LMFDB - Embedded newform 825.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.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:	$132.316651346$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+2.00000 q^{2} +9.00000 q^{3} -28.0000 q^{4} +18.0000 q^{6} -148.000 q^{7} -120.000 q^{8} +81.0000 q^{9} +121.000 q^{11} -252.000 q^{12} -574.000 q^{13} -296.000 q^{14} +656.000 q^{16} +722.000 q^{17} +162.000 q^{18} +2160.00 q^{19} -1332.00 q^{21} +242.000 q^{22} +2536.00 q^{23} -1080.00 q^{24} -1148.00 q^{26} +729.000 q^{27} +4144.00 q^{28} +4650.00 q^{29} +5032.00 q^{31} +5152.00 q^{32} +1089.00 q^{33} +1444.00 q^{34} -2268.00 q^{36} -8118.00 q^{37} +4320.00 q^{38} -5166.00 q^{39} -5138.00 q^{41} -2664.00 q^{42} -8304.00 q^{43} -3388.00 q^{44} +5072.00 q^{46} -24728.0 q^{47} +5904.00 q^{48} +5097.00 q^{49} +6498.00 q^{51} +16072.0 q^{52} +28746.0 q^{53} +1458.00 q^{54} +17760.0 q^{56} +19440.0 q^{57} +9300.00 q^{58} -5860.00 q^{59} -53658.0 q^{61} +10064.0 q^{62} -11988.0 q^{63} -10688.0 q^{64} +2178.00 q^{66} -30908.0 q^{67} -20216.0 q^{68} +22824.0 q^{69} -69648.0 q^{71} -9720.00 q^{72} +18446.0 q^{73} -16236.0 q^{74} -60480.0 q^{76} -17908.0 q^{77} -10332.0 q^{78} -25300.0 q^{79} +6561.00 q^{81} -10276.0 q^{82} +17556.0 q^{83} +37296.0 q^{84} -16608.0 q^{86} +41850.0 q^{87} -14520.0 q^{88} +132570. q^{89} +84952.0 q^{91} -71008.0 q^{92} +45288.0 q^{93} -49456.0 q^{94} +46368.0 q^{96} -70658.0 q^{97} +10194.0 q^{98} +9801.00 q^{99} +O(q^{100})\)

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	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	2.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−28.0000	−0.875000
$5$	0	0
$5$	0	0
$6$	18.0000	0.204124
$7$	−148.000	−1.14161	−0.570803	−	0.821087i	$-0.693368\pi$
$7$	−148.000	−1.14161	−0.570803	+	0.821087i	$0.693368\pi$
$8$	−120.000	−0.662913
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−252.000	−0.505181
$13$	−574.000	−0.942006	−0.471003	−	0.882132i	$-0.656108\pi$
$13$	−574.000	−0.942006	−0.471003	+	0.882132i	$0.656108\pi$
$14$	−296.000	−0.403619
$15$	0	0
$16$	656.000	0.640625
$17$	722.000	0.605919	0.302960	−	0.953003i	$-0.402025\pi$
$17$	722.000	0.605919	0.302960	+	0.953003i	$0.402025\pi$
$18$	162.000	0.117851
$19$	2160.00	1.37268	0.686341	−	0.727280i	$-0.259216\pi$
$19$	2160.00	1.37268	0.686341	+	0.727280i	$0.259216\pi$
$20$	0	0
$21$	−1332.00	−0.659107
$22$	242.000	0.106600
$23$	2536.00	0.999608	0.499804	−	0.866139i	$-0.333405\pi$
$23$	2536.00	0.999608	0.499804	+	0.866139i	$0.333405\pi$
$24$	−1080.00	−0.382733
$25$	0	0
$26$	−1148.00	−0.333049
$27$	729.000	0.192450
$28$	4144.00	0.998906
$29$	4650.00	1.02673	0.513367	−	0.858169i	$-0.328398\pi$
$29$	4650.00	1.02673	0.513367	+	0.858169i	$0.328398\pi$
$30$	0	0
$31$	5032.00	0.940451	0.470226	−	0.882546i	$-0.344172\pi$
$31$	5032.00	0.940451	0.470226	+	0.882546i	$0.344172\pi$
$32$	5152.00	0.889408
$33$	1089.00	0.174078
$34$	1444.00	0.214225
$35$	0	0
$36$	−2268.00	−0.291667
$37$	−8118.00	−0.974866	−0.487433	−	0.873161i	$-0.662067\pi$
$37$	−8118.00	−0.974866	−0.487433	+	0.873161i	$0.662067\pi$
$38$	4320.00	0.485316
$39$	−5166.00	−0.543867
$40$	0	0
$41$	−5138.00	−0.477347	−0.238674	−	0.971100i	$-0.576713\pi$
$41$	−5138.00	−0.477347	−0.238674	+	0.971100i	$0.576713\pi$
$42$	−2664.00	−0.233030
$43$	−8304.00	−0.684883	−0.342441	−	0.939539i	$-0.611254\pi$
$43$	−8304.00	−0.684883	−0.342441	+	0.939539i	$0.611254\pi$
$44$	−3388.00	−0.263822
$45$	0	0
$46$	5072.00	0.353415
$47$	−24728.0	−1.63284	−0.816421	−	0.577457i	$-0.804045\pi$
$47$	−24728.0	−1.63284	−0.816421	+	0.577457i	$0.804045\pi$
$48$	5904.00	0.369865
$49$	5097.00	0.303266
$50$	0	0
$51$	6498.00	0.349828
$52$	16072.0	0.824255
$53$	28746.0	1.40568	0.702842	−	0.711346i	$-0.251914\pi$
$53$	28746.0	1.40568	0.702842	+	0.711346i	$0.251914\pi$
$54$	1458.00	0.0680414
$55$	0	0
$56$	17760.0	0.756786
$57$	19440.0	0.792518
$58$	9300.00	0.363005
