LMFDB - Embedded newform 2450.4.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,4,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, 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("2450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2450.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:	$144.554679514$
Analytic rank:	$1$
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	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -7.00000 q^{3} +4.00000 q^{4} -14.0000 q^{6} +8.00000 q^{8} +22.0000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -7.00000 q^{3} +4.00000 q^{4} -14.0000 q^{6} +8.00000 q^{8} +22.0000 q^{9} +27.0000 q^{11} -28.0000 q^{12} +28.0000 q^{13} +16.0000 q^{16} -21.0000 q^{17} +44.0000 q^{18} -35.0000 q^{19} +54.0000 q^{22} -78.0000 q^{23} -56.0000 q^{24} +56.0000 q^{26} +35.0000 q^{27} -120.000 q^{29} -182.000 q^{31} +32.0000 q^{32} -189.000 q^{33} -42.0000 q^{34} +88.0000 q^{36} +146.000 q^{37} -70.0000 q^{38} -196.000 q^{39} -357.000 q^{41} -148.000 q^{43} +108.000 q^{44} -156.000 q^{46} +84.0000 q^{47} -112.000 q^{48} +147.000 q^{51} +112.000 q^{52} +702.000 q^{53} +70.0000 q^{54} +245.000 q^{57} -240.000 q^{58} +840.000 q^{59} +238.000 q^{61} -364.000 q^{62} +64.0000 q^{64} -378.000 q^{66} +461.000 q^{67} -84.0000 q^{68} +546.000 q^{69} -708.000 q^{71} +176.000 q^{72} +133.000 q^{73} +292.000 q^{74} -140.000 q^{76} -392.000 q^{78} +650.000 q^{79} -839.000 q^{81} -714.000 q^{82} +903.000 q^{83} -296.000 q^{86} +840.000 q^{87} +216.000 q^{88} -735.000 q^{89} -312.000 q^{92} +1274.00 q^{93} +168.000 q^{94} -224.000 q^{96} -1106.00 q^{97} +594.000 q^{99} +O(q^{100})\)

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
$3$	−7.00000	−1.34715	−0.673575	−	0.739119i	$-0.735242\pi$
$3$	−7.00000	−1.34715	−0.673575	+	0.739119i	$0.735242\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−14.0000	−0.952579
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	22.0000	0.814815
$10$	0	0
$11$	27.0000	0.740073	0.370037	−	0.929017i	$-0.379345\pi$
$11$	27.0000	0.740073	0.370037	+	0.929017i	$0.379345\pi$
$12$	−28.0000	−0.673575
$13$	28.0000	0.597369	0.298685	−	0.954352i	$-0.403452\pi$
$13$	28.0000	0.597369	0.298685	+	0.954352i	$0.403452\pi$
$14$	0	0
$15$	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	$0.547863\pi$
$18$	44.0000	0.576161
$19$	−35.0000	−0.422608	−0.211304	−	0.977420i	$-0.567771\pi$
$19$	−35.0000	−0.422608	−0.211304	+	0.977420i	$0.567771\pi$
$20$	0	0
$21$	0	0
$22$	54.0000	0.523311
$23$	−78.0000	−0.707136	−0.353568	−	0.935409i	$-0.615032\pi$
$23$	−78.0000	−0.707136	−0.353568	+	0.935409i	$0.615032\pi$
$24$	−56.0000	−0.476290
$25$	0	0
$26$	56.0000	0.422404
$27$	35.0000	0.249472
$28$	0	0
$29$	−120.000	−0.768395	−0.384197	−	0.923251i	$-0.625522\pi$
$29$	−120.000	−0.768395	−0.384197	+	0.923251i	$0.625522\pi$
$30$	0	0
$31$	−182.000	−1.05446	−0.527228	−	0.849724i	$-0.676769\pi$
$31$	−182.000	−1.05446	−0.527228	+	0.849724i	$0.676769\pi$
$32$	32.0000	0.176777
$33$	−189.000	−0.996990
$34$	−42.0000	−0.211851
$35$	0	0
$36$	88.0000	0.407407
$37$	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
$39$	−196.000	−0.804747
$40$	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	$0.458391\pi$
$48$	−112.000	−0.336788
$49$	0	0
$50$	0	0
$51$	147.000	0.403610
$52$	112.000	0.298685
$53$	702.000	1.81938	0.909690	−	0.415288i	$-0.136319\pi$
$53$	702.000	1.81938	0.909690	+	0.415288i	$0.136319\pi$
$54$	70.0000	0.176404
$55$	0	0
$56$	0	0
$57$	245.000	0.569317
$58$	−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	$-0.419643\pi$
$61$	238.000	0.499554	0.249777	+	0.968303i	$0.419643\pi$
$62$	−364.000	−0.745614
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	−378.000	−0.704979
$67$	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
$69$	546.000	0.952618
