LMFDB - Embedded newform 50.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,6,Mod(1,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("50.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	50.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:	$8.01919099065$
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	$\chi$	$=$	50.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} -11.0000 q^{3} +16.0000 q^{4} +44.0000 q^{6} -142.000 q^{7} -64.0000 q^{8} -122.000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} -11.0000 q^{3} +16.0000 q^{4} +44.0000 q^{6} -142.000 q^{7} -64.0000 q^{8} -122.000 q^{9} +777.000 q^{11} -176.000 q^{12} +884.000 q^{13} +568.000 q^{14} +256.000 q^{16} -27.0000 q^{17} +488.000 q^{18} +1145.00 q^{19} +1562.00 q^{21} -3108.00 q^{22} +1854.00 q^{23} +704.000 q^{24} -3536.00 q^{26} +4015.00 q^{27} -2272.00 q^{28} -4920.00 q^{29} +1802.00 q^{31} -1024.00 q^{32} -8547.00 q^{33} +108.000 q^{34} -1952.00 q^{36} +13178.0 q^{37} -4580.00 q^{38} -9724.00 q^{39} -15123.0 q^{41} -6248.00 q^{42} +7844.00 q^{43} +12432.0 q^{44} -7416.00 q^{46} -6732.00 q^{47} -2816.00 q^{48} +3357.00 q^{49} +297.000 q^{51} +14144.0 q^{52} +3414.00 q^{53} -16060.0 q^{54} +9088.00 q^{56} -12595.0 q^{57} +19680.0 q^{58} +33960.0 q^{59} +47402.0 q^{61} -7208.00 q^{62} +17324.0 q^{63} +4096.00 q^{64} +34188.0 q^{66} -13177.0 q^{67} -432.000 q^{68} -20394.0 q^{69} -7548.00 q^{71} +7808.00 q^{72} -59821.0 q^{73} -52712.0 q^{74} +18320.0 q^{76} -110334. q^{77} +38896.0 q^{78} +75830.0 q^{79} -14519.0 q^{81} +60492.0 q^{82} +46299.0 q^{83} +24992.0 q^{84} -31376.0 q^{86} +54120.0 q^{87} -49728.0 q^{88} -30585.0 q^{89} -125528. q^{91} +29664.0 q^{92} -19822.0 q^{93} +26928.0 q^{94} +11264.0 q^{96} +104018. q^{97} -13428.0 q^{98} -94794.0 q^{99} +O(q^{100})\)

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	−11.0000	−0.705650	−0.352825	−	0.935689i	$-0.614779\pi$
$3$	−11.0000	−0.705650	−0.352825	+	0.935689i	$0.614779\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	44.0000	0.498970
$7$	−142.000	−1.09533	−0.547663	−	0.836699i	$-0.684482\pi$
$7$	−142.000	−1.09533	−0.547663	+	0.836699i	$0.684482\pi$
$8$	−64.0000	−0.353553
$9$	−122.000	−0.502058
$10$	0	0
$11$	777.000	1.93615	0.968076	−	0.250658i	$-0.0806470\pi$
$11$	777.000	1.93615	0.968076	+	0.250658i	$0.0806470\pi$
$12$	−176.000	−0.352825
$13$	884.000	1.45075	0.725377	−	0.688352i	$-0.241665\pi$
$13$	884.000	1.45075	0.725377	+	0.688352i	$0.241665\pi$
$14$	568.000	0.774512
$15$	0	0
$16$	256.000	0.250000
$17$	−27.0000	−0.0226590	−0.0113295	−	0.999936i	$-0.503606\pi$
$17$	−27.0000	−0.0226590	−0.0113295	+	0.999936i	$0.503606\pi$
$18$	488.000	0.355008
$19$	1145.00	0.727648	0.363824	−	0.931468i	$-0.381471\pi$
$19$	1145.00	0.727648	0.363824	+	0.931468i	$0.381471\pi$
$20$	0	0
$21$	1562.00	0.772917
$22$	−3108.00	−1.36907
$23$	1854.00	0.730786	0.365393	−	0.930853i	$-0.380935\pi$
$23$	1854.00	0.730786	0.365393	+	0.930853i	$0.380935\pi$
$24$	704.000	0.249485
$25$	0	0
$26$	−3536.00	−1.02584
$27$	4015.00	1.05993
$28$	−2272.00	−0.547663
$29$	−4920.00	−1.08635	−0.543175	−	0.839619i	$-0.682778\pi$
$29$	−4920.00	−1.08635	−0.543175	+	0.839619i	$0.682778\pi$
$30$	0	0
$31$	1802.00	0.336783	0.168392	−	0.985720i	$-0.446143\pi$
$31$	1802.00	0.336783	0.168392	+	0.985720i	$0.446143\pi$
$32$	−1024.00	−0.176777
$33$	−8547.00	−1.36625
$34$	108.000	0.0160224
$35$	0	0
$36$	−1952.00	−0.251029
$37$	13178.0	1.58251	0.791253	−	0.611489i	$-0.209429\pi$
$37$	13178.0	1.58251	0.791253	+	0.611489i	$0.209429\pi$
$38$	−4580.00	−0.514525
$39$	−9724.00	−1.02373
$40$	0	0
$41$	−15123.0	−1.40501	−0.702503	−	0.711681i	$-0.747934\pi$
$41$	−15123.0	−1.40501	−0.702503	+	0.711681i	$0.747934\pi$
$42$	−6248.00	−0.546535
$43$	7844.00	0.646944	0.323472	−	0.946238i	$-0.395150\pi$
$43$	7844.00	0.646944	0.323472	+	0.946238i	$0.395150\pi$
$44$	12432.0	0.968076
$45$	0	0
$46$	−7416.00	−0.516744
$47$	−6732.00	−0.444528	−0.222264	−	0.974986i	$-0.571345\pi$
$47$	−6732.00	−0.444528	−0.222264	+	0.974986i	$0.571345\pi$
$48$	−2816.00	−0.176413
$49$	3357.00	0.199738
$50$	0	0
$51$	297.000	0.0159894
$52$	14144.0	0.725377
$53$	3414.00	0.166945	0.0834726	−	0.996510i	$-0.473399\pi$
$53$	3414.00	0.166945	0.0834726	+	0.996510i	$0.473399\pi$
$54$	−16060.0	−0.749482
$55$	0	0
$56$	9088.00	0.387256
$57$	−12595.0	−0.513465
$58$	19680.0	0.768166
$59$	33960.0	1.27010	0.635050	−	0.772471i	$-0.280980\pi$
$59$	33960.0	1.27010	0.635050	+	0.772471i	$0.280980\pi$
$60$	0	0
$61$	47402.0	1.63107	0.815534	−	0.578709i	$-0.196443\pi$
$61$	47402.0	1.63107	0.815534	+	0.578709i	$0.196443\pi$
$62$	−7208.00	−0.238142
$63$	17324.0	0.549917
$64$	4096.00	0.125000
$65$	0	0
$66$	34188.0	0.966082
$67$	−13177.0	−0.358616	−0.179308	−	0.983793i	$-0.557386\pi$
$67$	−13177.0	−0.358616	−0.179308	+	0.983793i	$0.557386\pi$
$68$	−432.000	−0.0113295
$69$	−20394.0	−0.515679
$70$	0	0
$71$	−7548.00	−0.177699	−0.0888497	−	0.996045i	$-0.528319\pi$
$71$	−7548.00	−0.177699	−0.0888497	+	0.996045i	$0.528319\pi$
$72$	7808.00	0.177504
$73$	−59821.0	−1.31385	−0.656926	−	0.753955i	$-0.728144\pi$
$73$	−59821.0	−1.31385	−0.656926	+	0.753955i	$0.728144\pi$
$74$	−52712.0	−1.11900
$75$	0	0