$59$	−5860.00	−0.219163	−0.109582	−	0.993978i	$-0.534951\pi$
$59$	−5860.00	−0.219163	−0.109582	+	0.993978i	$0.534951\pi$
$60$	0	0
$61$	−53658.0	−1.84633	−0.923166	−	0.384401i	$-0.874408\pi$
$61$	−53658.0	−1.84633	−0.923166	+	0.384401i	$0.874408\pi$
$62$	10064.0	0.332500
$63$	−11988.0	−0.380536
$64$	−10688.0	−0.326172
$65$	0	0
$66$	2178.00	0.0615457
$67$	−30908.0	−0.841170	−0.420585	−	0.907253i	$-0.638175\pi$
$67$	−30908.0	−0.841170	−0.420585	+	0.907253i	$0.638175\pi$
$68$	−20216.0	−0.530180
$69$	22824.0	0.577124
$70$	0	0
$71$	−69648.0	−1.63969	−0.819847	−	0.572583i	$-0.805942\pi$
$71$	−69648.0	−1.63969	−0.819847	+	0.572583i	$0.805942\pi$
$72$	−9720.00	−0.220971
$73$	18446.0	0.405131	0.202565	−	0.979269i	$-0.435072\pi$
$73$	18446.0	0.405131	0.202565	+	0.979269i	$0.435072\pi$
$74$	−16236.0	−0.344667
$75$	0	0
$76$	−60480.0	−1.20110
$77$	−17908.0	−0.344207
$78$	−10332.0	−0.192286
$79$	−25300.0	−0.456092	−0.228046	−	0.973650i	$-0.573234\pi$
$79$	−25300.0	−0.456092	−0.228046	+	0.973650i	$0.573234\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−10276.0	−0.168768
$83$	17556.0	0.279724	0.139862	−	0.990171i	$-0.455334\pi$
$83$	17556.0	0.279724	0.139862	+	0.990171i	$0.455334\pi$
$84$	37296.0	0.576719
$85$	0	0
$86$	−16608.0	−0.242143
$87$	41850.0	0.592785
$88$	−14520.0	−0.199876
$89$	132570.	1.77407	0.887034	−	0.461704i	$-0.152762\pi$
$89$	132570.	1.77407	0.887034	+	0.461704i	$0.152762\pi$
$90$	0	0
$91$	84952.0	1.07540
$92$	−71008.0	−0.874657
$93$	45288.0	0.542970
$94$	−49456.0	−0.577297
$95$	0	0
$96$	46368.0	0.513500
$97$	−70658.0	−0.762486	−0.381243	−	0.924475i	$-0.624504\pi$
$97$	−70658.0	−0.762486	−0.381243	+	0.924475i	$0.624504\pi$
$98$	10194.0	0.107221
$99$	9801.00	0.100504
$100$	0	0
$101$	−101998.	−0.994920	−0.497460	−	0.867487i	$-0.665734\pi$
$101$	−101998.	−0.994920	−0.497460	+	0.867487i	$0.665734\pi$
$102$	12996.0	0.123683
$103$	−130904.	−1.21579	−0.607897	−	0.794016i	$-0.707987\pi$
$103$	−130904.	−1.21579	−0.607897	+	0.794016i	$0.707987\pi$
$104$	68880.0	0.624467
$105$	0	0
$106$	57492.0	0.496984
$107$	141612.	1.19575	0.597875	−	0.801589i	$-0.296012\pi$
$107$	141612.	1.19575	0.597875	+	0.801589i	$0.296012\pi$
$108$	−20412.0	−0.168394
$109$	−239810.	−1.93331	−0.966654	−	0.256086i	$-0.917567\pi$
$109$	−239810.	−1.93331	−0.966654	+	0.256086i	$0.917567\pi$
$110$	0	0
$111$	−73062.0	−0.562839
$112$	−97088.0	−0.731342
$113$	42726.0	0.314772	0.157386	−	0.987537i	$-0.449693\pi$
$113$	42726.0	0.314772	0.157386	+	0.987537i	$0.449693\pi$
$114$	38880.0	0.280197
$115$	0	0
$116$	−130200.	−0.898392
$117$	−46494.0	−0.314002
$118$	−11720.0	−0.0774859
$119$	−106856.	−0.691722
$120$	0	0
$121$	14641.0	0.0909091
$122$	−107316.	−0.652777
$123$	−46242.0	−0.275597
$124$	−140896.	−0.822895
$125$	0	0
$126$	−23976.0	−0.134540
$127$	−51788.0	−0.284918	−0.142459	−	0.989801i	$-0.545501\pi$
$127$	−51788.0	−0.284918	−0.142459	+	0.989801i	$0.545501\pi$
$128$	−186240.	−1.00473
$129$	−74736.0	−0.395417
$130$	0	0
$131$	53652.0	0.273154	0.136577	−	0.990629i	$-0.456390\pi$
$131$	53652.0	0.273154	0.136577	+	0.990629i	$0.456390\pi$
$132$	−30492.0	−0.152318
$133$	−319680.	−1.56706
$134$	−61816.0	−0.297399
$135$	0	0
$136$	−86640.0	−0.401672
$137$	228862.	1.04177	0.520886	−	0.853627i	$-0.325602\pi$
$137$	228862.	1.04177	0.520886	+	0.853627i	$0.325602\pi$
$138$	45648.0	0.204044
$139$	374920.	1.64589	0.822947	−	0.568119i	$-0.192329\pi$
$139$	374920.	1.64589	0.822947	+	0.568119i	$0.192329\pi$
$140$	0	0
$141$	−222552.	−0.942722
$142$	−139296.	−0.579719
$143$	−69454.0	−0.284025
$144$	53136.0	0.213542
$145$	0	0
$146$	36892.0	0.143235
$147$	45873.0	0.175091
$148$	227304.	0.853007
$149$	−65830.0	−0.242917	−0.121459	−	0.992597i	$-0.538757\pi$
$149$	−65830.0	−0.242917	−0.121459	+	0.992597i	$0.538757\pi$
$150$	0	0
$151$	154052.	0.549826	0.274913	−	0.961469i	$-0.411351\pi$
$151$	154052.	0.549826	0.274913	+	0.961469i	$0.411351\pi$
$152$	−259200.	−0.909968
$153$	58482.0	0.201973