$70$	0	0
$71$	−708.000	−1.18344	−0.591719	−	0.806144i	$-0.701551\pi$
$71$	−708.000	−1.18344	−0.591719	+	0.806144i	$0.701551\pi$
$72$	176.000	0.288081
$73$	133.000	0.213239	0.106620	−	0.994300i	$-0.465997\pi$
$73$	133.000	0.213239	0.106620	+	0.994300i	$0.465997\pi$
$74$	292.000	0.458707
$75$	0	0
$76$	−140.000	−0.211304
$77$	0	0
$78$	−392.000	−0.569042
$79$	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
$81$	−839.000	−1.15089
$82$	−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
$87$	840.000	1.03514
$88$	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
$91$	0	0
$92$	−312.000	−0.353568
$93$	1274.00	1.42051
$94$	168.000	0.184339
$95$	0	0
$96$	−224.000	−0.238145
$97$	−1106.00	−1.15770	−0.578852	−	0.815433i	$-0.696499\pi$
$97$	−1106.00	−1.15770	−0.578852	+	0.815433i	$0.696499\pi$
$98$	0	0
$99$	594.000	0.603023
$100$	0	0
$101$	−462.000	−0.455156	−0.227578	−	0.973760i	$-0.573081\pi$
$101$	−462.000	−0.455156	−0.227578	+	0.973760i	$0.573081\pi$
$102$	294.000	0.285395
$103$	−812.000	−0.776784	−0.388392	−	0.921494i	$-0.626969\pi$
$103$	−812.000	−0.776784	−0.388392	+	0.921494i	$0.626969\pi$
$104$	224.000	0.211202
$105$	0	0
$106$	1404.00	1.28650
$107$	−789.000	−0.712855	−0.356428	−	0.934323i	$-0.616005\pi$
$107$	−789.000	−0.712855	−0.356428	+	0.934323i	$0.616005\pi$
$108$	140.000	0.124736
$109$	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
$111$	−1022.00	−0.873909
$112$	0	0
$113$	−2073.00	−1.72576	−0.862882	−	0.505405i	$-0.831343\pi$
$113$	−2073.00	−1.72576	−0.862882	+	0.505405i	$0.831343\pi$
$114$	490.000	0.402568
$115$	0	0
$116$	−480.000	−0.384197
$117$	616.000	0.486745
$118$	1680.00	1.31065
$119$	0	0
$120$	0	0
$121$	−602.000	−0.452292
$122$	476.000	0.353238
$123$	2499.00	1.83193
$124$	−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
$129$	1036.00	0.707091
$130$	0	0
$131$	−252.000	−0.168071	−0.0840357	−	0.996463i	$-0.526781\pi$
$131$	−252.000	−0.168071	−0.0840357	+	0.996463i	$0.526781\pi$
$132$	−756.000	−0.498495
$133$	0	0
$134$	922.000	0.594393
$135$	0	0
$136$	−168.000	−0.105926
$137$	1941.00	1.21044	0.605222	−	0.796057i	$-0.293084\pi$
$137$	1941.00	1.21044	0.605222	+	0.796057i	$0.293084\pi$
$138$	1092.00	0.673603
$139$	1645.00	1.00379	0.501896	−	0.864928i	$-0.332636\pi$
$139$	1645.00	1.00379	0.501896	+	0.864928i	$0.332636\pi$
$140$	0	0
$141$	−588.000	−0.351195
$142$	−1416.00	−0.836817
$143$	756.000	0.442097
$144$	352.000	0.203704
$145$	0	0
$146$	266.000	0.150783
$147$	0	0
$148$	584.000	0.324355
$149$	−1800.00	−0.989676	−0.494838	−	0.868985i	$-0.664773\pi$
$149$	−1800.00	−0.989676	−0.494838	+	0.868985i	$0.664773\pi$
$150$	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
$153$	−462.000	−0.244121
$154$	0	0
$155$	0	0
$156$	−784.000	−0.402373
$157$	−266.000	−0.135217	−0.0676086	−	0.997712i	$-0.521537\pi$
$157$	−266.000	−0.135217	−0.0676086	+	0.997712i	$0.521537\pi$
$158$	1300.00	0.654572
$159$	−4914.00	−2.45098
$160$	0	0
$161$	0	0
$162$	−1678.00	−0.813803
$163$	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
$171$	−770.000	−0.344347
$172$	−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	$0.366898\pi$
$174$	1680.00	0.731957
$175$	0	0
$176$	432.000	0.185018
$177$	−5880.00	−2.49699
$178$	−1470.00	−0.618995
$179$	135.000	0.0563708	0.0281854	−	0.999603i	$-0.491027\pi$
$179$	135.000	0.0563708	0.0281854	+	0.999603i	$0.491027\pi$
$180$	0	0
$181$	−2282.00	−0.937126	−0.468563	−	0.883430i	$-0.655228\pi$
$181$	−2282.00	−0.937126	−0.468563	+	0.883430i	$0.655228\pi$
$182$	0	0
$183$	−1666.00	−0.672974
$184$	−624.000	−0.250010
$185$	0	0
$186$	2548.00	1.00445
$187$	−567.000	−0.221728
$188$	336.000	0.130347
$189$	0	0
$190$	0	0
$191$	−1398.00	−0.529611	−0.264806	−	0.964302i	$-0.585308\pi$
$191$	−1398.00	−0.529611	−0.264806	+	0.964302i	$0.585308\pi$
$192$	−448.000	−0.168394
