$379$	8105.00	1.09849	0.549243	+	0.835663i	$0.314916\pi$
$380$	0	0
$381$	0	0
$382$	−2796.00	−0.374492
$383$	3318.00	0.442668	0.221334	−	0.975198i	$-0.428959\pi$
$383$	3318.00	0.442668	0.221334	+	0.975198i	$0.428959\pi$
$384$	0	0
$385$	0	0
$386$	−6634.00	−0.874771
$387$	0	0
$388$	4424.00	0.578852
$389$	13770.0	1.79477	0.897387	−	0.441245i	$-0.145463\pi$
$389$	13770.0	1.79477	0.897387	+	0.441245i	$0.145463\pi$
$390$	0	0
$391$	−1638.00	−0.211860
$392$	−6504.00	−0.838014
$393$	0	0
$394$	3372.00	0.431165
$395$	0	0
$396$	0	0
$397$	−3724.00	−0.470786	−0.235393	−	0.971900i	$-0.575638\pi$
$397$	−3724.00	−0.470786	−0.235393	+	0.971900i	$0.575638\pi$
$398$	3080.00	0.387906
$399$	0	0
$400$	0	0
$401$	−6117.00	−0.761767	−0.380883	−	0.924623i	$-0.624380\pi$
$401$	−6117.00	−0.761767	−0.380883	+	0.924623i	$0.624380\pi$
$402$	0	0
$403$	−5096.00	−0.629900
$404$	−1848.00	−0.227578
$405$	0	0
$406$	8160.00	0.997473
$407$	−3942.00	−0.480093
$408$	0	0
$409$	6125.00	0.740493	0.370247	−	0.928933i	$-0.379273\pi$
$409$	6125.00	0.740493	0.370247	+	0.928933i	$0.379273\pi$
$410$	0	0
$411$	0	0
$412$	3248.00	0.388392
$413$	−28560.0	−3.40277
$414$	0	0
$415$	0	0
$416$	896.000	0.105601
$417$	0	0
$418$	1890.00	0.221155
$419$	−1575.00	−0.183637	−0.0918184	−	0.995776i	$-0.529268\pi$
$419$	−1575.00	−0.183637	−0.0918184	+	0.995776i	$0.529268\pi$
$420$	0	0
$421$	−988.000	−0.114376	−0.0571879	−	0.998363i	$-0.518213\pi$
$421$	−988.000	−0.114376	−0.0571879	+	0.998363i	$0.518213\pi$
$422$	6086.00	0.702042
$423$	0	0
$424$	5616.00	0.643248
$425$	0	0
$426$	0	0
$427$	8092.00	0.917094
$428$	3156.00	0.356428
$429$	0	0
$430$	0	0
$431$	558.000	0.0623617	0.0311809	−	0.999514i	$-0.490073\pi$
$431$	558.000	0.0623617	0.0311809	+	0.999514i	$0.490073\pi$
$432$	0	0
$433$	−2443.00	−0.271139	−0.135569	−	0.990768i	$-0.543286\pi$
$433$	−2443.00	−0.271139	−0.135569	+	0.990768i	$0.543286\pi$
$434$	12376.0	1.36882
$435$	0	0
$436$	920.000	0.101055
$437$	2730.00	0.298841
$438$	0	0
$439$	12320.0	1.33941	0.669706	−	0.742627i	$-0.266420\pi$
$439$	12320.0	1.33941	0.669706	+	0.742627i	$0.266420\pi$
$440$	0	0
$441$	0	0
$442$	−1176.00	−0.126553
$443$	2343.00	0.251285	0.125643	−	0.992076i	$-0.459901\pi$
$443$	2343.00	0.251285	0.125643	+	0.992076i	$0.459901\pi$
$444$	0	0
$445$	0	0
$446$	−6664.00	−0.707510
$447$	0	0
$448$	−2176.00	−0.229478
$449$	−10905.0	−1.14619	−0.573094	−	0.819489i	$-0.694257\pi$
$449$	−10905.0	−1.14619	−0.573094	+	0.819489i	$0.694257\pi$
$450$	0	0
$451$	9639.00	1.00639
$452$	8292.00	0.862882
$453$	0	0
$454$	3192.00	0.329974
$455$	0	0
$456$	0	0
$457$	−3319.00	−0.339729	−0.169865	−	0.985467i	$-0.554333\pi$