$154$	−35816.0	−0.121696
$155$	0	0
$156$	144648.	0.475884
$157$	−287678.	−0.931446	−0.465723	−	0.884931i	$-0.654206\pi$
$157$	−287678.	−0.931446	−0.465723	+	0.884931i	$0.654206\pi$
$158$	−50600.0	−0.161253
$159$	258714.	0.811572
$160$	0	0
$161$	−375328.	−1.14116
$162$	13122.0	0.0392837
$163$	−105124.	−0.309908	−0.154954	−	0.987922i	$-0.549523\pi$
$163$	−105124.	−0.309908	−0.154954	+	0.987922i	$0.549523\pi$
$164$	143864.	0.417679
$165$	0	0
$166$	35112.0	0.0988975
$167$	−150528.	−0.417663	−0.208832	−	0.977952i	$-0.566966\pi$
$167$	−150528.	−0.417663	−0.208832	+	0.977952i	$0.566966\pi$
$168$	159840.	0.436930
$169$	−41817.0	−0.112625
$170$	0	0
$171$	174960.	0.457560
$172$	232512.	0.599272
$173$	2166.00	0.00550229	0.00275114	−	0.999996i	$-0.499124\pi$
$173$	2166.00	0.00550229	0.00275114	+	0.999996i	$0.499124\pi$
$174$	83700.0	0.209581
$175$	0	0
$176$	79376.0	0.193156
$177$	−52740.0	−0.126534
$178$	265140.	0.627228
$179$	672780.	1.56942	0.784712	−	0.619860i	$-0.212811\pi$
$179$	672780.	1.56942	0.784712	+	0.619860i	$0.212811\pi$
$180$	0	0
$181$	−526778.	−1.19517	−0.597587	−	0.801804i	$-0.703874\pi$
$181$	−526778.	−1.19517	−0.597587	+	0.801804i	$0.703874\pi$
$182$	169904.	0.380211
$183$	−482922.	−1.06598
$184$	−304320.	−0.662653
$185$	0	0
$186$	90576.0	0.191969
$187$	87362.0	0.182692
$188$	692384.	1.42874
$189$	−107892.	−0.219702
$190$	0	0
$191$	−305608.	−0.606152	−0.303076	−	0.952966i	$-0.598014\pi$
$191$	−305608.	−0.606152	−0.303076	+	0.952966i	$0.598014\pi$
$192$	−96192.0	−0.188315
$193$	−116434.	−0.225002	−0.112501	−	0.993652i	$-0.535886\pi$
$193$	−116434.	−0.225002	−0.112501	+	0.993652i	$0.535886\pi$
$194$	−141316.	−0.269580
$195$	0	0
$196$	−142716.	−0.265358
$197$	247742.	0.454814	0.227407	−	0.973800i	$-0.426975\pi$
$197$	247742.	0.454814	0.227407	+	0.973800i	$0.426975\pi$
$198$	19602.0	0.0355335
$199$	−513360.	−0.918945	−0.459472	−	0.888192i	$-0.651961\pi$
$199$	−513360.	−0.918945	−0.459472	+	0.888192i	$0.651961\pi$
$200$	0	0
$201$	−278172.	−0.485650
$202$	−203996.	−0.351757
$203$	−688200.	−1.17213
$204$	−181944.	−0.306099
$205$	0	0
$206$	−261808.	−0.429848
$207$	205416.	0.333203
$208$	−376544.	−0.603472
$209$	261360.	0.413879
$210$	0	0
$211$	−620688.	−0.959770	−0.479885	−	0.877331i	$-0.659322\pi$
$211$	−620688.	−0.959770	−0.479885	+	0.877331i	$0.659322\pi$
$212$	−804888.	−1.22997
$213$	−626832.	−0.946678
$214$	283224.	0.422762
$215$	0	0
$216$	−87480.0	−0.127578
$217$	−744736.	−1.07363
$218$	−479620.	−0.683528
$219$	166014.	0.233902
$220$	0	0
$221$	−414428.	−0.570780
$222$	−146124.	−0.198994
$223$	1.31802e6	1.77484	0.887419	−	0.460964i	$-0.152496\pi$
$223$	1.31802e6	1.77484	0.887419	+	0.460964i	$0.152496\pi$
$224$	−762496.	−1.01535
$225$	0	0
$226$	85452.0	0.111289
$227$	887412.	1.14304	0.571519	−	0.820589i	$-0.306354\pi$
$227$	887412.	1.14304	0.571519	+	0.820589i	$0.306354\pi$
$228$	−544320.	−0.693453
$229$	−237450.	−0.299215	−0.149608	−	0.988745i	$-0.547801\pi$
$229$	−237450.	−0.299215	−0.149608	+	0.988745i	$0.547801\pi$
$230$	0	0
$231$	−161172.	−0.198728
$232$	−558000.	−0.680635
$233$	914706.	1.10380	0.551902	−	0.833909i	$-0.313902\pi$
$233$	914706.	1.10380	0.551902	+	0.833909i	$0.313902\pi$
$234$	−92988.0	−0.111016
$235$	0	0
$236$	164080.	0.191768
$237$	−227700.	−0.263325
$238$	−213712.	−0.244561
$239$	1.40892e6	1.59548	0.797740	−	0.603001i	$-0.206029\pi$
$239$	1.40892e6	1.59548	0.797740	+	0.603001i	$0.206029\pi$
$240$	0	0
$241$	−826358.	−0.916486	−0.458243	−	0.888827i	$-0.651521\pi$
$241$	−826358.	−0.916486	−0.458243	+	0.888827i	$0.651521\pi$
$242$	29282.0	0.0321412
$243$	59049.0	0.0641500
$244$	1.50242e6	1.61554
$245$	0	0
$246$	−92484.0	−0.0974381
$247$	−1.23984e6	−1.29307
$248$	−603840.	−0.623437
$249$	158004.	0.161499
$250$	0	0
$251$	−1.60387e6	−1.60688	−0.803442	−	0.595384i	$-0.797000\pi$
$251$	−1.60387e6	−1.60688	−0.803442	+	0.595384i	$0.797000\pi$
$252$	335664.	0.332969
$253$	306856.	0.301393
$254$	−103576.	−0.100734