$193$	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
$196$	0	0
$197$	1686.00	0.609759	0.304880	−	0.952391i	$-0.401384\pi$
$197$	1686.00	0.609759	0.304880	+	0.952391i	$0.401384\pi$
$198$	1188.00	0.426401
$199$	1540.00	0.548581	0.274291	−	0.961647i	$-0.411557\pi$
$199$	1540.00	0.548581	0.274291	+	0.961647i	$0.411557\pi$
$200$	0	0
$201$	−3227.00	−1.13241
$202$	−924.000	−0.321844
$203$	0	0
$204$	588.000	0.201805
$205$	0	0
$206$	−1624.00	−0.549269
$207$	−1716.00	−0.576185
$208$	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
$213$	4956.00	1.59427
$214$	−1578.00	−0.504065
$215$	0	0
$216$	280.000	0.0882018
$217$	0	0
$218$	460.000	0.142913
$219$	−931.000	−0.287266
$220$	0	0
$221$	−588.000	−0.178974
$222$	−2044.00	−0.617947
$223$	−3332.00	−1.00057	−0.500285	−	0.865861i	$-0.666772\pi$
$223$	−3332.00	−1.00057	−0.500285	+	0.865861i	$0.666772\pi$
$224$	0	0
$225$	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	$0.574961\pi$
$228$	980.000	0.284658
$229$	−4340.00	−1.25238	−0.626191	−	0.779670i	$-0.715387\pi$
$229$	−4340.00	−1.25238	−0.626191	+	0.779670i	$0.715387\pi$
$230$	0	0
$231$	0	0
$232$	−960.000	−0.271668
$233$	−3018.00	−0.848565	−0.424283	−	0.905530i	$-0.639474\pi$
$233$	−3018.00	−0.848565	−0.424283	+	0.905530i	$0.639474\pi$
$234$	1232.00	0.344181
$235$	0	0
$236$	3360.00	0.926769
$237$	−4550.00	−1.24706
$238$	0	0
$239$	4440.00	1.20167	0.600836	−	0.799372i	$-0.294834\pi$
$239$	4440.00	1.20167	0.600836	+	0.799372i	$0.294834\pi$
$240$	0	0
$241$	3703.00	0.989756	0.494878	−	0.868962i	$-0.335213\pi$
$241$	3703.00	0.989756	0.494878	+	0.868962i	$0.335213\pi$
$242$	−1204.00	−0.319818
$243$	4928.00	1.30095
$244$	952.000	0.249777
$245$	0	0
$246$	4998.00	1.29537
$247$	−980.000	−0.252453
$248$	−1456.00	−0.372807
$249$	−6321.00	−1.60874
$250$	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	$0.812128\pi$
$258$	2072.00	0.499989
$259$	0	0
$260$	0	0
$261$	−2640.00	−0.626099
$262$	−504.000	−0.118844
$263$	−3438.00	−0.806069	−0.403035	−	0.915185i	$-0.632045\pi$
$263$	−3438.00	−0.806069	−0.403035	+	0.915185i	$0.632045\pi$
$264$	−1512.00	−0.352489
$265$	0	0
$266$	0	0
$267$	5145.00	1.17928
$268$	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	$-0.699010\pi$
$271$	−5222.00	−1.17053	−0.585266	+	0.810842i	$0.699010\pi$
$272$	−336.000	−0.0749007
$273$	0	0
$274$	3882.00	0.855913
$275$	0	0
$276$	2184.00	0.476309
$277$	−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
$279$	−4004.00	−0.859187
$280$	0	0
$281$	−3858.00	−0.819036	−0.409518	−	0.912302i	$-0.634303\pi$
$281$	−3858.00	−0.819036	−0.409518	+	0.912302i	$0.634303\pi$
$282$	−1176.00	−0.248333
$283$	−4277.00	−0.898379	−0.449190	−	0.893437i	$-0.648287\pi$
$283$	−4277.00	−0.898379	−0.449190	+	0.893437i	$0.648287\pi$
$284$	−2832.00	−0.591719
$285$	0	0
$286$	1512.00	0.312610
$287$	0	0
$288$	704.000	0.144040
$289$	−4472.00	−0.910238
$290$	0	0
$291$	7742.00	1.55960
$292$	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
$297$	945.000	0.184628
$298$	−3600.00	−0.699807
$299$	−2184.00	−0.422421
$300$	0	0
$301$	0	0
$302$	−6596.00	−1.25681
$303$	3234.00	0.613163
$304$	−560.000	−0.105652
$305$	0	0
$306$	−924.000	−0.172619
$307$	−5831.00	−1.08402	−0.542008	−	0.840373i	$-0.682336\pi$
$307$	−5831.00	−1.08402	−0.542008	+	0.840373i	$0.682336\pi$
$308$	0	0
$309$	5684.00	1.04644
$310$	0	0
$311$	2478.00	0.451815	0.225908	−	0.974149i	$-0.427465\pi$
$311$	2478.00	0.451815	0.225908	+	0.974149i	$0.427465\pi$
$312$	−1568.00	−0.284521
$313$	2758.00	0.498056	0.249028	−	0.968496i	$-0.419889\pi$
$313$	2758.00	0.498056	0.249028	+	0.968496i	$0.419889\pi$
$314$	−532.000	−0.0956130
$315$	0	0
$316$	2600.00	0.462853
$317$	636.000	0.112686	0.0563428	−	0.998411i	$-0.482056\pi$
$317$	636.000	0.112686	0.0563428	+	0.998411i	$0.482056\pi$
$318$	−9828.00	−1.73310
$319$	−3240.00	−0.568668
$320$	0	0