$76$	18320.0	0.363824
$77$	−110334.	−2.12072
$78$	38896.0	0.723883
$79$	75830.0	1.36702	0.683508	−	0.729943i	$-0.260454\pi$
$79$	75830.0	1.36702	0.683508	+	0.729943i	$0.260454\pi$
$80$	0	0
$81$	−14519.0	−0.245881
$82$	60492.0	0.993490
$83$	46299.0	0.737694	0.368847	−	0.929490i	$-0.379753\pi$
$83$	46299.0	0.737694	0.368847	+	0.929490i	$0.379753\pi$
$84$	24992.0	0.386458
$85$	0	0
$86$	−31376.0	−0.457458
$87$	54120.0	0.766584
$88$	−49728.0	−0.684533
$89$	−30585.0	−0.409292	−0.204646	−	0.978836i	$-0.565604\pi$
$89$	−30585.0	−0.409292	−0.204646	+	0.978836i	$0.565604\pi$
$90$	0	0
$91$	−125528.	−1.58905
$92$	29664.0	0.365393
$93$	−19822.0	−0.237651
$94$	26928.0	0.314329
$95$	0	0
$96$	11264.0	0.124743
$97$	104018.	1.12248	0.561241	−	0.827653i	$-0.310324\pi$
$97$	104018.	1.12248	0.561241	+	0.827653i	$0.310324\pi$
$98$	−13428.0	−0.141236
$99$	−94794.0	−0.972060
$100$	0	0
$101$	−23898.0	−0.233109	−0.116554	−	0.993184i	$-0.537185\pi$
$101$	−23898.0	−0.233109	−0.116554	+	0.993184i	$0.537185\pi$
$102$	−1188.00	−0.0113062
$103$	−22636.0	−0.210236	−0.105118	−	0.994460i	$-0.533522\pi$
$103$	−22636.0	−0.210236	−0.105118	+	0.994460i	$0.533522\pi$
$104$	−56576.0	−0.512919
$105$	0	0
$106$	−13656.0	−0.118048
$107$	60633.0	0.511976	0.255988	−	0.966680i	$-0.417599\pi$
$107$	60633.0	0.511976	0.255988	+	0.966680i	$0.417599\pi$
$108$	64240.0	0.529964
$109$	−7090.00	−0.0571584	−0.0285792	−	0.999592i	$-0.509098\pi$
$109$	−7090.00	−0.0571584	−0.0285792	+	0.999592i	$0.509098\pi$
$110$	0	0
$111$	−144958.	−1.11670
$112$	−36352.0	−0.273831
$113$	−128841.	−0.949201	−0.474600	−	0.880201i	$-0.657407\pi$
$113$	−128841.	−0.949201	−0.474600	+	0.880201i	$0.657407\pi$
$114$	50380.0	0.363075
$115$	0	0
$116$	−78720.0	−0.543175
$117$	−107848.	−0.728362
$118$	−135840.	−0.898096
$119$	3834.00	0.0248190
$120$	0	0
$121$	442678.	2.74868
$122$	−189608.	−1.15334
$123$	166353.	0.991443
$124$	28832.0	0.168392
$125$	0	0
$126$	−69296.0	−0.388850
$127$	141338.	0.777588	0.388794	−	0.921325i	$-0.372892\pi$
$127$	141338.	0.777588	0.388794	+	0.921325i	$0.372892\pi$
$128$	−16384.0	−0.0883883
$129$	−86284.0	−0.456516
$130$	0	0
$131$	80052.0	0.407562	0.203781	−	0.979016i	$-0.434677\pi$
$131$	80052.0	0.407562	0.203781	+	0.979016i	$0.434677\pi$
$132$	−136752.	−0.683123
$133$	−162590.	−0.797012
$134$	52708.0	0.253580
$135$	0	0
$136$	1728.00	0.00801118
$137$	32253.0	0.146814	0.0734072	−	0.997302i	$-0.476613\pi$
$137$	32253.0	0.146814	0.0734072	+	0.997302i	$0.476613\pi$
$138$	81576.0	0.364640
$139$	394865.	1.73345	0.866726	−	0.498785i	$-0.166220\pi$
$139$	394865.	1.73345	0.866726	+	0.498785i	$0.166220\pi$
$140$	0	0
$141$	74052.0	0.313682
$142$	30192.0	0.125652
$143$	686868.	2.80888
$144$	−31232.0	−0.125514
$145$	0	0
$146$	239284.	0.929034
$147$	−36927.0	−0.140945
$148$	210848.	0.791253
$149$	−491400.	−1.81330	−0.906650	−	0.421884i	$-0.861369\pi$
$149$	−491400.	−1.81330	−0.906650	+	0.421884i	$0.861369\pi$
$150$	0	0
$151$	200402.	0.715253	0.357626	−	0.933865i	$-0.383586\pi$
$151$	200402.	0.715253	0.357626	+	0.933865i	$0.383586\pi$
$152$	−73280.0	−0.257263
$153$	3294.00	0.0113761
$154$	441336.	1.49957
$155$	0	0
$156$	−155584.	−0.511863
$157$	−22942.0	−0.0742818	−0.0371409	−	0.999310i	$-0.511825\pi$
$157$	−22942.0	−0.0742818	−0.0371409	+	0.999310i	$0.511825\pi$
$158$	−303320.	−0.966626
$159$	−37554.0	−0.117805
$160$	0	0
$161$	−263268.	−0.800448
$162$	58076.0	0.173864
$163$	−336241.	−0.991246	−0.495623	−	0.868538i	$-0.665060\pi$
$163$	−336241.	−0.991246	−0.495623	+	0.868538i	$0.665060\pi$
$164$	−241968.	−0.702503
$165$	0	0
$166$	−185196.	−0.521629
$167$	59748.0	0.165780	0.0828900	−	0.996559i	$-0.473585\pi$
$167$	59748.0	0.165780	0.0828900	+	0.996559i	$0.473585\pi$
$168$	−99968.0	−0.273267
$169$	410163.	1.10469
$170$	0	0
$171$	−139690.	−0.365321
$172$	125504.	0.323472
$173$	−60696.0	−0.154186	−0.0770930	−	0.997024i	$-0.524564\pi$
$173$	−60696.0	−0.154186	−0.0770930	+	0.997024i	$0.524564\pi$
$174$	−216480.	−0.542057
$175$	0	0
$176$	198912.	0.484038
$177$	−373560.	−0.896246
$178$	122340.	0.289413
$179$	−7995.00	−0.0186503	−0.00932515	−	0.999957i	$-0.502968\pi$
$179$	−7995.00	−0.0186503	−0.00932515	+	0.999957i	$0.502968\pi$
$180$	0	0
$181$	−454798.	−1.03186	−0.515932	−	0.856630i	$-0.672554\pi$
$181$	−454798.	−1.03186	−0.515932	+	0.856630i	$0.672554\pi$
$182$	502112.	1.12363
$183$	−521422.	−1.15096
$184$	−118656.	−0.258372
$185$	0	0
$186$	79288.0	0.168045
$187$	−20979.0	−0.0438713
$188$	−107712.	−0.222264
$189$	−570130.	−1.16097
$190$	0	0
$191$	−428298.	−0.849499	−0.424749	−	0.905311i	$-0.639638\pi$
$191$	−428298.	−0.849499	−0.424749	+	0.905311i	$0.639638\pi$
$192$	−45056.0	−0.0882063
$193$	−835531.	−1.61462	−0.807308	−	0.590130i	$-0.799076\pi$
$193$	−835531.	−1.61462	−0.807308	+	0.590130i	$0.799076\pi$
$194$	−416072.	−0.793714
$195$	0	0
$196$	53712.0	0.0998691
$197$	678318.	1.24528	0.622641	−	0.782508i	$-0.286060\pi$
$197$	678318.	1.24528	0.622641	+	0.782508i	$0.286060\pi$
$198$	379176.	0.687350
$199$	−31900.0	−0.0571029	−0.0285514	−	0.999592i	$-0.509089\pi$
$199$	−31900.0	−0.0571029	−0.0285514	+	0.999592i	$0.509089\pi$
$200$	0	0
$201$	144947.	0.253057
$202$	95592.0	0.164833
$203$	698640.	1.18991