$255$	0	0
$256$	−30464.0	−0.0290527
$257$	−397618.	−0.375520	−0.187760	−	0.982215i	$-0.560123\pi$
$257$	−397618.	−0.375520	−0.187760	+	0.982215i	$0.560123\pi$
$258$	−149472.	−0.139801
$259$	1.20146e6	1.11291
$260$	0	0
$261$	376650.	0.342245
$262$	107304.	0.0965745
$263$	−2.13166e6	−1.90033	−0.950166	−	0.311745i	$-0.899087\pi$
$263$	−2.13166e6	−1.90033	−0.950166	+	0.311745i	$0.899087\pi$
$264$	−130680.	−0.115398
$265$	0	0
$266$	−639360.	−0.554040
$267$	1.19313e6	1.02426
$268$	865424.	0.736024
$269$	−725810.	−0.611564	−0.305782	−	0.952101i	$-0.598918\pi$
$269$	−725810.	−0.611564	−0.305782	+	0.952101i	$0.598918\pi$
$270$	0	0
$271$	−1.46787e6	−1.21413	−0.607063	−	0.794654i	$-0.707652\pi$
$271$	−1.46787e6	−1.21413	−0.607063	+	0.794654i	$0.707652\pi$
$272$	473632.	0.388167
$273$	764568.	0.620883
$274$	457724.	0.368322
$275$	0	0
$276$	−639072.	−0.504983
$277$	−1.52100e6	−1.19105	−0.595524	−	0.803338i	$-0.703056\pi$
$277$	−1.52100e6	−1.19105	−0.595524	+	0.803338i	$0.703056\pi$
$278$	749840.	0.581911
$279$	407592.	0.313484
$280$	0	0
$281$	464382.	0.350840	0.175420	−	0.984494i	$-0.443872\pi$
$281$	464382.	0.350840	0.175420	+	0.984494i	$0.443872\pi$
$282$	−445104.	−0.333303
$283$	415136.	0.308123	0.154062	−	0.988061i	$-0.450765\pi$
$283$	415136.	0.308123	0.154062	+	0.988061i	$0.450765\pi$
$284$	1.95014e6	1.43473
$285$	0	0
$286$	−138908.	−0.100418
$287$	760424.	0.544943
$288$	417312.	0.296469
$289$	−898573.	−0.632862
$290$	0	0
$291$	−635922.	−0.440222
$292$	−516488.	−0.354489
$293$	2.59321e6	1.76469	0.882344	−	0.470605i	$-0.155964\pi$
$293$	2.59321e6	1.76469	0.882344	+	0.470605i	$0.155964\pi$
$294$	91746.0	0.0619040
$295$	0	0
$296$	974160.	0.646251
$297$	88209.0	0.0580259
$298$	−131660.	−0.0858842
$299$	−1.45566e6	−0.941636
$300$	0	0
$301$	1.22899e6	0.781867
$302$	308104.	0.194393
$303$	−917982.	−0.574417
$304$	1.41696e6	0.879374
$305$	0	0
$306$	116964.	0.0714083
$307$	930832.	0.563671	0.281835	−	0.959463i	$-0.409057\pi$
$307$	930832.	0.563671	0.281835	+	0.959463i	$0.409057\pi$
$308$	501424.	0.301182
$309$	−1.17814e6	−0.701939
$310$	0	0
$311$	2.48527e6	1.45704	0.728522	−	0.685022i	$-0.240207\pi$
$311$	2.48527e6	1.45704	0.728522	+	0.685022i	$0.240207\pi$
$312$	619920.	0.360536
$313$	−1.31719e6	−0.759957	−0.379978	−	0.924995i	$-0.624069\pi$
$313$	−1.31719e6	−0.759957	−0.379978	+	0.924995i	$0.624069\pi$
$314$	−575356.	−0.329316
$315$	0	0
$316$	708400.	0.399081
$317$	−2.25540e6	−1.26059	−0.630297	−	0.776354i	$-0.717067\pi$
$317$	−2.25540e6	−1.26059	−0.630297	+	0.776354i	$0.717067\pi$
$318$	517428.	0.286934
$319$	562650.	0.309572
$320$	0	0
$321$	1.27451e6	0.690367
$322$	−750656.	−0.403461
$323$	1.55952e6	0.831734
$324$	−183708.	−0.0972222
$325$	0	0
$326$	−210248.	−0.109569
$327$	−2.15829e6	−1.11620
$328$	616560.	0.316440
$329$	3.65974e6	1.86406
$330$	0	0
$331$	−3.17071e6	−1.59069	−0.795346	−	0.606155i	$-0.792711\pi$
$331$	−3.17071e6	−1.59069	−0.795346	+	0.606155i	$0.792711\pi$
$332$	−491568.	−0.244759
$333$	−657558.	−0.324955
$334$	−301056.	−0.147666
$335$	0	0
$336$	−873792.	−0.422240
$337$	−1.27630e6	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$337$	−1.27630e6	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$338$	−83634.0	−0.0398191
$339$	384534.	0.181734
$340$	0	0
$341$	608872.	0.283557
$342$	349920.	0.161772
$343$	1.73308e6	0.795396
$344$	996480.	0.454017
$345$	0	0
$346$	4332.00	0.00194535
$347$	−3.69303e6	−1.64649	−0.823245	−	0.567687i	$-0.807838\pi$
$347$	−3.69303e6	−1.64649	−0.823245	+	0.567687i	$0.807838\pi$
$348$	−1.17180e6	−0.518687
$349$	1.70919e6	0.751150	0.375575	−	0.926792i	$-0.377445\pi$
$349$	1.70919e6	0.751150	0.375575	+	0.926792i	$0.377445\pi$
$350$	0	0
$351$	−418446.	−0.181289
$352$	623392.	0.268167
$353$	−4.36859e6	−1.86597	−0.932986	−	0.359914i	$-0.882806\pi$
$353$	−4.36859e6	−1.86597	−0.932986	+	0.359914i	$0.882806\pi$
$354$	−105480.	−0.0447365
$355$	0	0
$356$	−3.71196e6	−1.55231
$357$	−961704.	−0.399366