$321$	5523.00	0.960323
$322$	0	0
$323$	735.000	0.126615
$324$	−3356.00	−0.575446
$325$	0	0
$326$	2314.00	0.393131
$327$	−1610.00	−0.272273
$328$	−2856.00	−0.480781
$329$	0	0
$330$	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
$333$	3212.00	0.528578
$334$	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
$339$	14511.0	2.32487
$340$	0	0
$341$	−4914.00	−0.780375
$342$	−1540.00	−0.243490
$343$	0	0
$344$	−1184.00	−0.185573
$345$	0	0
$346$	3696.00	0.574272
$347$	−5349.00	−0.827520	−0.413760	−	0.910386i	$-0.635785\pi$
$347$	−5349.00	−0.827520	−0.413760	+	0.910386i	$0.635785\pi$
$348$	3360.00	0.517572
$349$	−11270.0	−1.72857	−0.864283	−	0.503007i	$-0.832227\pi$
$349$	−11270.0	−1.72857	−0.864283	+	0.503007i	$0.832227\pi$
$350$	0	0
$351$	980.000	0.149027
$352$	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	$0.531295\pi$
$354$	−11760.0	−1.76564
$355$	0	0
$356$	−2940.00	−0.437696
$357$	0	0
$358$	270.000	0.0398602
$359$	−750.000	−0.110260	−0.0551302	−	0.998479i	$-0.517557\pi$
$359$	−750.000	−0.110260	−0.0551302	+	0.998479i	$0.517557\pi$
$360$	0	0
$361$	−5634.00	−0.821403
$362$	−4564.00	−0.662648
$363$	4214.00	0.609305
$364$	0	0
$365$	0	0
$366$	−3332.00	−0.475865
$367$	1204.00	0.171249	0.0856244	−	0.996327i	$-0.472712\pi$
$367$	1204.00	0.171249	0.0856244	+	0.996327i	$0.472712\pi$
$368$	−1248.00	−0.176784
$369$	−7854.00	−1.10803
$370$	0	0
$371$	0	0
$372$	5096.00	0.710256
$373$	−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
$381$	7798.00	1.04857
$382$	−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	$0.428959\pi$
$384$	−896.000	−0.119072
$385$	0	0
$386$	6634.00	0.874771
$387$	−3256.00	−0.427679
$388$	−4424.00	−0.578852
$389$	−13770.0	−1.79477	−0.897387	−	0.441245i	$-0.854537\pi$
$389$	−13770.0	−1.79477	−0.897387	+	0.441245i	$0.854537\pi$
$390$	0	0
$391$	1638.00	0.211860
$392$	0	0
$393$	1764.00	0.226417
$394$	3372.00	0.431165
$395$	0	0
$396$	2376.00	0.301511
$397$	3724.00	0.470786	0.235393	−	0.971900i	$-0.424362\pi$
$397$	3724.00	0.470786	0.235393	+	0.971900i	$0.424362\pi$
$398$	3080.00	0.387906
$399$	0	0
$400$	0	0
$401$	6117.00	0.761767	0.380883	−	0.924623i	$-0.375620\pi$
$401$	6117.00	0.761767	0.380883	+	0.924623i	$0.375620\pi$
$402$	−6454.00	−0.800737
$403$	−5096.00	−0.629900
$404$	−1848.00	−0.227578
$405$	0	0
$406$	0	0
$407$	3942.00	0.480093
$408$	1176.00	0.142698
$409$	−6125.00	−0.740493	−0.370247	−	0.928933i	$-0.620727\pi$
$409$	−6125.00	−0.740493	−0.370247	+	0.928933i	$0.620727\pi$
$410$	0	0
$411$	−13587.0	−1.63065
$412$	−3248.00	−0.388392
$413$	0	0
$414$	−3432.00	−0.407424
$415$	0	0
$416$	896.000	0.105601
$417$	−11515.0	−1.35226
$418$	−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
$423$	1848.00	0.212418
$424$	5616.00	0.643248
$425$	0	0
$426$	9912.00	1.12732
$427$	0	0
$428$	−3156.00	−0.356428
$429$	−5292.00	−0.595571
$430$	0	0
$431$	−558.000	−0.0623617	−0.0311809	−	0.999514i	$-0.509927\pi$
$431$	−558.000	−0.0623617	−0.0311809	+	0.999514i	$0.509927\pi$
$432$	560.000	0.0623681
$433$	2443.00	0.271139	0.135569	−	0.990768i	$-0.456714\pi$
$433$	2443.00	0.271139	0.135569	+	0.990768i	$0.456714\pi$
$434$	0	0
$435$	0	0
$436$	920.000	0.101055
$437$	2730.00	0.298841
$438$	−1862.00	−0.203127
$439$	−12320.0	−1.33941	−0.669706	−	0.742627i	$-0.733580\pi$
$439$	−12320.0	−1.33941	−0.669706	+	0.742627i	$0.733580\pi$
$440$	0	0
$441$	0	0
$442$	−1176.00	−0.126553
$443$	−2343.00	−0.251285	−0.125643	−	0.992076i	$-0.540099\pi$
$443$	−2343.00	−0.251285	−0.125643	+	0.992076i	$0.540099\pi$
$444$	−4088.00	−0.436955
$445$	0	0
$446$	−6664.00	−0.707510
$447$	12600.0	1.33324
$448$	0	0
$449$	10905.0	1.14619	0.573094	−	0.819489i	$-0.305743\pi$
$449$	10905.0	1.14619	0.573094	+	0.819489i	$0.305743\pi$
$450$	0	0
$451$	−9639.00	−1.00639
$452$	−8292.00	−0.862882
$453$	23086.0	2.39443
$454$	−3192.00	−0.329974
$455$	0	0