$204$	4752.00	0.00799468
$205$	0	0
$206$	90544.0	0.148659
$207$	−226188.	−0.366897
$208$	226304.	0.362689
$209$	889665.	1.40884
$210$	0	0
$211$	−423673.	−0.655126	−0.327563	−	0.944829i	$-0.606227\pi$
$211$	−423673.	−0.655126	−0.327563	+	0.944829i	$0.606227\pi$
$212$	54624.0	0.0834726
$213$	83028.0	0.125394
$214$	−242532.	−0.362022
$215$	0	0
$216$	−256960.	−0.374741
$217$	−255884.	−0.368887
$218$	28360.0	0.0404171
$219$	658031.	0.927120
$220$	0	0
$221$	−23868.0	−0.0328727
$222$	579832.	0.789623
$223$	398204.	0.536221	0.268110	−	0.963388i	$-0.413601\pi$
$223$	398204.	0.536221	0.268110	+	0.963388i	$0.413601\pi$
$224$	145408.	0.193628
$225$	0	0
$226$	515364.	0.671186
$227$	−1.25761e6	−1.61988	−0.809938	−	0.586515i	$-0.800500\pi$
$227$	−1.25761e6	−1.61988	−0.809938	+	0.586515i	$0.800500\pi$
$228$	−201520.	−0.256733
$229$	203780.	0.256787	0.128393	−	0.991723i	$-0.459018\pi$
$229$	203780.	0.256787	0.128393	+	0.991723i	$0.459018\pi$
$230$	0	0
$231$	1.21367e6	1.49648
$232$	314880.	0.384083
$233$	823974.	0.994314	0.497157	−	0.867661i	$-0.334377\pi$
$233$	823974.	0.994314	0.497157	+	0.867661i	$0.334377\pi$
$234$	431392.	0.515030
$235$	0	0
$236$	543360.	0.635050
$237$	−834130.	−0.964635
$238$	−15336.0	−0.0175497
$239$	−555960.	−0.629577	−0.314788	−	0.949162i	$-0.601934\pi$
$239$	−555960.	−0.629577	−0.314788	+	0.949162i	$0.601934\pi$
$240$	0	0
$241$	523577.	0.580681	0.290341	−	0.956923i	$-0.406231\pi$
$241$	523577.	0.580681	0.290341	+	0.956923i	$0.406231\pi$
$242$	−1.77071e6	−1.94361
$243$	−815936.	−0.886422
$244$	758432.	0.815534
$245$	0	0
$246$	−665412.	−0.701056
$247$	1.01218e6	1.05564
$248$	−115328.	−0.119071
$249$	−509289.	−0.520554
$250$	0	0
$251$	113127.	0.113340	0.0566698	−	0.998393i	$-0.481952\pi$
$251$	113127.	0.113340	0.0566698	+	0.998393i	$0.481952\pi$
$252$	277184.	0.274958
$253$	1.44056e6	1.41491
$254$	−565352.	−0.549838
$255$	0	0
$256$	65536.0	0.0625000
$257$	872958.	0.824443	0.412221	−	0.911084i	$-0.364753\pi$
$257$	872958.	0.824443	0.412221	+	0.911084i	$0.364753\pi$
$258$	345136.	0.322806
$259$	−1.87128e6	−1.73336
$260$	0	0
$261$	600240.	0.545411
$262$	−320208.	−0.288190
$263$	−1.64647e6	−1.46779	−0.733894	−	0.679264i	$-0.762299\pi$
$263$	−1.64647e6	−1.46779	−0.733894	+	0.679264i	$0.762299\pi$
$264$	547008.	0.483041
$265$	0	0
$266$	650360.	0.563572
$267$	336435.	0.288817
$268$	−210832.	−0.179308
$269$	1.78872e6	1.50717	0.753584	−	0.657352i	$-0.228323\pi$
$269$	1.78872e6	1.50717	0.753584	+	0.657352i	$0.228323\pi$
$270$	0	0
$271$	1.12140e6	0.927552	0.463776	−	0.885953i	$-0.346494\pi$
$271$	1.12140e6	0.927552	0.463776	+	0.885953i	$0.346494\pi$
$272$	−6912.00	−0.00566476
$273$	1.38081e6	1.12131
$274$	−129012.	−0.103813
$275$	0	0
$276$	−326304.	−0.257840
$277$	−598312.	−0.468520	−0.234260	−	0.972174i	$-0.575267\pi$
$277$	−598312.	−0.468520	−0.234260	+	0.972174i	$0.575267\pi$
$278$	−1.57946e6	−1.22574
$279$	−219844.	−0.169085
$280$	0	0
$281$	−1.53050e6	−1.15629	−0.578145	−	0.815934i	$-0.696223\pi$
$281$	−1.53050e6	−1.15629	−0.578145	+	0.815934i	$0.696223\pi$
$282$	−296208.	−0.221806
$283$	−1.79700e6	−1.33377	−0.666887	−	0.745159i	$-0.732374\pi$
$283$	−1.79700e6	−1.33377	−0.666887	+	0.745159i	$0.732374\pi$
$284$	−120768.	−0.0888497
$285$	0	0
$286$	−2.74747e6	−1.98618
$287$	2.14747e6	1.53894
$288$	124928.	0.0887521
$289$	−1.41913e6	−0.999487
$290$	0	0
$291$	−1.14420e6	−0.792079
$292$	−957136.	−0.656926
$293$	754494.	0.513437	0.256718	−	0.966486i	$-0.417359\pi$
$293$	754494.	0.513437	0.256718	+	0.966486i	$0.417359\pi$
$294$	147708.	0.0996634
$295$	0	0
$296$	−843392.	−0.559500
$297$	3.11965e6	2.05218
$298$	1.96560e6	1.28220
$299$	1.63894e6	1.06019
$300$	0	0
$301$	−1.11385e6	−0.708614
$302$	−801608.	−0.505760
$303$	262878.	0.164493
$304$	293120.	0.181912
$305$	0	0
$306$	−13176.0	−0.00804415
$307$	−1.96627e6	−1.19068	−0.595342	−	0.803472i	$-0.702983\pi$
$307$	−1.96627e6	−1.19068	−0.595342	+	0.803472i	$0.702983\pi$
$308$	−1.76534e6	−1.06036
$309$	248996.	0.148353
$310$	0	0
$311$	−599298.	−0.351352	−0.175676	−	0.984448i	$-0.556211\pi$
$311$	−599298.	−0.351352	−0.175676	+	0.984448i	$0.556211\pi$
$312$	622336.	0.361942
$313$	−721366.	−0.416193	−0.208097	−	0.978108i	$-0.566727\pi$
$313$	−721366.	−0.416193	−0.208097	+	0.978108i	$0.566727\pi$
$314$	91768.0	0.0525251
$315$	0	0
$316$	1.21328e6	0.683508
$317$	102348.	0.0572046	0.0286023	−	0.999591i	$-0.490894\pi$
$317$	102348.	0.0572046	0.0286023	+	0.999591i	$0.490894\pi$
$318$	150216.	0.0833007
$319$	−3.82284e6	−2.10334
$320$	0	0
$321$	−666963.	−0.361276
$322$	1.05307e6	0.566003
$323$	−30915.0	−0.0164878
$324$	−232304.	−0.122940
$325$	0	0
$326$	1.34496e6	0.700917
$327$	77990.0	0.0403338
$328$	967872.	0.496745
$329$	955944.	0.486903
$330$	0	0
$331$	1.31048e6	0.657445	0.328722	−	0.944427i	$-0.393382\pi$
$331$	1.31048e6	0.657445	0.328722	+	0.944427i	$0.393382\pi$
$332$	740784.	0.368847
$333$	−1.60772e6	−0.794509
$334$	−238992.	−0.117224
$335$	0	0
$336$	399872.	0.193229
$337$	−804397.	−0.385830	−0.192915	−	0.981215i	$-0.561794\pi$
$337$	−804397.	−0.385830	−0.192915	+	0.981215i	$0.561794\pi$
$338$	−1.64065e6	−0.781133
$339$	1.41725e6	0.669804
$340$	0	0
$341$	1.40015e6	0.652063
$342$	558760.	0.258321