$457$	−3319.00	−0.339729	−0.169865	+	0.985467i	$0.554333\pi$
$458$	−8680.00	−0.885567
$459$	0	0
$460$	0	0
$461$	6468.00	0.653459	0.326730	−	0.945118i	$-0.394053\pi$
$461$	6468.00	0.653459	0.326730	+	0.945118i	$0.394053\pi$
$462$	0	0
$463$	11972.0	1.20170	0.600849	−	0.799363i	$-0.294829\pi$
$463$	11972.0	1.20170	0.600849	+	0.799363i	$0.294829\pi$
$464$	1920.00	0.192099
$465$	0	0
$466$	−6036.00	−0.600026
$467$	−6636.00	−0.657553	−0.328777	−	0.944408i	$-0.606636\pi$
$467$	−6636.00	−0.657553	−0.328777	+	0.944408i	$0.606636\pi$
$468$	0	0
$469$	−15674.0	−1.54319
$470$	0	0
$471$	0	0
$472$	−6720.00	−0.655324
$473$	3996.00	0.388449
$474$	0	0
$475$	0	0
$476$	2856.00	0.275010
$477$	0	0
$478$	8880.00	0.849711
$479$	630.000	0.0600949	0.0300474	−	0.999548i	$-0.490434\pi$
$479$	630.000	0.0600949	0.0300474	+	0.999548i	$0.490434\pi$
$480$	0	0
$481$	−4088.00	−0.387519
$482$	7406.00	0.699863
$483$	0	0
$484$	−2408.00	−0.226146
$485$	0	0
$486$	0	0
$487$	10646.0	0.990588	0.495294	−	0.868725i	$-0.335060\pi$
$487$	10646.0	0.990588	0.495294	+	0.868725i	$0.335060\pi$
$488$	1904.00	0.176619
$489$	0	0
$490$	0	0
$491$	2388.00	0.219489	0.109744	−	0.993960i	$-0.464997\pi$
$491$	2388.00	0.219489	0.109744	+	0.993960i	$0.464997\pi$
$492$	0	0
$493$	−2520.00	−0.230213
$494$	1960.00	0.178511
$495$	0	0
$496$	2912.00	0.263614
$497$	−24072.0	−2.17259
$498$	0	0
$499$	−10540.0	−0.945562	−0.472781	−	0.881180i	$-0.656750\pi$
$499$	−10540.0	−0.945562	−0.472781	+	0.881180i	$0.656750\pi$
$500$	0	0
$501$	0	0
$502$	14154.0	1.25841
$503$	−6972.00	−0.618024	−0.309012	−	0.951058i	$-0.599998\pi$
$503$	−6972.00	−0.618024	−0.309012	+	0.951058i	$0.599998\pi$
$504$	0	0
$505$	0	0
$506$	4212.00	0.370052
$507$	0	0
$508$	−4456.00	−0.389179
$509$	9030.00	0.786341	0.393171	−	0.919466i	$-0.371378\pi$
$509$	9030.00	0.786341	0.393171	+	0.919466i	$0.371378\pi$
$510$	0	0
$511$	4522.00	0.391471
$512$	−512.000	−0.0441942
$513$	0	0
$514$	13692.0	1.17496
$515$	0	0
$516$	0	0
$517$	−2268.00	−0.192933
$518$	9928.00	0.842107
$519$	0	0
$520$	0	0
$521$	−10437.0	−0.877645	−0.438823	−	0.898574i	$-0.644604\pi$
$521$	−10437.0	−0.877645	−0.438823	+	0.898574i	$0.644604\pi$
$522$	0	0
$523$	−8113.00	−0.678311	−0.339156	−	0.940730i	$-0.610141\pi$
$523$	−8113.00	−0.678311	−0.339156	+	0.940730i	$0.610141\pi$
$524$	−1008.00	−0.0840357
$525$	0	0
$526$	−6876.00	−0.569977
$527$	−3822.00	−0.315918
$528$	0	0
$529$	−6083.00	−0.499959
$530$	0	0
$531$	0	0
$532$	−4760.00	−0.387918
$533$	9996.00	0.812336
$534$	0	0
$535$	0	0
$536$	−3688.00	−0.297197
$537$	0	0
$538$	−3360.00	−0.269256
$539$	−21951.0	−1.75417
$540$	0	0
$541$	−14848.0	−1.17997	−0.589986	−	0.807413i	$-0.700867\pi$