$358$	1.34556e6	0.554875
$359$	−3.51284e6	−1.43854	−0.719271	−	0.694730i	$-0.755524\pi$
$359$	−3.51284e6	−1.43854	−0.719271	+	0.694730i	$0.755524\pi$
$360$	0	0
$361$	2.18950e6	0.884254
$362$	−1.05356e6	−0.422558
$363$	131769.	0.0524864
$364$	−2.37866e6	−0.940975
$365$	0	0
$366$	−965844.	−0.376881
$367$	2.15259e6	0.834251	0.417125	−	0.908849i	$-0.363038\pi$
$367$	2.15259e6	0.834251	0.417125	+	0.908849i	$0.363038\pi$
$368$	1.66362e6	0.640374
$369$	−416178.	−0.159116
$370$	0	0
$371$	−4.25441e6	−1.60474
$372$	−1.26806e6	−0.475099
$373$	2.24247e6	0.834553	0.417276	−	0.908780i	$-0.362985\pi$
$373$	2.24247e6	0.834553	0.417276	+	0.908780i	$0.362985\pi$
$374$	174724.	0.0645912
$375$	0	0
$376$	2.96736e6	1.08243
$377$	−2.66910e6	−0.967189
$378$	−215784.	−0.0776765
$379$	−2.40986e6	−0.861775	−0.430887	−	0.902406i	$-0.641799\pi$
$379$	−2.40986e6	−0.861775	−0.430887	+	0.902406i	$0.641799\pi$
$380$	0	0
$381$	−466092.	−0.164497
$382$	−611216.	−0.214307
$383$	1.01066e6	0.352052	0.176026	−	0.984386i	$-0.443676\pi$
$383$	1.01066e6	0.352052	0.176026	+	0.984386i	$0.443676\pi$
$384$	−1.67616e6	−0.580079
$385$	0	0
$386$	−232868.	−0.0795503
$387$	−672624.	−0.228294
$388$	1.97842e6	0.667175
$389$	1.27779e6	0.428140	0.214070	−	0.976818i	$-0.431328\pi$
$389$	1.27779e6	0.428140	0.214070	+	0.976818i	$0.431328\pi$
$390$	0	0
$391$	1.83099e6	0.605682
$392$	−611640.	−0.201039
$393$	482868.	0.157706
$394$	495484.	0.160801
$395$	0	0
$396$	−274428.	−0.0879408
$397$	−5.45400e6	−1.73676	−0.868378	−	0.495903i	$-0.834837\pi$
$397$	−5.45400e6	−1.73676	−0.868378	+	0.495903i	$0.834837\pi$
$398$	−1.02672e6	−0.324896
$399$	−2.87712e6	−0.904744
$400$	0	0
$401$	−1.48980e6	−0.462665	−0.231332	−	0.972875i	$-0.574308\pi$
$401$	−1.48980e6	−0.462665	−0.231332	+	0.972875i	$0.574308\pi$
$402$	−556344.	−0.171703
$403$	−2.88837e6	−0.885911
$404$	2.85594e6	0.870555
$405$	0	0
$406$	−1.37640e6	−0.414409
$407$	−982278.	−0.293933
$408$	−779760.	−0.231905
$409$	−4.39899e6	−1.30030	−0.650152	−	0.759804i	$-0.725295\pi$
$409$	−4.39899e6	−1.30030	−0.650152	+	0.759804i	$0.725295\pi$
$410$	0	0
$411$	2.05976e6	0.601467
$412$	3.66531e6	1.06382
$413$	867280.	0.250198
$414$	410832.	0.117805
$415$	0	0
$416$	−2.95725e6	−0.837827
$417$	3.37428e6	0.950257
$418$	522720.	0.146328
$419$	−280420.	−0.0780322	−0.0390161	−	0.999239i	$-0.512422\pi$
$419$	−280420.	−0.0780322	−0.0390161	+	0.999239i	$0.512422\pi$
$420$	0	0
$421$	817462.	0.224782	0.112391	−	0.993664i	$-0.464149\pi$
$421$	817462.	0.224782	0.112391	+	0.993664i	$0.464149\pi$
$422$	−1.24138e6	−0.339330
$423$	−2.00297e6	−0.544281
$424$	−3.44952e6	−0.931846
$425$	0	0
$426$	−1.25366e6	−0.334701
$427$	7.94138e6	2.10779
$428$	−3.96514e6	−1.04628
$429$	−625086.	−0.163982
$430$	0	0
$431$	1.88599e6	0.489043	0.244521	−	0.969644i	$-0.421369\pi$
$431$	1.88599e6	0.489043	0.244521	+	0.969644i	$0.421369\pi$
$432$	478224.	0.123288
$433$	−5.84067e6	−1.49707	−0.748537	−	0.663093i	$-0.769243\pi$
$433$	−5.84067e6	−1.49707	−0.748537	+	0.663093i	$0.769243\pi$
$434$	−1.48947e6	−0.379584
$435$	0	0
$436$	6.71468e6	1.69164
$437$	5.47776e6	1.37214
$438$	332028.	0.0826969
$439$	−509540.	−0.126188	−0.0630938	−	0.998008i	$-0.520097\pi$
$439$	−509540.	−0.126188	−0.0630938	+	0.998008i	$0.520097\pi$
$440$	0	0
$441$	412857.	0.101089
$442$	−828856.	−0.201801
$443$	−4.10268e6	−0.993250	−0.496625	−	0.867965i	$-0.665428\pi$
$443$	−4.10268e6	−0.993250	−0.496625	+	0.867965i	$0.665428\pi$
$444$	2.04574e6	0.492484
$445$	0	0
$446$	2.63603e6	0.627500
$447$	−592470.	−0.140248
$448$	1.58182e6	0.372360
$449$	513410.	0.120185	0.0600923	−	0.998193i	$-0.480861\pi$
$449$	513410.	0.120185	0.0600923	+	0.998193i	$0.480861\pi$
$450$	0	0
$451$	−621698.	−0.143926
$452$	−1.19633e6	−0.275426
$453$	1.38647e6	0.317442
$454$	1.77482e6	0.404125
$455$	0	0
$456$	−2.33280e6	−0.525370
$457$	−1.22738e6	−0.274908	−0.137454	−	0.990508i	$-0.543892\pi$
$457$	−1.22738e6	−0.274908	−0.137454	+	0.990508i	$0.543892\pi$