$343$	1.90990e6	0.876547
$344$	−502016.	−0.228729
$345$	0	0
$346$	242784.	0.109026
$347$	−2.88321e6	−1.28544	−0.642720	−	0.766101i	$-0.722194\pi$
$347$	−2.88321e6	−1.28544	−0.642720	+	0.766101i	$0.722194\pi$
$348$	865920.	0.383292
$349$	1.27355e6	0.559696	0.279848	−	0.960044i	$-0.409716\pi$
$349$	1.27355e6	0.559696	0.279848	+	0.960044i	$0.409716\pi$
$350$	0	0
$351$	3.54926e6	1.53769
$352$	−795648.	−0.342266
$353$	2.83061e6	1.20905	0.604524	−	0.796587i	$-0.293363\pi$
$353$	2.83061e6	1.20905	0.604524	+	0.796587i	$0.293363\pi$
$354$	1.49424e6	0.633742
$355$	0	0
$356$	−489360.	−0.204646
$357$	−42174.0	−0.0175136
$358$	31980.0	0.0131878
$359$	−981090.	−0.401766	−0.200883	−	0.979615i	$-0.564381\pi$
$359$	−981090.	−0.401766	−0.200883	+	0.979615i	$0.564381\pi$
$360$	0	0
$361$	−1.16507e6	−0.470528
$362$	1.81919e6	0.729637
$363$	−4.86946e6	−1.93961
$364$	−2.00845e6	−0.794524
$365$	0	0
$366$	2.08569e6	0.813854
$367$	−4.19105e6	−1.62427	−0.812134	−	0.583470i	$-0.801694\pi$
$367$	−4.19105e6	−1.62427	−0.812134	+	0.583470i	$0.801694\pi$
$368$	474624.	0.182696
$369$	1.84501e6	0.705394
$370$	0	0
$371$	−484788.	−0.182859
$372$	−317152.	−0.118826
$373$	3.23455e6	1.20377	0.601883	−	0.798584i	$-0.294417\pi$
$373$	3.23455e6	1.20377	0.601883	+	0.798584i	$0.294417\pi$
$374$	83916.0	0.0310217
$375$	0	0
$376$	430848.	0.157165
$377$	−4.34928e6	−1.57603
$378$	2.28052e6	0.820927
$379$	1.39036e6	0.497196	0.248598	−	0.968607i	$-0.420030\pi$
$379$	1.39036e6	0.497196	0.248598	+	0.968607i	$0.420030\pi$
$380$	0	0
$381$	−1.55472e6	−0.548705
$382$	1.71319e6	0.600686
$383$	1.14197e6	0.397795	0.198897	−	0.980020i	$-0.436264\pi$
$383$	1.14197e6	0.397795	0.198897	+	0.980020i	$0.436264\pi$
$384$	180224.	0.0623713
$385$	0	0
$386$	3.34212e6	1.14171
$387$	−956968.	−0.324803
$388$	1.66429e6	0.561241
$389$	3.46299e6	1.16032	0.580159	−	0.814503i	$-0.302990\pi$
$389$	3.46299e6	1.16032	0.580159	+	0.814503i	$0.302990\pi$
$390$	0	0
$391$	−50058.0	−0.0165589
$392$	−214848.	−0.0706181
$393$	−880572.	−0.287596
$394$	−2.71327e6	−0.880547
$395$	0	0
$396$	−1.51670e6	−0.486030
$397$	5.94007e6	1.89154	0.945769	−	0.324839i	$-0.105310\pi$
$397$	5.94007e6	1.89154	0.945769	+	0.324839i	$0.105310\pi$
$398$	127600.	0.0403778
$399$	1.78849e6	0.562412
$400$	0	0
$401$	−2.27412e6	−0.706241	−0.353121	−	0.935578i	$-0.614879\pi$
$401$	−2.27412e6	−0.706241	−0.353121	+	0.935578i	$0.614879\pi$
$402$	−579788.	−0.178939
$403$	1.59297e6	0.488590
$404$	−382368.	−0.116554
$405$	0	0
$406$	−2.79456e6	−0.841392
$407$	1.02393e7	3.06397
$408$	−19008.0	−0.00565309
$409$	−4.29552e6	−1.26972	−0.634859	−	0.772628i	$-0.718942\pi$
$409$	−4.29552e6	−1.26972	−0.634859	+	0.772628i	$0.718942\pi$
$410$	0	0
$411$	−354783.	−0.103600
$412$	−362176.	−0.105118
$413$	−4.82232e6	−1.39117
$414$	904752.	0.259435
$415$	0	0
$416$	−905216.	−0.256460
$417$	−4.34352e6	−1.22321
$418$	−3.55866e6	−0.996198
$419$	1.79705e6	0.500062	0.250031	−	0.968238i	$-0.419559\pi$
$419$	1.79705e6	0.500062	0.250031	+	0.968238i	$0.419559\pi$
$420$	0	0
$421$	−257548.	−0.0708195	−0.0354098	−	0.999373i	$-0.511274\pi$
$421$	−257548.	−0.0708195	−0.0354098	+	0.999373i	$0.511274\pi$
$422$	1.69469e6	0.463244
$423$	821304.	0.223179
$424$	−218496.	−0.0590240
$425$	0	0
$426$	−332112.	−0.0886667
$427$	−6.73108e6	−1.78655
$428$	970128.	0.255988
$429$	−7.55555e6	−1.98209
$430$	0	0
$431$	2.22910e6	0.578012	0.289006	−	0.957327i	$-0.406675\pi$
$431$	2.22910e6	0.578012	0.289006	+	0.957327i	$0.406675\pi$
$432$	1.02784e6	0.264982
$433$	−4.20585e6	−1.07804	−0.539019	−	0.842294i	$-0.681205\pi$
$433$	−4.20585e6	−1.07804	−0.539019	+	0.842294i	$0.681205\pi$
$434$	1.02354e6	0.260843
$435$	0	0
$436$	−113440.	−0.0285792
$437$	2.12283e6	0.531755
$438$	−2.63212e6	−0.655573
$439$	352640.	0.0873314	0.0436657	−	0.999046i	$-0.486096\pi$
$439$	352640.	0.0873314	0.0436657	+	0.999046i	$0.486096\pi$
$440$	0	0
$441$	−409554.	−0.100280
$442$	95472.0	0.0232445
$443$	1.28362e6	0.310761	0.155381	−	0.987855i	$-0.450340\pi$
$443$	1.28362e6	0.310761	0.155381	+	0.987855i	$0.450340\pi$
$444$	−2.31933e6	−0.558348
$445$	0	0
$446$	−1.59282e6	−0.379165
$447$	5.40540e6	1.27956
$448$	−581632.	−0.136916
$449$	−2.10398e6	−0.492521	−0.246260	−	0.969204i	$-0.579202\pi$
$449$	−2.10398e6	−0.492521	−0.246260	+	0.969204i	$0.579202\pi$
$450$	0	0
$451$	−1.17506e7	−2.72031
$452$	−2.06146e6	−0.474600
$453$	−2.20442e6	−0.504719
$454$	5.03045e6	1.14543
$455$	0	0
$456$	806080.	0.181537
$457$	825233.	0.184836	0.0924179	−	0.995720i	$-0.470540\pi$
$457$	825233.	0.184836	0.0924179	+	0.995720i	$0.470540\pi$
$458$	−815120.	−0.181576
$459$	−108405.	−0.0240169
$460$	0	0
$461$	4.50145e6	0.986507	0.493254	−	0.869886i	$-0.335807\pi$
$461$	4.50145e6	0.986507	0.493254	+	0.869886i	$0.335807\pi$
$462$	−4.85470e6	−1.05817
$463$	−1.44212e6	−0.312642	−0.156321	−	0.987706i	$-0.549963\pi$
$463$	−1.44212e6	−0.312642	−0.156321	+	0.987706i	$0.549963\pi$
$464$	−1.25952e6	−0.271588
$465$	0	0
$466$	−3.29590e6	−0.703086
$467$	393348.	0.0834612	0.0417306	−	0.999129i	$-0.486713\pi$
$467$	393348.	0.0834612	0.0417306	+	0.999129i	$0.486713\pi$
$468$	−1.72557e6	−0.364181
$469$	1.87113e6	0.392801
$470$	0	0
$471$	252362.	0.0524169