$541$	−14848.0	−1.17997	−0.589986	+	0.807413i	$0.700867\pi$
$542$	−10444.0	−0.827690
$543$	0	0
$544$	672.000	0.0529628
$545$	0	0
$546$	0	0
$547$	−20329.0	−1.58904	−0.794520	−	0.607238i	$-0.792278\pi$
$547$	−20329.0	−1.58904	−0.794520	+	0.607238i	$0.792278\pi$
$548$	−7764.00	−0.605222
$549$	0	0
$550$	0	0
$551$	4200.00	0.324730
$552$	0	0
$553$	−22100.0	−1.69944
$554$	2768.00	0.212276
$555$	0	0
$556$	−6580.00	−0.501896
$557$	15324.0	1.16571	0.582853	−	0.812577i	$-0.301936\pi$
$557$	15324.0	1.16571	0.582853	+	0.812577i	$0.301936\pi$
$558$	0	0
$559$	4144.00	0.313547
$560$	0	0
$561$	0	0
$562$	−7716.00	−0.579146
$563$	9408.00	0.704263	0.352131	−	0.935951i	$-0.385457\pi$
$563$	9408.00	0.704263	0.352131	+	0.935951i	$0.385457\pi$
$564$	0	0
$565$	0	0
$566$	−8554.00	−0.635250
$567$	0	0
$568$	−5664.00	−0.418409
$569$	24375.0	1.79588	0.897938	−	0.440122i	$-0.145065\pi$
$569$	24375.0	1.79588	0.897938	+	0.440122i	$0.145065\pi$
$570$	0	0
$571$	−988.000	−0.0724107	−0.0362054	−	0.999344i	$-0.511527\pi$
$571$	−988.000	−0.0724107	−0.0362054	+	0.999344i	$0.511527\pi$
$572$	3024.00	0.221049
$573$	0	0
$574$	−24276.0	−1.76526
$575$	0	0
$576$	0	0
$577$	−16429.0	−1.18535	−0.592676	−	0.805441i	$-0.701929\pi$
$577$	−16429.0	−1.18535	−0.592676	+	0.805441i	$0.701929\pi$
$578$	8944.00	0.643636
$579$	0	0
$580$	0	0
$581$	−30702.0	−2.19231
$582$	0	0
$583$	18954.0	1.34647
$584$	1064.00	0.0753915
$585$	0	0
$586$	12684.0	0.894149
$587$	13839.0	0.973078	0.486539	−	0.873659i	$-0.338259\pi$
$587$	13839.0	0.973078	0.486539	+	0.873659i	$0.338259\pi$
$588$	0	0
$589$	6370.00	0.445622
$590$	0	0
$591$	0	0
$592$	2336.00	0.162177
$593$	−14007.0	−0.969981	−0.484990	−	0.874520i	$-0.661177\pi$
$593$	−14007.0	−0.969981	−0.484990	+	0.874520i	$0.661177\pi$
$594$	0	0
$595$	0	0
$596$	7200.00	0.494838
$597$	0	0
$598$	4368.00	0.298697
$599$	−21090.0	−1.43859	−0.719294	−	0.694706i	$-0.755535\pi$
$599$	−21090.0	−1.43859	−0.719294	+	0.694706i	$0.755535\pi$
$600$	0	0
$601$	5747.00	0.390058	0.195029	−	0.980797i	$-0.437520\pi$
$601$	5747.00	0.390058	0.195029	+	0.980797i	$0.437520\pi$
$602$	−10064.0	−0.681359
$603$	0	0
$604$	−13192.0	−0.888700
$605$	0	0
$606$	0	0
$607$	17696.0	1.18329	0.591646	−	0.806198i	$-0.298478\pi$
$607$	17696.0	1.18329	0.591646	+	0.806198i	$0.298478\pi$
$608$	−1120.00	−0.0747072
$609$	0	0
$610$	0	0
$611$	−2352.00	−0.155731
$612$	0	0
$613$	26102.0	1.71982	0.859910	−	0.510445i	$-0.170519\pi$
$613$	26102.0	1.71982	0.859910	+	0.510445i	$0.170519\pi$
$614$	−11662.0	−0.766515
$615$	0	0
$616$	−7344.00	−0.480354
$617$	4194.00	0.273653	0.136827	−	0.990595i	$-0.456310\pi$
$617$	4194.00	0.273653	0.136827	+	0.990595i	$0.456310\pi$
$618$	0	0