$456$	1960.00	0.201284
$457$	−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
$459$	−735.000	−0.0747426
$460$	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	$0.606636\pi$
$468$	2464.00	0.243373
$469$	0	0
$470$	0	0
$471$	1862.00	0.182158
$472$	6720.00	0.655324
$473$	−3996.00	−0.388449
$474$	−9100.00	−0.881808
$475$	0	0
$476$	0	0
$477$	15444.0	1.48246
$478$	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
$486$	9856.00	0.919912
$487$	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
$489$	−8099.00	−0.748976
$490$	0	0
$491$	−2388.00	−0.219489	−0.109744	−	0.993960i	$-0.535003\pi$
$491$	−2388.00	−0.219489	−0.109744	+	0.993960i	$0.535003\pi$
$492$	9996.00	0.915964
$493$	2520.00	0.230213
$494$	−1960.00	−0.178511
$495$	0	0
$496$	−2912.00	−0.263614
$497$	0	0
$498$	−12642.0	−1.13755
$499$	−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
$501$	−12348.0	−1.10113
$502$	−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
$507$	9891.00	0.866420
$508$	−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
$511$	0	0
$512$	512.000	0.0441942
$513$	−1225.00	−0.105429
$514$	−13692.0	−1.17496
$515$	0	0
$516$	4144.00	0.353545
$517$	2268.00	0.192933
$518$	0	0
$519$	−12936.0	−1.09408
$520$	0	0
$521$	−10437.0	−0.877645	−0.438823	−	0.898574i	$-0.644604\pi$
$521$	−10437.0	−0.877645	−0.438823	+	0.898574i	$0.644604\pi$
$522$	−5280.00	−0.442719
$523$	8113.00	0.678311	0.339156	−	0.940730i	$-0.389859\pi$
$523$	8113.00	0.678311	0.339156	+	0.940730i	$0.389859\pi$
$524$	−1008.00	−0.0840357
$525$	0	0
$526$	−6876.00	−0.569977
$527$	3822.00	0.315918
$528$	−3024.00	−0.249248
$529$	−6083.00	−0.499959
$530$	0	0
$531$	18480.0	1.51029
$532$	0	0
$533$	−9996.00	−0.812336
$534$	10290.0	0.833880
$535$	0	0
$536$	3688.00	0.297197
$537$	−945.000	−0.0759400
$538$	3360.00	0.269256
$539$	0	0
$540$	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
$543$	15974.0	1.26245
$544$	−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
$549$	5236.00	0.407044
$550$	0	0
$551$	4200.00	0.324730
$552$	4368.00	0.336801
$553$	0	0
$554$	−2768.00	−0.212276
$555$	0	0
$556$	6580.00	0.501896
$557$	−15324.0	−1.16571	−0.582853	−	0.812577i	$-0.698064\pi$
$557$	−15324.0	−1.16571	−0.582853	+	0.812577i	$0.698064\pi$
$558$	−8008.00	−0.607537
$559$	−4144.00	−0.313547
$560$	0	0
$561$	3969.00	0.298701
$562$	−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	$0.385457\pi$
$564$	−2352.00	−0.175598
$565$	0	0
$566$	−8554.00	−0.635250
$567$	0	0
$568$	−5664.00	−0.418409
$569$	−24375.0	−1.79588	−0.897938	−	0.440122i	$-0.854935\pi$
$569$	−24375.0	−1.79588	−0.897938	+	0.440122i	$0.854935\pi$
$570$	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
$573$	9786.00	0.713466
$574$	0	0
$575$	0	0
$576$	1408.00	0.101852
$577$	16429.0	1.18535	0.592676	−	0.805441i	$-0.298071\pi$
$577$	16429.0	1.18535	0.592676	+	0.805441i	$0.298071\pi$
$578$	−8944.00	−0.643636
$579$	−23219.0	−1.66658
$580$	0	0
$581$	0	0
$582$	15484.0	1.10280
$583$	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
$591$	−11802.0	−0.821437
$592$	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	$0.661177\pi$
$594$	1890.00	0.130552
$595$	0	0
$596$	−7200.00	−0.494838
$597$	−10780.0	−0.739022
$598$	−4368.00	−0.298697
$599$	21090.0	1.43859	0.719294	−	0.694706i	$-0.244465\pi$
$599$	21090.0	1.43859	0.719294	+	0.694706i	$0.244465\pi$
$600$	0	0
$601$	−5747.00	−0.390058	−0.195029	−	0.980797i	$-0.562480\pi$
$601$	−5747.00	−0.390058	−0.195029	+	0.980797i	$0.562480\pi$
$602$	0	0
$603$	10142.0	0.684932
$604$	−13192.0	−0.888700
$605$	0	0
$606$	6468.00	0.433572
$607$	−17696.0	−1.18329	−0.591646	−	0.806198i	$-0.701522\pi$
$607$	−17696.0	−1.18329	−0.591646	+	0.806198i	$0.701522\pi$
$608$	−1120.00	−0.0747072
$609$	0	0
$610$	0	0
$611$	2352.00	0.155731
$612$	−1848.00	−0.122060
$613$	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