$472$	−2.17344e6	−0.449048
$473$	6.09479e6	1.25258
$474$	3.33652e6	0.682100
$475$	0	0
$476$	61344.0	0.0124095
$477$	−416508.	−0.0838161
$478$	2.22384e6	0.445178
$479$	−9.17697e6	−1.82751	−0.913757	−	0.406262i	$-0.866832\pi$
$479$	−9.17697e6	−1.82751	−0.913757	+	0.406262i	$0.866832\pi$
$480$	0	0
$481$	1.16494e7	2.29583
$482$	−2.09431e6	−0.410604
$483$	2.89595e6	0.564837
$484$	7.08285e6	1.37434
$485$	0	0
$486$	3.26374e6	0.626795
$487$	6.60598e6	1.26216	0.631080	−	0.775717i	$-0.282612\pi$
$487$	6.60598e6	1.26216	0.631080	+	0.775717i	$0.282612\pi$
$488$	−3.03373e6	−0.576670
$489$	3.69865e6	0.699473
$490$	0	0
$491$	38052.0	0.00712318	0.00356159	−	0.999994i	$-0.498866\pi$
$491$	38052.0	0.00712318	0.00356159	+	0.999994i	$0.498866\pi$
$492$	2.66165e6	0.495722
$493$	132840.	0.0246157
$494$	−4.04872e6	−0.746449
$495$	0	0
$496$	461312.	0.0841958
$497$	1.07182e6	0.194639
$498$	2.03716e6	0.368087
$499$	6.85670e6	1.23272	0.616359	−	0.787465i	$-0.288607\pi$
$499$	6.85670e6	1.23272	0.616359	+	0.787465i	$0.288607\pi$
$500$	0	0
$501$	−657228.	−0.116983
$502$	−452508.	−0.0801433
$503$	8.20016e6	1.44512	0.722558	−	0.691311i	$-0.242966\pi$
$503$	8.20016e6	1.44512	0.722558	+	0.691311i	$0.242966\pi$
$504$	−1.10874e6	−0.194425
$505$	0	0
$506$	−5.76223e6	−1.00049
$507$	−4.51179e6	−0.779524
$508$	2.26141e6	0.388794
$509$	4.06581e6	0.695589	0.347794	−	0.937571i	$-0.386931\pi$
$509$	4.06581e6	0.695589	0.347794	+	0.937571i	$0.386931\pi$
$510$	0	0
$511$	8.49458e6	1.43910
$512$	−262144.	−0.0441942
$513$	4.59718e6	0.771254
$514$	−3.49183e6	−0.582969
$515$	0	0
$516$	−1.38054e6	−0.228258
$517$	−5.23076e6	−0.860674
$518$	7.48510e6	1.22567
$519$	667656.	0.108801
$520$	0	0
$521$	5.28408e6	0.852854	0.426427	−	0.904522i	$-0.359772\pi$
$521$	5.28408e6	0.852854	0.426427	+	0.904522i	$0.359772\pi$
$522$	−2.40096e6	−0.385664
$523$	2.53383e6	0.405063	0.202532	−	0.979276i	$-0.435083\pi$
$523$	2.53383e6	0.405063	0.202532	+	0.979276i	$0.435083\pi$
$524$	1.28083e6	0.203781
$525$	0	0
$526$	6.58586e6	1.03788
$527$	−48654.0	−0.00763119
$528$	−2.18803e6	−0.341561
$529$	−2.99903e6	−0.465952
$530$	0	0
$531$	−4.14312e6	−0.637663
$532$	−2.60144e6	−0.398506
$533$	−1.33687e7	−2.03832
$534$	−1.34574e6	−0.204225
$535$	0	0
$536$	843328.	0.126790
$537$	87945.0	0.0131606
$538$	−7.15488e6	−1.06573
$539$	2.60839e6	0.386723
$540$	0	0
$541$	498752.	0.0732641	0.0366321	−	0.999329i	$-0.488337\pi$
$541$	498752.	0.0732641	0.0366321	+	0.999329i	$0.488337\pi$
$542$	−4.48561e6	−0.655878
$543$	5.00278e6	0.728135
$544$	27648.0	0.00400559
$545$	0	0
$546$	−5.52323e6	−0.792888
$547$	3.00269e6	0.429084	0.214542	−	0.976715i	$-0.431174\pi$
$547$	3.00269e6	0.429084	0.214542	+	0.976715i	$0.431174\pi$
$548$	516048.	0.0734072
$549$	−5.78304e6	−0.818890
$550$	0	0
$551$	−5.63340e6	−0.790481
$552$	1.30522e6	0.182320
$553$	−1.07679e7	−1.49733
$554$	2.39325e6	0.331294
$555$	0	0
$556$	6.31784e6	0.866726
$557$	−1.27373e7	−1.73956	−0.869779	−	0.493441i	$-0.835739\pi$
$557$	−1.27373e7	−1.73956	−0.869779	+	0.493441i	$0.835739\pi$
$558$	879376.	0.119561
$559$	6.93410e6	0.938556
$560$	0	0
$561$	230769.	0.0309578
$562$	6.12199e6	0.817621
$563$	−5.97082e6	−0.793894	−0.396947	−	0.917841i	$-0.629930\pi$
$563$	−5.97082e6	−0.793894	−0.396947	+	0.917841i	$0.629930\pi$
$564$	1.18483e6	0.156841
$565$	0	0
$566$	7.18800e6	0.943121
$567$	2.06170e6	0.269319
$568$	483072.	0.0628262
$569$	−9.26906e6	−1.20020	−0.600102	−	0.799924i	$-0.704873\pi$
$569$	−9.26906e6	−1.20020	−0.600102	+	0.799924i	$0.704873\pi$
$570$	0	0
$571$	−3.89535e6	−0.499984	−0.249992	−	0.968248i	$-0.580428\pi$
$571$	−3.89535e6	−0.499984	−0.249992	+	0.968248i	$0.580428\pi$
$572$	1.09899e7	1.40444
$573$	4.71128e6	0.599449
$574$	−8.58986e6	−1.08819
$575$	0	0
$576$	−499712.	−0.0627572
$577$	7.29416e6	0.912086	0.456043	−	0.889958i	$-0.349266\pi$
$577$	7.29416e6	0.912086	0.456043	+	0.889958i	$0.349266\pi$
$578$	5.67651e6	0.706744
$579$	9.19084e6	1.13935
$580$	0	0
$581$	−6.57446e6	−0.808015
$582$	4.57679e6	0.560085
$583$	2.65268e6	0.323231
$584$	3.82854e6	0.464517
$585$	0	0
$586$	−3.01798e6	−0.363054
$587$	−8.72820e6	−1.04551	−0.522756	−	0.852482i	$-0.675096\pi$
$587$	−8.72820e6	−1.04551	−0.522756	+	0.852482i	$0.675096\pi$
$588$	−590832.	−0.0704727
$589$	2.06329e6	0.245060
$590$	0	0
$591$	−7.46150e6	−0.878734
$592$	3.37357e6	0.395626
$593$	−1.30963e7	−1.52937	−0.764683	−	0.644407i	$-0.777104\pi$
$593$	−1.30963e7	−1.52937	−0.764683	+	0.644407i	$0.777104\pi$
$594$	−1.24786e7	−1.45111
$595$	0	0
$596$	−7.86240e6	−0.906650
$597$	350900.	0.0402947
$598$	−6.55574e6	−0.749668
$599$	−1.30168e7	−1.48231	−0.741155	−	0.671334i	$-0.765721\pi$
$599$	−1.30168e7	−1.48231	−0.741155	+	0.671334i	$0.765721\pi$
$600$	0	0
$601$	−9.93997e6	−1.12253	−0.561266	−	0.827635i	$-0.689686\pi$
$601$	−9.93997e6	−1.12253	−0.561266	+	0.827635i	$0.689686\pi$
$602$	4.45539e6	0.501066
$603$	1.60759e6	0.180046
$604$	3.20643e6	0.357626
$605$	0	0
$606$	−1.05151e6	−0.116314
$607$	1.56438e7	1.72334	0.861670	−	0.507470i	$-0.169419\pi$
$607$	1.56438e7	1.72334	0.861670	+	0.507470i	$0.169419\pi$
$608$	−1.17248e6	−0.128631
$609$	−7.68504e6	−0.839659
$610$	0	0
$611$	−5.95109e6	−0.644901
$612$	52704.0	0.00568807