$619$	−7420.00	−0.481801	−0.240901	−	0.970550i	$-0.577443\pi$
$619$	−7420.00	−0.481801	−0.240901	+	0.970550i	$0.577443\pi$
$620$	0	0
$621$	0	0
$622$	−4956.00	−0.319482
$623$	24990.0	1.60707
$624$	0	0
$625$	0	0
$626$	5516.00	0.352178
$627$	0	0
$628$	1064.00	0.0676086
$629$	−3066.00	−0.194355
$630$	0	0
$631$	−5818.00	−0.367054	−0.183527	−	0.983015i	$-0.558751\pi$
$631$	−5818.00	−0.367054	−0.183527	+	0.983015i	$0.558751\pi$
$632$	−5200.00	−0.327286
$633$	0	0
$634$	1272.00	0.0796807
$635$	0	0
$636$	0	0
$637$	−22764.0	−1.41592
$638$	6480.00	0.402109
$639$	0	0
$640$	0	0
$641$	29478.0	1.81640	0.908199	−	0.418539i	$-0.137458\pi$
$641$	29478.0	1.81640	0.908199	+	0.418539i	$0.137458\pi$
$642$	0	0
$643$	5852.00	0.358912	0.179456	−	0.983766i	$-0.442566\pi$
$643$	5852.00	0.358912	0.179456	+	0.983766i	$0.442566\pi$
$644$	−10608.0	−0.649090
$645$	0	0
$646$	1470.00	0.0895300
$647$	29484.0	1.79155	0.895777	−	0.444503i	$-0.146620\pi$
$647$	29484.0	1.79155	0.895777	+	0.444503i	$0.146620\pi$
$648$	0	0
$649$	−22680.0	−1.37175
$650$	0	0
$651$	0	0
$652$	4628.00	0.277985
$653$	3498.00	0.209628	0.104814	−	0.994492i	$-0.466575\pi$
$653$	3498.00	0.209628	0.104814	+	0.994492i	$0.466575\pi$
$654$	0	0
$655$	0	0
$656$	−5712.00	−0.339964
$657$	0	0
$658$	5712.00	0.338415
$659$	−7905.00	−0.467276	−0.233638	−	0.972324i	$-0.575063\pi$
$659$	−7905.00	−0.467276	−0.233638	+	0.972324i	$0.575063\pi$
$660$	0	0
$661$	27272.0	1.60478	0.802389	−	0.596802i	$-0.203562\pi$
$661$	27272.0	1.60478	0.802389	+	0.596802i	$0.203562\pi$
$662$	−13774.0	−0.808673
$663$	0	0
$664$	−7224.00	−0.422207
$665$	0	0
$666$	0	0
$667$	9360.00	0.543359
$668$	7056.00	0.408690
$669$	0	0
$670$	0	0
$671$	6426.00	0.369706
$672$	0	0
$673$	12602.0	0.721800	0.360900	−	0.932605i	$-0.382470\pi$
$673$	12602.0	0.721800	0.360900	+	0.932605i	$0.382470\pi$
$674$	−14662.0	−0.837922
$675$	0	0
$676$	−5652.00	−0.321575
$677$	−25536.0	−1.44967	−0.724836	−	0.688921i	$-0.758085\pi$
$677$	−25536.0	−1.44967	−0.724836	+	0.688921i	$0.758085\pi$
$678$	0	0
$679$	−37604.0	−2.12534
$680$	0	0
$681$	0	0
$682$	9828.00	0.551809
$683$	−2127.00	−0.119162	−0.0595808	−	0.998223i	$-0.518976\pi$
$683$	−2127.00	−0.119162	−0.0595808	+	0.998223i	$0.518976\pi$
$684$	0	0
$685$	0	0
$686$	31960.0	1.77877
$687$	0	0
$688$	−2368.00	−0.131220
$689$	19656.0	1.08684
$690$	0	0
$691$	−8953.00	−0.492892	−0.246446	−	0.969157i	$-0.579263\pi$
$691$	−8953.00	−0.492892	−0.246446	+	0.969157i	$0.579263\pi$
$692$	7392.00	0.406072
$693$	0	0
$694$	−10698.0	−0.585145
$695$	0	0
$696$	0	0
$697$	7497.00	0.407416
$698$	−22540.0	−1.22228
$699$	0	0
$700$	0	0
$701$	−23172.0	−1.24849	−0.624247	−	0.781227i	$-0.714594\pi$