$458$	−474900.	−0.105789
$459$	526338.	0.116609
$460$	0	0
$461$	−6.41000e6	−1.40477	−0.702386	−	0.711797i	$-0.747882\pi$
$461$	−6.41000e6	−1.40477	−0.702386	+	0.711797i	$0.747882\pi$
$462$	−322344.	−0.0702611
$463$	−6.63030e6	−1.43741	−0.718705	−	0.695315i	$-0.755265\pi$
$463$	−6.63030e6	−1.43741	−0.718705	+	0.695315i	$0.755265\pi$
$464$	3.05040e6	0.657751
$465$	0	0
$466$	1.82941e6	0.390253
$467$	4.14769e6	0.880064	0.440032	−	0.897982i	$-0.354967\pi$
$467$	4.14769e6	0.880064	0.440032	+	0.897982i	$0.354967\pi$
$468$	1.30183e6	0.274752
$469$	4.57438e6	0.960286
$470$	0	0
$471$	−2.58910e6	−0.537770
$472$	703200.	0.145286
$473$	−1.00478e6	−0.206500
$474$	−455400.	−0.0930995
$475$	0	0
$476$	2.99197e6	0.605257
$477$	2.32843e6	0.468561
$478$	2.81784e6	0.564088
$479$	−5.05132e6	−1.00593	−0.502963	−	0.864308i	$-0.667757\pi$
$479$	−5.05132e6	−1.00593	−0.502963	+	0.864308i	$0.667757\pi$
$480$	0	0
$481$	4.65973e6	0.918329
$482$	−1.65272e6	−0.324027
$483$	−3.37795e6	−0.658849
$484$	−409948.	−0.0795455
$485$	0	0
$486$	118098.	0.0226805
$487$	−2.66221e6	−0.508651	−0.254325	−	0.967119i	$-0.581853\pi$
$487$	−2.66221e6	−0.508651	−0.254325	+	0.967119i	$0.581853\pi$
$488$	6.43896e6	1.22396
$489$	−946116.	−0.178925
$490$	0	0
$491$	−5.54659e6	−1.03830	−0.519149	−	0.854684i	$-0.673751\pi$
$491$	−5.54659e6	−1.03830	−0.519149	+	0.854684i	$0.673751\pi$
$492$	1.29478e6	0.241147
$493$	3.35730e6	0.622118
$494$	−2.47968e6	−0.457171
$495$	0	0
$496$	3.30099e6	0.602477
$497$	1.03079e7	1.87189
$498$	316008.	0.0570985
$499$	−6820.00	−0.00122612	−0.000613060	−	1.00000i	$-0.500195\pi$
$499$	−6820.00	−0.00122612	−0.000613060		1.00000i	$0.500195\pi$
$500$	0	0
$501$	−1.35475e6	−0.241138
$502$	−3.20774e6	−0.568119
$503$	451136.	0.0795037	0.0397519	−	0.999210i	$-0.487343\pi$
$503$	451136.	0.0795037	0.0397519	+	0.999210i	$0.487343\pi$
$504$	1.43856e6	0.252262
$505$	0	0
$506$	613712.	0.106559
$507$	−376353.	−0.0650243
$508$	1.45006e6	0.249303
$509$	393390.	0.0673021	0.0336511	−	0.999434i	$-0.489287\pi$
$509$	393390.	0.0673021	0.0336511	+	0.999434i	$0.489287\pi$
$510$	0	0
$511$	−2.73001e6	−0.462500
$512$	5.89875e6	0.994455
$513$	1.57464e6	0.264173
$514$	−795236.	−0.132766
$515$	0	0
$516$	2.09261e6	0.345990
$517$	−2.99209e6	−0.492321
$518$	2.40293e6	0.393474
$519$	19494.0	0.00317675
$520$	0	0
$521$	3.28432e6	0.530092	0.265046	−	0.964236i	$-0.414613\pi$
$521$	3.28432e6	0.530092	0.265046	+	0.964236i	$0.414613\pi$
$522$	753300.	0.121002
$523$	1.68266e6	0.268993	0.134497	−	0.990914i	$-0.457058\pi$
$523$	1.68266e6	0.268993	0.134497	+	0.990914i	$0.457058\pi$
$524$	−1.50226e6	−0.239010
$525$	0	0
$526$	−4.26333e6	−0.671869
$527$	3.63310e6	0.569838
$528$	714384.	0.111518
$529$	−5047.00	−0.000784141 0
$530$	0	0
$531$	−474660.	−0.0730544
$532$	8.95104e6	1.37118
$533$	2.94921e6	0.449664
$534$	2.38626e6	0.362130
$535$	0	0
$536$	3.70896e6	0.557622
$537$	6.05502e6	0.906108
$538$	−1.45162e6	−0.216221
$539$	616737.	0.0914383
$540$	0	0
$541$	9.48158e6	1.39280	0.696398	−	0.717656i	$-0.254785\pi$
$541$	9.48158e6	1.39280	0.696398	+	0.717656i	$0.254785\pi$
$542$	−2.93574e6	−0.429258
$543$	−4.74100e6	−0.690034
$544$	3.71974e6	0.538909
$545$	0	0
$546$	1.52914e6	0.219515
$547$	6.09239e6	0.870602	0.435301	−	0.900285i	$-0.356642\pi$
$547$	6.09239e6	0.870602	0.435301	+	0.900285i	$0.356642\pi$
$548$	−6.40814e6	−0.911550
$549$	−4.34630e6	−0.615444
$550$	0	0
$551$	1.00440e7	1.40938
$552$	−2.73888e6	−0.382583
$553$	3.74440e6	0.520678
$554$	−3.04200e6	−0.421099
$555$	0	0
$556$	−1.04978e7	−1.44016
$557$	−8.49594e6	−1.16031	−0.580154	−	0.814507i	$-0.697008\pi$
$557$	−8.49594e6	−1.16031	−0.580154	+	0.814507i	$0.697008\pi$
$558$	815184.	0.110833
$559$	4.76650e6	0.645163
$560$	0	0
$561$	786258.	0.105477
$562$	928764.	0.124041
$563$	7.02216e6	0.933683	0.466842	−	0.884341i	$-0.345392\pi$
$563$	7.02216e6	0.933683	0.466842	+	0.884341i	$0.345392\pi$
$564$	6.23146e6	0.824882
$565$	0	0
$566$	830272.	0.108938