$613$	9.33793e6	1.00369	0.501845	−	0.864958i	$-0.332655\pi$
$613$	9.33793e6	1.00369	0.501845	+	0.864958i	$0.332655\pi$
$614$	7.86507e6	0.841941
$615$	0	0
$616$	7.06138e6	0.749786
$617$	−5.06680e6	−0.535823	−0.267911	−	0.963444i	$-0.586333\pi$
$617$	−5.06680e6	−0.535823	−0.267911	+	0.963444i	$0.586333\pi$
$618$	−995984.	−0.104901
$619$	1.37670e7	1.44415	0.722077	−	0.691813i	$-0.243188\pi$
$619$	1.37670e7	1.44415	0.722077	+	0.691813i	$0.243188\pi$
$620$	0	0
$621$	7.44381e6	0.774580
$622$	2.39719e6	0.248443
$623$	4.34307e6	0.448308
$624$	−2.48934e6	−0.255931
$625$	0	0
$626$	2.88546e6	0.294293
$627$	−9.78632e6	−0.994146
$628$	−367072.	−0.0371409
$629$	−355806.	−0.0358580
$630$	0	0
$631$	2.07060e6	0.207025	0.103513	−	0.994628i	$-0.466992\pi$
$631$	2.07060e6	0.207025	0.103513	+	0.994628i	$0.466992\pi$
$632$	−4.85312e6	−0.483313
$633$	4.66040e6	0.462290
$634$	−409392.	−0.0404498
$635$	0	0
$636$	−600864.	−0.0589025
$637$	2.96759e6	0.289771
$638$	1.52914e7	1.48729
$639$	920856.	0.0892153
$640$	0	0
$641$	−1.79114e7	−1.72181	−0.860903	−	0.508768i	$-0.830101\pi$
$641$	−1.79114e7	−1.72181	−0.860903	+	0.508768i	$0.830101\pi$
$642$	2.66785e6	0.255461
$643$	−1.71414e7	−1.63500	−0.817500	−	0.575929i	$-0.804641\pi$
$643$	−1.71414e7	−1.63500	−0.817500	+	0.575929i	$0.804641\pi$
$644$	−4.21229e6	−0.400224
$645$	0	0
$646$	123660.	0.0116586
$647$	−8.48773e6	−0.797133	−0.398567	−	0.917139i	$-0.630492\pi$
$647$	−8.48773e6	−0.797133	−0.398567	+	0.917139i	$0.630492\pi$
$648$	929216.	0.0869319
$649$	2.63869e7	2.45910
$650$	0	0
$651$	2.81472e6	0.260306
$652$	−5.37986e6	−0.495623
$653$	−2.45479e6	−0.225284	−0.112642	−	0.993636i	$-0.535931\pi$
$653$	−2.45479e6	−0.225284	−0.112642	+	0.993636i	$0.535931\pi$
$654$	−311960.	−0.0285203
$655$	0	0
$656$	−3.87149e6	−0.351252
$657$	7.29816e6	0.659630
$658$	−3.82378e6	−0.344293
$659$	−5.91557e6	−0.530619	−0.265309	−	0.964163i	$-0.585474\pi$
$659$	−5.91557e6	−0.530619	−0.265309	+	0.964163i	$0.585474\pi$
$660$	0	0
$661$	4.33095e6	0.385549	0.192775	−	0.981243i	$-0.438251\pi$
$661$	4.33095e6	0.385549	0.192775	+	0.981243i	$0.438251\pi$
$662$	−5.24191e6	−0.464884
$663$	262548.	0.0231966
$664$	−2.96314e6	−0.260814
$665$	0	0
$666$	6.43086e6	0.561803
$667$	−9.12168e6	−0.793890
$668$	955968.	0.0828900
$669$	−4.38024e6	−0.378384
$670$	0	0
$671$	3.68314e7	3.15799
$672$	−1.59949e6	−0.136634
$673$	−9.13985e6	−0.777860	−0.388930	−	0.921267i	$-0.627155\pi$
$673$	−9.13985e6	−0.777860	−0.388930	+	0.921267i	$0.627155\pi$
$674$	3.21759e6	0.272823
$675$	0	0
$676$	6.56261e6	0.552344
$677$	4.57229e6	0.383409	0.191704	−	0.981453i	$-0.438599\pi$
$677$	4.57229e6	0.383409	0.191704	+	0.981453i	$0.438599\pi$
$678$	−5.66900e6	−0.473623
$679$	−1.47706e7	−1.22948
$680$	0	0
$681$	1.38337e7	1.14307
$682$	−5.60062e6	−0.461078
$683$	1.53221e7	1.25681	0.628403	−	0.777888i	$-0.283709\pi$
$683$	1.53221e7	1.25681	0.628403	+	0.777888i	$0.283709\pi$
$684$	−2.23504e6	−0.182661
$685$	0	0
$686$	−7.63960e6	−0.619813
$687$	−2.24158e6	−0.181202
$688$	2.00806e6	0.161736
$689$	3.01798e6	0.242196
$690$	0	0
$691$	7.02548e6	0.559733	0.279866	−	0.960039i	$-0.409710\pi$
$691$	7.02548e6	0.559733	0.279866	+	0.960039i	$0.409710\pi$
$692$	−971136.	−0.0770930
$693$	1.34607e7	1.06472
$694$	1.15328e7	0.908944
$695$	0	0
$696$	−3.46368e6	−0.271028
$697$	408321.	0.0318361
$698$	−5.09420e6	−0.395765
$699$	−9.06371e6	−0.701638
$700$	0	0
$701$	7.91125e6	0.608065	0.304033	−	0.952662i	$-0.401667\pi$
$701$	7.91125e6	0.608065	0.304033	+	0.952662i	$0.401667\pi$
$702$	−1.41970e7	−1.08731
$703$	1.50888e7	1.15151
$704$	3.18259e6	0.242019
$705$	0	0
$706$	−1.13225e7	−0.854927
$707$	3.39352e6	0.255330
$708$	−5.97696e6	−0.448123
$709$	−1.54485e7	−1.15418	−0.577088	−	0.816682i	$-0.695811\pi$
$709$	−1.54485e7	−1.15418	−0.577088	+	0.816682i	$0.695811\pi$
$710$	0	0
$711$	−9.25126e6	−0.686320
$712$	1.95744e6	0.144707
$713$	3.34091e6	0.246116
$714$	168696.	0.0123840
$715$	0	0
$716$	−127920.	−0.00932515
$717$	6.11556e6	0.444261
$718$	3.92436e6	0.284091
$719$	2.30544e7	1.66315	0.831574	−	0.555414i	$-0.187440\pi$
$719$	2.30544e7	1.66315	0.831574	+	0.555414i	$0.187440\pi$
$720$	0	0
$721$	3.21431e6	0.230277
$722$	4.66030e6	0.332714
$723$	−5.75935e6	−0.409758
$724$	−7.27677e6	−0.515932
$725$	0	0
$726$	1.94778e7	1.37151
$727$	1.62905e7	1.14314	0.571568	−	0.820555i	$-0.306335\pi$
$727$	1.62905e7	1.14314	0.571568	+	0.820555i	$0.306335\pi$
$728$	8.03379e6	0.561813
$729$	1.25034e7	0.871384
$730$	0	0
$731$	−211788.	−0.0146591
$732$	−8.34275e6	−0.575482
$733$	1.28279e7	0.881853	0.440927	−	0.897543i	$-0.354650\pi$
$733$	1.28279e7	0.881853	0.440927	+	0.897543i	$0.354650\pi$
$734$	1.67642e7	1.14853
$735$	0	0
$736$	−1.89850e6	−0.129186
$737$	−1.02385e7	−0.694335
$738$	−7.38002e6	−0.498789
$739$	−1.24535e7	−0.838840	−0.419420	−	0.907792i	$-0.637766\pi$
$739$	−1.24535e7	−0.838840	−0.419420	+	0.907792i	$0.637766\pi$
$740$	0	0
$741$	−1.11340e7	−0.744912
$742$	1.93915e6	0.129301
$743$	−2.63247e7	−1.74941	−0.874705	−	0.484656i	$-0.838945\pi$
$743$	−2.63247e7	−1.74941	−0.874705	+	0.484656i	$0.838945\pi$
$744$	1.26861e6	0.0840224
$745$	0	0
$746$	−1.29382e7	−0.851192
$747$	−5.64848e6	−0.370365
$748$	−335664.	−0.0219357
$749$	−8.60989e6	−0.560780