$701$	−23172.0	−1.24849	−0.624247	+	0.781227i	$0.714594\pi$
$702$	0	0
$703$	5110.00	0.274150
$704$	−1728.00	−0.0925092
$705$	0	0
$706$	2604.00	0.138814
$707$	15708.0	0.835587
$708$	0	0
$709$	9620.00	0.509572	0.254786	−	0.966997i	$-0.417995\pi$
$709$	9620.00	0.509572	0.254786	+	0.966997i	$0.417995\pi$
$710$	0	0
$711$	0	0
$712$	5880.00	0.309498
$713$	14196.0	0.745644
$714$	0	0
$715$	0	0
$716$	−540.000	−0.0281854
$717$	0	0
$718$	−1500.00	−0.0779659
$719$	−4830.00	−0.250527	−0.125263	−	0.992124i	$-0.539978\pi$
$719$	−4830.00	−0.250527	−0.125263	+	0.992124i	$0.539978\pi$
$720$	0	0
$721$	−27608.0	−1.42604
$722$	11268.0	0.580819
$723$	0	0
$724$	9128.00	0.468563
$725$	0	0
$726$	0	0
$727$	11816.0	0.602794	0.301397	−	0.953499i	$-0.402547\pi$
$727$	11816.0	0.602794	0.301397	+	0.953499i	$0.402547\pi$
$728$	−7616.00	−0.387730
$729$	0	0
$730$	0	0
$731$	3108.00	0.157255
$732$	0	0
$733$	23492.0	1.18376	0.591881	−	0.806026i	$-0.298386\pi$
$733$	23492.0	1.18376	0.591881	+	0.806026i	$0.298386\pi$
$734$	2408.00	0.121091
$735$	0	0
$736$	−2496.00	−0.125005
$737$	−12447.0	−0.622105
$738$	0	0
$739$	35300.0	1.75715	0.878573	−	0.477607i	$-0.158496\pi$
$739$	35300.0	1.75715	0.878573	+	0.477607i	$0.158496\pi$
$740$	0	0
$741$	0	0
$742$	−47736.0	−2.36178
$743$	−16242.0	−0.801967	−0.400983	−	0.916085i	$-0.631331\pi$
$743$	−16242.0	−0.801967	−0.400983	+	0.916085i	$0.631331\pi$
$744$	0	0
$745$	0	0
$746$	2396.00	0.117592
$747$	0	0
$748$	2268.00	0.110864
$749$	−26826.0	−1.30868
$750$	0	0
$751$	10712.0	0.520488	0.260244	−	0.965543i	$-0.416197\pi$
$751$	10712.0	0.520488	0.260244	+	0.965543i	$0.416197\pi$
$752$	1344.00	0.0651737
$753$	0	0
$754$	6720.00	0.324573
$755$	0	0
$756$	0	0
$757$	−13504.0	−0.648364	−0.324182	−	0.945995i	$-0.605089\pi$
$757$	−13504.0	−0.648364	−0.324182	+	0.945995i	$0.605089\pi$
$758$	−16210.0	−0.776746
$759$	0	0
$760$	0	0
$761$	18123.0	0.863283	0.431641	−	0.902045i	$-0.357935\pi$
$761$	18123.0	0.863283	0.431641	+	0.902045i	$0.357935\pi$
$762$	0	0
$763$	−7820.00	−0.371039
$764$	5592.00	0.264806
$765$	0	0
$766$	−6636.00	−0.313014
$767$	−23520.0	−1.10725
$768$	0	0
$769$	9485.00	0.444783	0.222391	−	0.974957i	$-0.428614\pi$
$769$	9485.00	0.444783	0.222391	+	0.974957i	$0.428614\pi$
$770$	0	0
$771$	0	0
$772$	13268.0	0.618557
$773$	31248.0	1.45396	0.726981	−	0.686658i	$-0.240923\pi$
$773$	31248.0	1.45396	0.726981	+	0.686658i	$0.240923\pi$
$774$	0	0
$775$	0	0
$776$	−8848.00	−0.409310
$777$	0	0
$778$	−27540.0	−1.26910
$779$	−12495.0	−0.574685
$780$	0	0
$781$	−19116.0	−0.875831
$782$	3276.00	0.149808
$783$	0	0
$784$	13008.0	0.592566
$785$	0	0
$786$	0	0
$787$	−15484.0	−0.701328	−0.350664	−	0.936501i	$-0.614044\pi$