$567$	−971028.	−0.126845
$568$	8.35776e6	1.08697
$569$	9.41847e6	1.21955	0.609775	−	0.792574i	$-0.291260\pi$
$569$	9.41847e6	1.21955	0.609775	+	0.792574i	$0.291260\pi$
$570$	0	0
$571$	7.29699e6	0.936599	0.468299	−	0.883570i	$-0.344867\pi$
$571$	7.29699e6	0.936599	0.468299	+	0.883570i	$0.344867\pi$
$572$	1.94471e6	0.248522
$573$	−2.75047e6	−0.349962
$574$	1.52085e6	0.192666
$575$	0	0
$576$	−865728.	−0.108724
$577$	3.29590e6	0.412131	0.206065	−	0.978538i	$-0.433934\pi$
$577$	3.29590e6	0.412131	0.206065	+	0.978538i	$0.433934\pi$
$578$	−1.79715e6	−0.223750
$579$	−1.04791e6	−0.129905
$580$	0	0
$581$	−2.59829e6	−0.319335
$582$	−1.27184e6	−0.155642
$583$	3.47827e6	0.423830
$584$	−2.21352e6	−0.268566
$585$	0	0
$586$	5.18641e6	0.623911
$587$	−4.39827e6	−0.526849	−0.263425	−	0.964680i	$-0.584852\pi$
$587$	−4.39827e6	−0.526849	−0.263425	+	0.964680i	$0.584852\pi$
$588$	−1.28444e6	−0.153205
$589$	1.08691e7	1.29094
$590$	0	0
$591$	2.22968e6	0.262587
$592$	−5.32541e6	−0.624523
$593$	−9.21781e6	−1.07644	−0.538222	−	0.842803i	$-0.680904\pi$
$593$	−9.21781e6	−1.07644	−0.538222	+	0.842803i	$0.680904\pi$
$594$	176418.	0.0205152
$595$	0	0
$596$	1.84324e6	0.212553
$597$	−4.62024e6	−0.530553
$598$	−2.91133e6	−0.332919
$599$	3.77140e6	0.429473	0.214736	−	0.976672i	$-0.431111\pi$
$599$	3.77140e6	0.429473	0.214736	+	0.976672i	$0.431111\pi$
$600$	0	0
$601$	4.19724e6	0.473999	0.237000	−	0.971510i	$-0.423836\pi$
$601$	4.19724e6	0.473999	0.237000	+	0.971510i	$0.423836\pi$
$602$	2.45798e6	0.276432
$603$	−2.50355e6	−0.280390
$604$	−4.31346e6	−0.481097
$605$	0	0
$606$	−1.83596e6	−0.203087
$607$	1.00133e6	0.110308	0.0551539	−	0.998478i	$-0.482435\pi$
$607$	1.00133e6	0.110308	0.0551539	+	0.998478i	$0.482435\pi$
$608$	1.11283e7	1.22087
$609$	−6.19380e6	−0.676728
$610$	0	0
$611$	1.41939e7	1.53815
$612$	−1.63750e6	−0.176727
$613$	7.38239e6	0.793498	0.396749	−	0.917927i	$-0.370138\pi$
$613$	7.38239e6	0.793498	0.396749	+	0.917927i	$0.370138\pi$
$614$	1.86166e6	0.199288
$615$	0	0
$616$	2.14896e6	0.228179
$617$	1.54025e7	1.62884	0.814418	−	0.580279i	$-0.197056\pi$
$617$	1.54025e7	1.62884	0.814418	+	0.580279i	$0.197056\pi$
$618$	−2.35627e6	−0.248173
$619$	−7.12402e6	−0.747306	−0.373653	−	0.927569i	$-0.621895\pi$
$619$	−7.12402e6	−0.747306	−0.373653	+	0.927569i	$0.621895\pi$
$620$	0	0
$621$	1.84874e6	0.192375
$622$	4.97054e6	0.515143
$623$	−1.96204e7	−2.02529
$624$	−3.38890e6	−0.348415
$625$	0	0
$626$	−2.63439e6	−0.268685
$627$	2.35224e6	0.238953
$628$	8.05498e6	0.815015
$629$	−5.86120e6	−0.590690
$630$	0	0
$631$	1.16696e7	1.16677	0.583383	−	0.812197i	$-0.301729\pi$
$631$	1.16696e7	1.16677	0.583383	+	0.812197i	$0.301729\pi$
$632$	3.03600e6	0.302349
$633$	−5.58619e6	−0.554124
$634$	−4.51080e6	−0.445687
$635$	0	0
$636$	−7.24399e6	−0.710126
$637$	−2.92568e6	−0.285679
$638$	1.12530e6	0.109450
$639$	−5.64149e6	−0.546565
$640$	0	0
$641$	−1.10271e7	−1.06003	−0.530014	−	0.847989i	$-0.677813\pi$
$641$	−1.10271e7	−1.06003	−0.530014	+	0.847989i	$0.677813\pi$
$642$	2.54902e6	0.244082
$643$	9.56024e6	0.911887	0.455944	−	0.890009i	$-0.349302\pi$
$643$	9.56024e6	0.911887	0.455944	+	0.890009i	$0.349302\pi$
$644$	1.05092e7	0.998514
$645$	0	0
$646$	3.11904e6	0.294063
$647$	1.09942e7	1.03253	0.516263	−	0.856430i	$-0.327323\pi$
$647$	1.09942e7	1.03253	0.516263	+	0.856430i	$0.327323\pi$
$648$	−787320.	−0.0736570
$649$	−709060.	−0.0660802
$650$	0	0
$651$	−6.70262e6	−0.619858
$652$	2.94347e6	0.271170
$653$	295346.	0.0271049	0.0135525	−	0.999908i	$-0.495686\pi$
$653$	295346.	0.0271049	0.0135525	+	0.999908i	$0.495686\pi$
$654$	−4.31658e6	−0.394635
$655$	0	0
$656$	−3.37053e6	−0.305801
$657$	1.49413e6	0.135044
$658$	7.31949e6	0.659046
$659$	−1.65613e7	−1.48553	−0.742766	−	0.669551i	$-0.766486\pi$
$659$	−1.65613e7	−1.48553	−0.742766	+	0.669551i	$0.766486\pi$
$660$	0	0
$661$	1.97042e6	0.175411	0.0877053	−	0.996146i	$-0.472047\pi$
$661$	1.97042e6	0.175411	0.0877053	+	0.996146i	$0.472047\pi$