$750$	0	0
$751$	−1.74994e7	−1.13220	−0.566102	−	0.824335i	$-0.691549\pi$
$751$	−1.74994e7	−1.13220	−0.566102	+	0.824335i	$0.691549\pi$
$752$	−1.72339e6	−0.111132
$753$	−1.24440e6	−0.0799782
$754$	1.73971e7	1.11442
$755$	0	0
$756$	−9.12208e6	−0.580483
$757$	3.46381e6	0.219692	0.109846	−	0.993949i	$-0.464964\pi$
$757$	3.46381e6	0.219692	0.109846	+	0.993949i	$0.464964\pi$
$758$	−5.56142e6	−0.351571
$759$	−1.58461e7	−0.998433
$760$	0	0
$761$	−1.26175e7	−0.789792	−0.394896	−	0.918726i	$-0.629219\pi$
$761$	−1.26175e7	−0.789792	−0.394896	+	0.918726i	$0.629219\pi$
$762$	6.21887e6	0.387993
$763$	1.00678e6	0.0626070
$764$	−6.85277e6	−0.424749
$765$	0	0
$766$	−4.56790e6	−0.281284
$767$	3.00206e7	1.84260
$768$	−720896.	−0.0441031
$769$	5.70804e6	0.348074	0.174037	−	0.984739i	$-0.444319\pi$
$769$	5.70804e6	0.348074	0.174037	+	0.984739i	$0.444319\pi$
$770$	0	0
$771$	−9.60254e6	−0.581768
$772$	−1.33685e7	−0.807308
$773$	1.20827e7	0.727303	0.363652	−	0.931535i	$-0.381530\pi$
$773$	1.20827e7	0.727303	0.363652	+	0.931535i	$0.381530\pi$
$774$	3.82787e6	0.229670
$775$	0	0
$776$	−6.65715e6	−0.396857
$777$	2.05840e7	1.22315
$778$	−1.38520e7	−0.820469
$779$	−1.73158e7	−1.02235
$780$	0	0
$781$	−5.86480e6	−0.344053
$782$	200232.	0.0117089
$783$	−1.97538e7	−1.15145
$784$	859392.	0.0499346
$785$	0	0
$786$	3.52229e6	0.203361
$787$	−1.37636e7	−0.792126	−0.396063	−	0.918223i	$-0.629624\pi$
$787$	−1.37636e7	−0.792126	−0.396063	+	0.918223i	$0.629624\pi$
$788$	1.08531e7	0.622641
$789$	1.81111e7	1.03575
$790$	0	0
$791$	1.82954e7	1.03968
$792$	6.06682e6	0.343675
$793$	4.19034e7	2.36628
$794$	−2.37603e7	−1.33752
$795$	0	0
$796$	−510400.	−0.0285514
$797$	−8.77738e6	−0.489462	−0.244731	−	0.969591i	$-0.578700\pi$
$797$	−8.77738e6	−0.489462	−0.244731	+	0.969591i	$0.578700\pi$
$798$	−7.15396e6	−0.397685
$799$	181764.	0.0100726
$800$	0	0
$801$	3.73137e6	0.205488
$802$	9.09649e6	0.499388
$803$	−4.64809e7	−2.54382
$804$	2.31915e6	0.126529
$805$	0	0
$806$	−6.37187e6	−0.345485
$807$	−1.96759e7	−1.06353
$808$	1.52947e6	0.0824163
$809$	−1.02046e7	−0.548181	−0.274091	−	0.961704i	$-0.588377\pi$
$809$	−1.02046e7	−0.548181	−0.274091	+	0.961704i	$0.588377\pi$
$810$	0	0
$811$	−1.17375e6	−0.0626647	−0.0313323	−	0.999509i	$-0.509975\pi$
$811$	−1.17375e6	−0.0626647	−0.0313323	+	0.999509i	$0.509975\pi$
$812$	1.11782e7	0.594954
$813$	−1.23354e7	−0.654527
$814$	−4.09572e7	−2.16655
$815$	0	0
$816$	76032.0	0.00399734
$817$	8.98138e6	0.470747
$818$	1.71821e7	0.897826
$819$	1.53144e7	0.797794
$820$	0	0
$821$	−1.98062e7	−1.02552	−0.512759	−	0.858533i	$-0.671377\pi$
$821$	−1.98062e7	−1.02552	−0.512759	+	0.858533i	$0.671377\pi$
$822$	1.41913e6	0.0732560
$823$	−3.06722e7	−1.57850	−0.789251	−	0.614070i	$-0.789531\pi$
$823$	−3.06722e7	−1.57850	−0.789251	+	0.614070i	$0.789531\pi$
$824$	1.44870e6	0.0743296
$825$	0	0
$826$	1.92893e7	0.983707
$827$	2.55520e7	1.29915	0.649577	−	0.760296i	$-0.274946\pi$
$827$	2.55520e7	1.29915	0.649577	+	0.760296i	$0.274946\pi$
$828$	−3.61901e6	−0.183448
$829$	−9.19402e6	−0.464643	−0.232321	−	0.972639i	$-0.574632\pi$
$829$	−9.19402e6	−0.464643	−0.232321	+	0.972639i	$0.574632\pi$
$830$	0	0
$831$	6.58143e6	0.330611
$832$	3.62086e6	0.181344
$833$	−90639.0	−0.00452588
$834$	1.73741e7	0.864940
$835$	0	0
$836$	1.42346e7	0.704419
$837$	7.23503e6	0.356966
$838$	−7.18818e6	−0.353597
$839$	−1.56910e7	−0.769564	−0.384782	−	0.923008i	$-0.625723\pi$
$839$	−1.56910e7	−0.769564	−0.384782	+	0.923008i	$0.625723\pi$
$840$	0	0
$841$	3.69525e6	0.180158
$842$	1.03019e6	0.0500770
$843$	1.68355e7	0.815937
$844$	−6.77877e6	−0.327563
$845$	0	0
$846$	−3.28522e6	−0.157811
$847$	−6.28603e7	−3.01070
$848$	873984.	0.0417363
$849$	1.97670e7	0.941178
$850$	0	0
$851$	2.44320e7	1.15647
$852$	1.32845e6	0.0626968
$853$	1.60111e6	0.0753442	0.0376721	−	0.999290i	$-0.488006\pi$
$853$	1.60111e6	0.0753442	0.0376721	+	0.999290i	$0.488006\pi$
$854$	2.69243e7	1.26328
$855$	0	0
$856$	−3.88051e6	−0.181011
$857$	−1.64613e7	−0.765616	−0.382808	−	0.923828i	$-0.625043\pi$
$857$	−1.64613e7	−0.765616	−0.382808	+	0.923828i	$0.625043\pi$
$858$	3.02222e7	1.40155
$859$	1.96736e7	0.909705	0.454853	−	0.890567i	$-0.349692\pi$
$859$	1.96736e7	0.909705	0.454853	+	0.890567i	$0.349692\pi$
$860$	0	0
$861$	−2.36221e7	−1.08595
$862$	−8.91641e6	−0.408716
$863$	3.68068e7	1.68229	0.841145	−	0.540810i	$-0.181882\pi$
$863$	3.68068e7	1.68229	0.841145	+	0.540810i	$0.181882\pi$
$864$	−4.11136e6	−0.187370
$865$	0	0
$866$	1.68234e7	0.762288
$867$	1.56104e7	0.705288
$868$	−4.09414e6	−0.184444
$869$	5.89199e7	2.64675
$870$	0	0
$871$	−1.16485e7	−0.520264
$872$	453760.	0.0202085
$873$	−1.26902e7	−0.563550
$874$	−8.49132e6	−0.376008
$875$	0	0
$876$	1.05285e7	0.463560
$877$	−2.69596e7	−1.18363	−0.591813	−	0.806075i	$-0.701588\pi$
$877$	−2.69596e7	−1.18363	−0.591813	+	0.806075i	$0.701588\pi$
$878$	−1.41056e6	−0.0617526
$879$	−8.29943e6	−0.362307
$880$	0	0
$881$	3.47335e7	1.50768	0.753839	−	0.657059i	$-0.228200\pi$
$881$	3.47335e7	1.50768	0.753839	+	0.657059i	$0.228200\pi$
$882$	1.63822e6	0.0709087
$883$	2.16187e7	0.933101	0.466551	−	0.884494i	$-0.345497\pi$
$883$	2.16187e7	0.933101	0.466551	+	0.884494i	$0.345497\pi$
$884$	−381888.	−0.0164363