$787$	−15484.0	−0.701328	−0.350664	+	0.936501i	$0.614044\pi$
$788$	−6744.00	−0.304880
$789$	0	0
$790$	0	0
$791$	−70482.0	−3.16821
$792$	0	0
$793$	6664.00	0.298418
$794$	7448.00	0.332896
$795$	0	0
$796$	−6160.00	−0.274291
$797$	−34146.0	−1.51758	−0.758791	−	0.651334i	$-0.774210\pi$
$797$	−34146.0	−1.51758	−0.758791	+	0.651334i	$0.774210\pi$
$798$	0	0
$799$	−1764.00	−0.0781049
$800$	0	0
$801$	0	0
$802$	12234.0	0.538650
$803$	3591.00	0.157813
$804$	0	0
$805$	0	0
$806$	10192.0	0.445407
$807$	0	0
$808$	3696.00	0.160922
$809$	36030.0	1.56582	0.782909	−	0.622136i	$-0.213735\pi$
$809$	36030.0	1.56582	0.782909	+	0.622136i	$0.213735\pi$
$810$	0	0
$811$	−6748.00	−0.292175	−0.146088	−	0.989272i	$-0.546668\pi$
$811$	−6748.00	−0.292175	−0.146088	+	0.989272i	$0.546668\pi$
$812$	−16320.0	−0.705320
$813$	0	0
$814$	7884.00	0.339477
$815$	0	0
$816$	0	0
$817$	−5180.00	−0.221818
$818$	−12250.0	−0.523608
$819$	0	0
$820$	0	0
$821$	5598.00	0.237968	0.118984	−	0.992896i	$-0.462036\pi$
$821$	5598.00	0.237968	0.118984	+	0.992896i	$0.462036\pi$
$822$	0	0
$823$	5732.00	0.242776	0.121388	−	0.992605i	$-0.461265\pi$
$823$	5732.00	0.242776	0.121388	+	0.992605i	$0.461265\pi$
$824$	−6496.00	−0.274635
$825$	0	0
$826$	57120.0	2.40612
$827$	39999.0	1.68186	0.840932	−	0.541141i	$-0.182007\pi$
$827$	39999.0	1.68186	0.840932	+	0.541141i	$0.182007\pi$
$828$	0	0
$829$	16940.0	0.709711	0.354856	−	0.934921i	$-0.384530\pi$
$829$	16940.0	0.709711	0.354856	+	0.934921i	$0.384530\pi$
$830$	0	0
$831$	0	0
$832$	−1792.00	−0.0746712
$833$	−17073.0	−0.710137
$834$	0	0
$835$	0	0
$836$	−3780.00	−0.156380
$837$	0	0
$838$	3150.00	0.129851
$839$	45360.0	1.86651	0.933255	−	0.359216i	$-0.116956\pi$
$839$	45360.0	1.86651	0.933255	+	0.359216i	$0.116956\pi$
$840$	0	0
$841$	−9989.00	−0.409570
$842$	1976.00	0.0808758
$843$	0	0
$844$	−12172.0	−0.496419
$845$	0	0
$846$	0	0
$847$	20468.0	0.830329
$848$	−11232.0	−0.454845
$849$	0	0
$850$	0	0
$851$	11388.0	0.458726
$852$	0	0
$853$	−43918.0	−1.76286	−0.881432	−	0.472310i	$-0.843420\pi$
$853$	−43918.0	−1.76286	−0.881432	+	0.472310i	$0.843420\pi$
$854$	−16184.0	−0.648484
$855$	0	0
$856$	−6312.00	−0.252032
$857$	3339.00	0.133090	0.0665450	−	0.997783i	$-0.478802\pi$
$857$	3339.00	0.133090	0.0665450	+	0.997783i	$0.478802\pi$
$858$	0	0
$859$	−36925.0	−1.46666	−0.733332	−	0.679870i	$-0.762036\pi$
$859$	−36925.0	−1.46666	−0.733332	+	0.679870i	$0.762036\pi$
$860$	0	0
$861$	0	0
$862$	−1116.00	−0.0440964
$863$	40608.0	1.60175	0.800876	−	0.598830i	$-0.204367\pi$
$863$	40608.0	1.60175	0.800876	+	0.598830i	$0.204367\pi$
$864$	0	0
$865$	0	0
$866$	4886.00	0.191724
$867$	0	0
$868$	−24752.0	−0.967900
$869$	−17550.0	−0.685090