$616$	0	0
$617$	−4194.00	−0.273653	−0.136827	−	0.990595i	$-0.543690\pi$
$617$	−4194.00	−0.273653	−0.136827	+	0.990595i	$0.543690\pi$
$618$	11368.0	0.739948
$619$	7420.00	0.481801	0.240901	−	0.970550i	$-0.422557\pi$
$619$	7420.00	0.481801	0.240901	+	0.970550i	$0.422557\pi$
$620$	0	0
$621$	−2730.00	−0.176411
$622$	4956.00	0.319482
$623$	0	0
$624$	−3136.00	−0.201187
$625$	0	0
$626$	5516.00	0.352178
$627$	6615.00	0.421336
$628$	−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
$633$	21301.0	1.33750
$634$	1272.00	0.0796807
$635$	0	0
$636$	−19656.0	−1.22549
$637$	0	0
$638$	−6480.00	−0.402109
$639$	−15576.0	−0.964283
$640$	0	0
$641$	−29478.0	−1.81640	−0.908199	−	0.418539i	$-0.862542\pi$
$641$	−29478.0	−1.81640	−0.908199	+	0.418539i	$0.862542\pi$
$642$	11046.0	0.679051
$643$	−5852.00	−0.358912	−0.179456	−	0.983766i	$-0.557434\pi$
$643$	−5852.00	−0.358912	−0.179456	+	0.983766i	$0.557434\pi$
$644$	0	0
$645$	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	$0.146620\pi$
$648$	−6712.00	−0.406902
$649$	22680.0	1.37175
$650$	0	0
$651$	0	0
$652$	4628.00	0.277985
$653$	−3498.00	−0.209628	−0.104814	−	0.994492i	$-0.533425\pi$
$653$	−3498.00	−0.209628	−0.104814	+	0.994492i	$0.533425\pi$
$654$	−3220.00	−0.192526
$655$	0	0
$656$	−5712.00	−0.339964
$657$	2926.00	0.173751
$658$	0	0
$659$	7905.00	0.467276	0.233638	−	0.972324i	$-0.424937\pi$
$659$	7905.00	0.467276	0.233638	+	0.972324i	$0.424937\pi$
$660$	0	0
$661$	−27272.0	−1.60478	−0.802389	−	0.596802i	$-0.796438\pi$
$661$	−27272.0	−1.60478	−0.802389	+	0.596802i	$0.796438\pi$
$662$	13774.0	0.808673
$663$	4116.00	0.241104
$664$	7224.00	0.422207
$665$	0	0
$666$	6424.00	0.373761
$667$	9360.00	0.543359
$668$	7056.00	0.408690
$669$	23324.0	1.34792
$670$	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	$0.758085\pi$
$678$	29022.0	1.64393
$679$	0	0
$680$	0	0
$681$	11172.0	0.628652
$682$	−9828.00	−0.551809
$683$	2127.00	0.119162	0.0595808	−	0.998223i	$-0.481024\pi$
$683$	2127.00	0.119162	0.0595808	+	0.998223i	$0.481024\pi$
$684$	−3080.00	−0.172174
$685$	0	0
$686$	0	0
$687$	30380.0	1.68715
$688$	−2368.00	−0.131220
$689$	19656.0	1.08684
$690$	0	0
$691$	8953.00	0.492892	0.246446	−	0.969157i	$-0.420737\pi$
$691$	8953.00	0.492892	0.246446	+	0.969157i	$0.420737\pi$
$692$	7392.00	0.406072
$693$	0	0
$694$	−10698.0	−0.585145
$695$	0	0
$696$	6720.00	0.365978
$697$	7497.00	0.407416
$698$	−22540.0	−1.22228
$699$	21126.0	1.14315
$700$	0	0
$701$	23172.0	1.24849	0.624247	−	0.781227i	$-0.285406\pi$
$701$	23172.0	1.24849	0.624247	+	0.781227i	$0.285406\pi$
$702$	1960.00	0.105378
$703$	−5110.00	−0.274150
$704$	1728.00	0.0925092
$705$	0	0
$706$	−2604.00	−0.138814
$707$	0	0
$708$	−23520.0	−1.24850
$709$	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
$711$	14300.0	0.754278
$712$	−5880.00	−0.309498
$713$	14196.0	0.745644
$714$	0	0
$715$	0	0
$716$	540.000	0.0281854
$717$	−31080.0	−1.61883
$718$	−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
$721$	0	0
$722$	−11268.0	−0.580819
$723$	−25921.0	−1.33335
$724$	−9128.00	−0.468563
$725$	0	0
$726$	8428.00	0.430844
$727$	−11816.0	−0.602794	−0.301397	−	0.953499i	$-0.597453\pi$
$727$	−11816.0	−0.602794	−0.301397	+	0.953499i	$0.597453\pi$
$728$	0	0
$729$	−11843.0	−0.601687
$730$	0	0
$731$	3108.00	0.157255
$732$	−6664.00	−0.336487
$733$	−23492.0	−1.18376	−0.591881	−	0.806026i	$-0.701614\pi$
$733$	−23492.0	−1.18376	−0.591881	+	0.806026i	$0.701614\pi$
$734$	2408.00	0.121091
$735$	0	0
$736$	−2496.00	−0.125005
$737$	12447.0	0.622105
$738$	−15708.0	−0.783495
$739$	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
$741$	6860.00	0.340092
$742$	0	0
$743$	16242.0	0.801967	0.400983	−	0.916085i	$-0.368669\pi$
$743$	16242.0	0.801967	0.400983	+	0.916085i	$0.368669\pi$
$744$	10192.0	0.502227
$745$	0	0
$746$	−2396.00	−0.117592