$885$	0	0
$886$	−5.13448e6	−0.219741
$887$	4.48163e6	0.191261	0.0956306	−	0.995417i	$-0.469513\pi$
$887$	4.48163e6	0.191261	0.0956306	+	0.995417i	$0.469513\pi$
$888$	9.27731e6	0.394811
$889$	−2.00700e7	−0.851712
$890$	0	0
$891$	−1.12813e7	−0.476062
$892$	6.37126e6	0.268110
$893$	−7.70814e6	−0.323460
$894$	−2.16216e7	−0.904782
$895$	0	0
$896$	2.32653e6	0.0968140
$897$	−1.80283e7	−0.748124
$898$	8.41590e6	0.348265
$899$	−8.86584e6	−0.365865
$900$	0	0
$901$	−92178.0	−0.00378282
$902$	4.70023e7	1.92355
$903$	1.22523e7	0.500034
$904$	8.24582e6	0.335593
$905$	0	0
$906$	8.81769e6	0.356890
$907$	−3.36639e7	−1.35877	−0.679385	−	0.733782i	$-0.737753\pi$
$907$	−3.36639e7	−1.35877	−0.679385	+	0.733782i	$0.737753\pi$
$908$	−2.01218e7	−0.809938
$909$	2.91556e6	0.117034
$910$	0	0
$911$	−1.03175e6	−0.0411887	−0.0205943	−	0.999788i	$-0.506556\pi$
$911$	−1.03175e6	−0.0411887	−0.0205943	+	0.999788i	$0.506556\pi$
$912$	−3.22432e6	−0.128366
$913$	3.59743e7	1.42829
$914$	−3.30093e6	−0.130699
$915$	0	0
$916$	3.26048e6	0.128393
$917$	−1.13674e7	−0.446413
$918$	433620.	0.0169825
$919$	4.10147e6	0.160196	0.0800978	−	0.996787i	$-0.474477\pi$
$919$	4.10147e6	0.160196	0.0800978	+	0.996787i	$0.474477\pi$
$920$	0	0
$921$	2.16289e7	0.840207
$922$	−1.80058e7	−0.697566
$923$	−6.67243e6	−0.257798
$924$	1.94188e7	0.748242
$925$	0	0
$926$	5.76846e6	0.221071
$927$	2.76159e6	0.105550
$928$	5.03808e6	0.192042
$929$	7.71603e6	0.293329	0.146664	−	0.989186i	$-0.453146\pi$
$929$	7.71603e6	0.293329	0.146664	+	0.989186i	$0.453146\pi$
$930$	0	0
$931$	3.84376e6	0.145339
$932$	1.31836e7	0.497157
$933$	6.59228e6	0.247931
$934$	−1.57339e6	−0.0590160
$935$	0	0
$936$	6.90227e6	0.257515
$937$	4.38458e7	1.63147	0.815735	−	0.578426i	$-0.196333\pi$
$937$	4.38458e7	1.63147	0.815735	+	0.578426i	$0.196333\pi$
$938$	−7.48454e6	−0.277752
$939$	7.93503e6	0.293687
$940$	0	0
$941$	−1.00215e7	−0.368944	−0.184472	−	0.982838i	$-0.559058\pi$
$941$	−1.00215e7	−0.368944	−0.184472	+	0.982838i	$0.559058\pi$
$942$	−1.00945e6	−0.0370644
$943$	−2.80380e7	−1.02676
$944$	8.69376e6	0.317525
$945$	0	0
$946$	−2.43792e7	−0.885708
$947$	2.79530e7	1.01287	0.506434	−	0.862279i	$-0.330963\pi$
$947$	2.79530e7	1.01287	0.506434	+	0.862279i	$0.330963\pi$
$948$	−1.33461e7	−0.482317
$949$	−5.28818e7	−1.90608
$950$	0	0
$951$	−1.12583e6	−0.0403665
$952$	−245376.	−0.00877485
$953$	−2.31811e7	−0.826803	−0.413401	−	0.910549i	$-0.635659\pi$
$953$	−2.31811e7	−0.826803	−0.413401	+	0.910549i	$0.635659\pi$
$954$	1.66603e6	0.0592669
$955$	0	0
$956$	−8.89536e6	−0.314788
$957$	4.20512e7	1.48422
$958$	3.67079e7	1.29225
$959$	−4.57993e6	−0.160810
$960$	0	0
$961$	−2.53819e7	−0.886577
$962$	−4.65974e7	−1.62339
$963$	−7.39723e6	−0.257041
$964$	8.37723e6	0.290341
$965$	0	0
$966$	−1.15838e7	−0.399400
$967$	−1.58435e6	−0.0544861	−0.0272430	−	0.999629i	$-0.508673\pi$
$967$	−1.58435e6	−0.0544861	−0.0272430	+	0.999629i	$0.508673\pi$
$968$	−2.83314e7	−0.971806
$969$	340065.	0.0116346
$970$	0	0
$971$	3.44552e7	1.17275	0.586376	−	0.810039i	$-0.300554\pi$
$971$	3.44552e7	1.17275	0.586376	+	0.810039i	$0.300554\pi$
$972$	−1.30550e7	−0.443211
$973$	−5.60708e7	−1.89869
$974$	−2.64239e7	−0.892483
$975$	0	0
$976$	1.21349e7	0.407767
$977$	−2.93599e7	−0.984052	−0.492026	−	0.870581i	$-0.663743\pi$
$977$	−2.93599e7	−0.984052	−0.492026	+	0.870581i	$0.663743\pi$
$978$	−1.47946e7	−0.494602
$979$	−2.37645e7	−0.792452
$980$	0	0
$981$	864980.	0.0286968
$982$	−152208.	−0.00503685
$983$	8.93957e6	0.295075	0.147538	−	0.989056i	$-0.452865\pi$
$983$	8.93957e6	0.295075	0.147538	+	0.989056i	$0.452865\pi$
$984$	−1.06466e7	−0.350528
$985$	0	0
$986$	−531360.	−0.0174059
$987$	−1.05154e7	−0.343583
$988$	1.61949e7	0.527819
$989$	1.45428e7	0.472777
$990$	0	0
$991$	−1.78899e7	−0.578660	−0.289330	−	0.957229i	$-0.593433\pi$
$991$	−1.78899e7	−0.578660	−0.289330	+	0.957229i	$0.593433\pi$
$992$	−1.84525e6	−0.0595354
$993$	−1.44152e7	−0.463926
$994$	−4.28726e6	−0.137630
$995$	0	0
$996$	−8.14862e6	−0.260277
$997$	3.30517e7	1.05307	0.526533	−	0.850155i	$-0.323492\pi$
$997$	3.30517e7	1.05307	0.526533	+	0.850155i	$0.323492\pi$
$998$	−2.74268e7	−0.871663
$999$	5.29097e7	1.67734

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.6.a.a.1.1	✓	1
3.2	odd	2		450.6.a.n.1.1		1
4.3	odd	2		400.6.a.j.1.1		1
5.2	odd	4		50.6.b.c.49.1		2
5.3	odd	4		50.6.b.c.49.2		2
5.4	even	2		50.6.a.f.1.1	yes	1
15.2	even	4		450.6.c.a.199.2		2
15.8	even	4		450.6.c.a.199.1		2
15.14	odd	2		450.6.a.j.1.1		1
20.3	even	4		400.6.c.g.49.2		2
20.7	even	4		400.6.c.g.49.1		2
20.19	odd	2		400.6.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.6.a.a.1.1	✓	1	1.1	even	1	trivial
50.6.a.f.1.1	yes	1	5.4	even	2
50.6.b.c.49.1		2	5.2	odd	4
50.6.b.c.49.2		2	5.3	odd	4
400.6.a.e.1.1		1	20.19	odd	2
400.6.a.j.1.1		1	4.3	odd	2
400.6.c.g.49.1		2	20.7	even	4
400.6.c.g.49.2		2	20.3	even	4
450.6.a.j.1.1		1	15.14	odd	2
450.6.a.n.1.1		1	3.2	odd	2
450.6.c.a.199.1		2	15.8	even	4
450.6.c.a.199.2		2	15.2	even	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:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	50.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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