$870$	0	0
$871$	−12908.0	−0.502148
$872$	−1840.00	−0.0714567
$873$	0	0
$874$	−5460.00	−0.211313
$875$	0	0
$876$	0	0
$877$	35156.0	1.35363	0.676815	−	0.736153i	$-0.263360\pi$
$877$	35156.0	1.35363	0.676815	+	0.736153i	$0.263360\pi$
$878$	−24640.0	−0.947107
$879$	0	0
$880$	0	0
$881$	−2142.00	−0.0819135	−0.0409568	−	0.999161i	$-0.513041\pi$
$881$	−2142.00	−0.0819135	−0.0409568	+	0.999161i	$0.513041\pi$
$882$	0	0
$883$	−19153.0	−0.729954	−0.364977	−	0.931016i	$-0.618923\pi$
$883$	−19153.0	−0.729954	−0.364977	+	0.931016i	$0.618923\pi$
$884$	2352.00	0.0894868
$885$	0	0
$886$	−4686.00	−0.177685
$887$	24444.0	0.925309	0.462655	−	0.886539i	$-0.346897\pi$
$887$	24444.0	0.925309	0.462655	+	0.886539i	$0.346897\pi$
$888$	0	0
$889$	37876.0	1.42893
$890$	0	0
$891$	0	0
$892$	13328.0	0.500285
$893$	2940.00	0.110172
$894$	0	0
$895$	0	0
$896$	4352.00	0.162266
$897$	0	0
$898$	21810.0	0.810478
$899$	21840.0	0.810239
$900$	0	0
$901$	14742.0	0.545091
$902$	−19278.0	−0.711627
$903$	0	0
$904$	−16584.0	−0.610150
$905$	0	0
$906$	0	0
$907$	20036.0	0.733500	0.366750	−	0.930320i	$-0.380470\pi$
$907$	20036.0	0.733500	0.366750	+	0.930320i	$0.380470\pi$
$908$	−6384.00	−0.233327
$909$	0	0
$910$	0	0
$911$	33468.0	1.21717	0.608586	−	0.793488i	$-0.291737\pi$
$911$	33468.0	1.21717	0.608586	+	0.793488i	$0.291737\pi$
$912$	0	0
$913$	−24381.0	−0.883782
$914$	6638.00	0.240225
$915$	0	0
$916$	17360.0	0.626191
$917$	8568.00	0.308550
$918$	0	0
$919$	35090.0	1.25953	0.629767	−	0.776784i	$-0.283150\pi$
$919$	35090.0	1.25953	0.629767	+	0.776784i	$0.283150\pi$
$920$	0	0
$921$	0	0
$922$	−12936.0	−0.462066
$923$	−19824.0	−0.706950
$924$	0	0
$925$	0	0
$926$	−23944.0	−0.849729
$927$	0	0
$928$	−3840.00	−0.135834
$929$	32130.0	1.13472	0.567358	−	0.823471i	$-0.307966\pi$
$929$	32130.0	1.13472	0.567358	+	0.823471i	$0.307966\pi$
$930$	0	0
$931$	28455.0	1.00169
$932$	12072.0	0.424283
$933$	0	0
$934$	13272.0	0.464960
$935$	0	0
$936$	0	0
$937$	20531.0	0.715815	0.357907	−	0.933757i	$-0.383490\pi$
$937$	20531.0	0.715815	0.357907	+	0.933757i	$0.383490\pi$
$938$	31348.0	1.09120
$939$	0	0
$940$	0	0
$941$	−7812.00	−0.270631	−0.135316	−	0.990803i	$-0.543205\pi$
$941$	−7812.00	−0.270631	−0.135316	+	0.990803i	$0.543205\pi$
$942$	0	0
$943$	−27846.0	−0.961602
$944$	13440.0	0.463384
$945$	0	0
$946$	−7992.00	−0.274675
$947$	−23256.0	−0.798013	−0.399007	−	0.916948i	$-0.630645\pi$
$947$	−23256.0	−0.798013	−0.399007	+	0.916948i	$0.630645\pi$
$948$	0	0
$949$	3724.00	0.127383
$950$	0	0
$951$	0	0
$952$	−5712.00	−0.194461
$953$	−14097.0	−0.479167	−0.239584	−	0.970876i	$-0.577011\pi$
$953$	−14097.0	−0.479167	−0.239584	+	0.970876i	$0.577011\pi$
$954$	0	0