$747$	19866.0	0.973037
$748$	−2268.00	−0.110864
$749$	0	0
$750$	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
$753$	49539.0	2.39748
$754$	−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
$759$	14742.0	0.705008
$760$	0	0
$761$	18123.0	0.863283	0.431641	−	0.902045i	$-0.357935\pi$
$761$	18123.0	0.863283	0.431641	+	0.902045i	$0.357935\pi$
$762$	15596.0	0.741448
$763$	0	0
$764$	−5592.00	−0.264806
$765$	0	0
$766$	6636.00	0.313014
$767$	23520.0	1.10725
$768$	−1792.00	−0.0841969
$769$	−9485.00	−0.444783	−0.222391	−	0.974957i	$-0.571386\pi$
$769$	−9485.00	−0.444783	−0.222391	+	0.974957i	$0.571386\pi$
$770$	0	0
$771$	47922.0	2.23848
$772$	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	$0.240923\pi$
$774$	−6512.00	−0.302415
$775$	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
$783$	−4200.00	−0.191693
$784$	0	0
$785$	0	0
$786$	3528.00	0.160101
$787$	15484.0	0.701328	0.350664	−	0.936501i	$-0.385956\pi$
$787$	15484.0	0.701328	0.350664	+	0.936501i	$0.385956\pi$
$788$	6744.00	0.304880
$789$	24066.0	1.08590
$790$	0	0
$791$	0	0
$792$	4752.00	0.213201
$793$	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
$801$	−16170.0	−0.713282
$802$	12234.0	0.538650
$803$	3591.00	0.157813
$804$	−12908.0	−0.566207
$805$	0	0
$806$	−10192.0	−0.445407
$807$	−11760.0	−0.512976
$808$	−3696.00	−0.160922
$809$	−36030.0	−1.56582	−0.782909	−	0.622136i	$-0.786265\pi$
$809$	−36030.0	−1.56582	−0.782909	+	0.622136i	$0.786265\pi$
$810$	0	0
$811$	6748.00	0.292175	0.146088	−	0.989272i	$-0.453332\pi$
$811$	6748.00	0.292175	0.146088	+	0.989272i	$0.453332\pi$
$812$	0	0
$813$	36554.0	1.57688
$814$	7884.00	0.339477
$815$	0	0
$816$	2352.00	0.100903
$817$	5180.00	0.221818
$818$	−12250.0	−0.523608
$819$	0	0
$820$	0	0
$821$	−5598.00	−0.237968	−0.118984	−	0.992896i	$-0.537964\pi$
$821$	−5598.00	−0.237968	−0.118984	+	0.992896i	$0.537964\pi$
$822$	−27174.0	−1.15304
$823$	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
$826$	0	0
$827$	−39999.0	−1.68186	−0.840932	−	0.541141i	$-0.817993\pi$
$827$	−39999.0	−1.68186	−0.840932	+	0.541141i	$0.817993\pi$
$828$	−6864.00	−0.288092
$829$	−16940.0	−0.709711	−0.354856	−	0.934921i	$-0.615470\pi$
$829$	−16940.0	−0.709711	−0.354856	+	0.934921i	$0.615470\pi$
$830$	0	0
$831$	9688.00	0.404420
$832$	1792.00	0.0746712
$833$	0	0
$834$	−23030.0	−0.956191
$835$	0	0
$836$	−3780.00	−0.156380
$837$	−6370.00	−0.263058
$838$	−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
$843$	27006.0	1.10336
$844$	−12172.0	−0.496419
$845$	0	0
$846$	3696.00	0.150202
$847$	0	0
$848$	11232.0	0.454845
$849$	29939.0	1.21025
$850$	0	0
$851$	−11388.0	−0.458726
$852$	19824.0	0.797135
$853$	43918.0	1.76286	0.881432	−	0.472310i	$-0.156580\pi$
$853$	43918.0	1.76286	0.881432	+	0.472310i	$0.156580\pi$
$854$	0	0
$855$	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	$0.478802\pi$
$858$	−10584.0	−0.421133
$859$	36925.0	1.46666	0.733332	−	0.679870i	$-0.237964\pi$
$859$	36925.0	1.46666	0.733332	+	0.679870i	$0.237964\pi$
$860$	0	0
$861$	0	0
$862$	−1116.00	−0.0440964
$863$	−40608.0	−1.60175	−0.800876	−	0.598830i	$-0.795633\pi$
$863$	−40608.0	−1.60175	−0.800876	+	0.598830i	$0.795633\pi$
$864$	1120.00	0.0441009
$865$	0	0
$866$	4886.00	0.191724
$867$	31304.0	1.22623
$868$	0	0
$869$	17550.0	0.685090
$870$	0	0
$871$	12908.0	0.502148
$872$	1840.00	0.0714567
$873$	−24332.0	−0.943314
$874$	5460.00	0.211313
$875$	0	0
$876$	−3724.00	−0.143633
$877$	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
$879$	44394.0	1.70350
$880$	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	$0.346897\pi$
$888$	−8176.00	−0.308974
$889$	0	0
$890$	0	0
$891$	−22653.0	−0.851744
$892$	−13328.0	−0.500285
$893$	−2940.00	−0.110172
$894$	25200.0	0.942745
$895$	0	0
$896$	0	0
$897$	15288.0	0.569065
$898$	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