$955$	0	0
$956$	−17760.0	−0.600836
$957$	0	0
$958$	−1260.00	−0.0424935
$959$	65994.0	2.22217
$960$	0	0
$961$	3333.00	0.111879
$962$	8176.00	0.274017
$963$	0	0
$964$	−14812.0	−0.494878
$965$	0	0
$966$	0	0
$967$	−34144.0	−1.13547	−0.567734	−	0.823212i	$-0.692180\pi$
$967$	−34144.0	−1.13547	−0.567734	+	0.823212i	$0.692180\pi$
$968$	4816.00	0.159909
$969$	0	0
$970$	0	0
$971$	32613.0	1.07786	0.538929	−	0.842351i	$-0.318829\pi$
$971$	32613.0	1.07786	0.538929	+	0.842351i	$0.318829\pi$
$972$	0	0
$973$	55930.0	1.84279
$974$	−21292.0	−0.700451
$975$	0	0
$976$	−3808.00	−0.124888
$977$	43359.0	1.41983	0.709917	−	0.704286i	$-0.248733\pi$
$977$	43359.0	1.41983	0.709917	+	0.704286i	$0.248733\pi$
$978$	0	0
$979$	19845.0	0.647854
$980$	0	0
$981$	0	0
$982$	−4776.00	−0.155202
$983$	28518.0	0.925313	0.462657	−	0.886538i	$-0.346896\pi$
$983$	28518.0	0.925313	0.462657	+	0.886538i	$0.346896\pi$
$984$	0	0
$985$	0	0
$986$	5040.00	0.162785
$987$	0	0
$988$	−3920.00	−0.126227
$989$	−11544.0	−0.371161
$990$	0	0
$991$	18122.0	0.580892	0.290446	−	0.956891i	$-0.406196\pi$
$991$	18122.0	0.580892	0.290446	+	0.956891i	$0.406196\pi$
$992$	−5824.00	−0.186403
$993$	0	0
$994$	48144.0	1.53625
$995$	0	0
$996$	0	0
$997$	−28924.0	−0.918789	−0.459394	−	0.888232i	$-0.651934\pi$
$997$	−28924.0	−0.918789	−0.459394	+	0.888232i	$0.651934\pi$
$998$	21080.0	0.668613
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.4.a.a.1.1		1
3.2	odd	2		50.4.a.e.1.1	yes	1
5.2	odd	4		450.4.c.c.199.1		2
5.3	odd	4		450.4.c.c.199.2		2
5.4	even	2		450.4.a.t.1.1		1
12.11	even	2		400.4.a.d.1.1		1
15.2	even	4		50.4.b.b.49.2		2
15.8	even	4		50.4.b.b.49.1		2
15.14	odd	2		50.4.a.a.1.1	✓	1
21.20	even	2		2450.4.a.y.1.1		1
24.5	odd	2		1600.4.a.f.1.1		1
24.11	even	2		1600.4.a.bv.1.1		1
60.23	odd	4		400.4.c.d.49.1		2
60.47	odd	4		400.4.c.d.49.2		2
60.59	even	2		400.4.a.r.1.1		1
105.104	even	2		2450.4.a.t.1.1		1
120.29	odd	2		1600.4.a.bu.1.1		1
120.59	even	2		1600.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.4.a.a.1.1	✓	1	15.14	odd	2
50.4.a.e.1.1	yes	1	3.2	odd	2
50.4.b.b.49.1		2	15.8	even	4
50.4.b.b.49.2		2	15.2	even	4
400.4.a.d.1.1		1	12.11	even	2
400.4.a.r.1.1		1	60.59	even	2
400.4.c.d.49.1		2	60.23	odd	4
400.4.c.d.49.2		2	60.47	odd	4
450.4.a.a.1.1		1	1.1	even	1	trivial
450.4.a.t.1.1		1	5.4	even	2
450.4.c.c.199.1		2	5.2	odd	4
450.4.c.c.199.2		2	5.3	odd	4
1600.4.a.f.1.1		1	24.5	odd	2
1600.4.a.g.1.1		1	120.59	even	2
1600.4.a.bu.1.1		1	120.29	odd	2
1600.4.a.bv.1.1		1	24.11	even	2
2450.4.a.t.1.1		1	105.104	even	2
2450.4.a.y.1.1		1	21.20	even	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

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