$906$	46172.0	1.69311
$907$	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
$909$	−10164.0	−0.370868
$910$	0	0
$911$	−33468.0	−1.21717	−0.608586	−	0.793488i	$-0.708263\pi$
$911$	−33468.0	−1.21717	−0.608586	+	0.793488i	$0.708263\pi$
$912$	3920.00	0.142329
$913$	24381.0	0.883782
$914$	−6638.00	−0.240225
$915$	0	0
$916$	−17360.0	−0.626191
$917$	0	0
$918$	−1470.00	−0.0528510
$919$	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
$921$	40817.0	1.46033
$922$	12936.0	0.462066
$923$	−19824.0	−0.706950
$924$	0	0
$925$	0	0
$926$	23944.0	0.849729
$927$	−17864.0	−0.632935
$928$	−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
$931$	0	0
$932$	−12072.0	−0.424283
$933$	−17346.0	−0.608663
$934$	−13272.0	−0.464960
$935$	0	0
$936$	4928.00	0.172091
$937$	−20531.0	−0.715815	−0.357907	−	0.933757i	$-0.616510\pi$
$937$	−20531.0	−0.715815	−0.357907	+	0.933757i	$0.616510\pi$
$938$	0	0
$939$	−19306.0	−0.670956
$940$	0	0
$941$	−7812.00	−0.270631	−0.135316	−	0.990803i	$-0.543205\pi$
$941$	−7812.00	−0.270631	−0.135316	+	0.990803i	$0.543205\pi$
$942$	3724.00	0.128805
$943$	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	$-0.369355\pi$
$947$	23256.0	0.798013	0.399007	+	0.916948i	$0.369355\pi$
$948$	−18200.0	−0.623532
$949$	3724.00	0.127383
$950$	0	0
$951$	−4452.00	−0.151804
$952$	0	0
$953$	14097.0	0.479167	0.239584	−	0.970876i	$-0.422989\pi$
$953$	14097.0	0.479167	0.239584	+	0.970876i	$0.422989\pi$
$954$	30888.0	1.04826
$955$	0	0
$956$	17760.0	0.600836
$957$	22680.0	0.766082
$958$	1260.00	0.0424935
$959$	0	0
$960$	0	0
$961$	3333.00	0.111879
$962$	8176.00	0.274017
$963$	−17358.0	−0.580845
$964$	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
$969$	−5145.00	−0.170569
$970$	0	0
$971$	32613.0	1.07786	0.538929	−	0.842351i	$-0.318829\pi$
$971$	32613.0	1.07786	0.538929	+	0.842351i	$0.318829\pi$
$972$	19712.0	0.650476
$973$	0	0
$974$	21292.0	0.700451
$975$	0	0
$976$	3808.00	0.124888
$977$	−43359.0	−1.41983	−0.709917	−	0.704286i	$-0.751267\pi$
$977$	−43359.0	−1.41983	−0.709917	+	0.704286i	$0.751267\pi$
$978$	−16198.0	−0.529606
$979$	−19845.0	−0.647854
$980$	0	0
$981$	5060.00	0.164682
$982$	−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	$0.346896\pi$
$984$	19992.0	0.647685
$985$	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
$993$	−48209.0	−1.54065
$994$	0	0
$995$	0	0
$996$	−25284.0	−0.804372
$997$	28924.0	0.918789	0.459394	−	0.888232i	$-0.348066\pi$
$997$	28924.0	0.918789	0.459394	+	0.888232i	$0.348066\pi$
$998$	−21080.0	−0.668613
$999$	5110.00	0.161835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.4.a.y.1.1		1
5.4	even	2		2450.4.a.t.1.1		1
7.6	odd	2		50.4.a.e.1.1	yes	1
21.20	even	2		450.4.a.a.1.1		1
28.27	even	2		400.4.a.d.1.1		1
35.13	even	4		50.4.b.b.49.1		2
35.27	even	4		50.4.b.b.49.2		2
35.34	odd	2		50.4.a.a.1.1	✓	1
56.13	odd	2		1600.4.a.f.1.1		1
56.27	even	2		1600.4.a.bv.1.1		1
105.62	odd	4		450.4.c.c.199.1		2
105.83	odd	4		450.4.c.c.199.2		2
105.104	even	2		450.4.a.t.1.1		1
140.27	odd	4		400.4.c.d.49.2		2
140.83	odd	4		400.4.c.d.49.1		2
140.139	even	2		400.4.a.r.1.1		1
280.69	odd	2		1600.4.a.bu.1.1		1
280.139	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	35.34	odd	2
50.4.a.e.1.1	yes	1	7.6	odd	2
50.4.b.b.49.1		2	35.13	even	4
50.4.b.b.49.2		2	35.27	even	4
400.4.a.d.1.1		1	28.27	even	2
400.4.a.r.1.1		1	140.139	even	2
400.4.c.d.49.1		2	140.83	odd	4
400.4.c.d.49.2		2	140.27	odd	4
450.4.a.a.1.1		1	21.20	even	2
450.4.a.t.1.1		1	105.104	even	2
450.4.c.c.199.1		2	105.62	odd	4
450.4.c.c.199.2		2	105.83	odd	4
1600.4.a.f.1.1		1	56.13	odd	2
1600.4.a.g.1.1		1	280.139	even	2
1600.4.a.bu.1.1		1	280.69	odd	2
1600.4.a.bv.1.1		1	56.27	even	2
2450.4.a.t.1.1		1	5.4	even	2
2450.4.a.y.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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