$662$	−6.34142e6	−0.562395
$663$	−3.72985e6	−0.329540
$664$	−2.10672e6	−0.185433
$665$	0	0
$666$	−1.31512e6	−0.114889
$667$	1.17924e7	1.02633
$668$	4.21478e6	0.365455
$669$	1.18621e7	1.02470
$670$	0	0
$671$	−6.49262e6	−0.556690
$672$	−6.86246e6	−0.586215
$673$	1.63733e6	0.139347	0.0696735	−	0.997570i	$-0.477804\pi$
$673$	1.63733e6	0.139347	0.0696735	+	0.997570i	$0.477804\pi$
$674$	−2.55260e6	−0.216437
$675$	0	0
$676$	1.17088e6	0.0985472
$677$	6.35878e6	0.533215	0.266607	−	0.963805i	$-0.414097\pi$
$677$	6.35878e6	0.533215	0.266607	+	0.963805i	$0.414097\pi$
$678$	769068.	0.0642526
$679$	1.04574e7	0.870460
$680$	0	0
$681$	7.98671e6	0.659933
$682$	1.21774e6	0.100252
$683$	−1.11033e7	−0.910751	−0.455376	−	0.890299i	$-0.650495\pi$
$683$	−1.11033e7	−0.910751	−0.455376	+	0.890299i	$0.650495\pi$
$684$	−4.89888e6	−0.400365
$685$	0	0
$686$	3.46616e6	0.281215
$687$	−2.13705e6	−0.172752
$688$	−5.44742e6	−0.438753
$689$	−1.65002e7	−1.32416
$690$	0	0
$691$	1.70189e7	1.35592	0.677962	−	0.735097i	$-0.262864\pi$
$691$	1.70189e7	1.35592	0.677962	+	0.735097i	$0.262864\pi$
$692$	−60648.0	−0.00481450
$693$	−1.45055e6	−0.114736
$694$	−7.38606e6	−0.582122
$695$	0	0
$696$	−5.02200e6	−0.392965
$697$	−3.70964e6	−0.289234
$698$	3.41838e6	0.265572
$699$	8.23235e6	0.637281
$700$	0	0
$701$	1.58021e7	1.21456	0.607280	−	0.794488i	$-0.292260\pi$
$701$	1.58021e7	1.21456	0.607280	+	0.794488i	$0.292260\pi$
$702$	−836892.	−0.0640954
$703$	−1.75349e7	−1.33818
$704$	−1.29325e6	−0.0983445
$705$	0	0
$706$	−8.73719e6	−0.659720
$707$	1.50957e7	1.13581
$708$	1.47672e6	0.110717
$709$	1.24834e7	0.932643	0.466322	−	0.884615i	$-0.345579\pi$
$709$	1.24834e7	0.932643	0.466322	+	0.884615i	$0.345579\pi$
$710$	0	0
$711$	−2.04930e6	−0.152031
$712$	−1.59084e7	−1.17605
$713$	1.27612e7	0.940083
$714$	−1.92341e6	−0.141197
$715$	0	0
$716$	−1.88378e7	−1.37325
$717$	1.26803e7	0.921151
$718$	−7.02568e6	−0.508601
$719$	2.00724e6	0.144803	0.0724014	−	0.997376i	$-0.476934\pi$
$719$	2.00724e6	0.144803	0.0724014	+	0.997376i	$0.476934\pi$
$720$	0	0
$721$	1.93738e7	1.38796
$722$	4.37900e6	0.312631
$723$	−7.43722e6	−0.529133
$724$	1.47498e7	1.04578
$725$	0	0
$726$	263538.	0.0185567
$727$	−6.97301e6	−0.489310	−0.244655	−	0.969610i	$-0.578675\pi$
$727$	−6.97301e6	−0.489310	−0.244655	+	0.969610i	$0.578675\pi$
$728$	−1.01942e7	−0.712896
$729$	531441.	0.0370370
$730$	0	0
$731$	−5.99549e6	−0.414984
$732$	1.35218e7	0.932733
$733$	2.34965e7	1.61527	0.807633	−	0.589685i	$-0.200748\pi$
$733$	2.34965e7	1.61527	0.807633	+	0.589685i	$0.200748\pi$
$734$	4.30518e6	0.294952
$735$	0	0
$736$	1.30655e7	0.889059
$737$	−3.73987e6	−0.253622
$738$	−832356.	−0.0562559
$739$	−1.39901e7	−0.942346	−0.471173	−	0.882041i	$-0.656169\pi$
$739$	−1.39901e7	−0.942346	−0.471173	+	0.882041i	$0.656169\pi$
$740$	0	0
$741$	−1.11586e7	−0.746556
$742$	−8.50882e6	−0.567361
$743$	−2.42745e7	−1.61316	−0.806582	−	0.591123i	$-0.798685\pi$
$743$	−2.42745e7	−1.61316	−0.806582	+	0.591123i	$0.798685\pi$
$744$	−5.43456e6	−0.359942
$745$	0	0
$746$	4.48493e6	0.295059
$747$	1.42204e6	0.0932415
$748$	−2.44614e6	−0.159855
$749$	−2.09586e7	−1.36508
$750$	0	0
$751$	1.53660e7	0.994170	0.497085	−	0.867702i	$-0.334404\pi$
$751$	1.53660e7	0.994170	0.497085	+	0.867702i	$0.334404\pi$
$752$	−1.62216e7	−1.04604
$753$	−1.44348e7	−0.927734
$754$	−5.33820e6	−0.341953
$755$	0	0
$756$	3.02098e6	0.192240
$757$	−2.07605e7	−1.31674	−0.658368	−	0.752697i	$-0.728753\pi$
$757$	−2.07605e7	−1.31674	−0.658368	+	0.752697i	$0.728753\pi$
$758$	−4.81972e6	−0.304683
$759$	2.76170e6	0.174009
$760$	0	0
$761$	5.83810e6	0.365435	0.182717	−	0.983165i	$-0.441511\pi$
$761$	5.83810e6	0.365435	0.182717	+	0.983165i	$0.441511\pi$
$762$	−932184.	−0.0581586
$763$	3.54919e7	2.20708
$764$	8.55702e6	0.530383
$765$	0	0
$766$	2.02131e6	0.124469
$767$	3.36364e6	0.206453
$768$	−274176.	−0.0167736
$769$	−1.39197e7	−0.848818	−0.424409	−	0.905471i	$-0.639518\pi$
$769$	−1.39197e7	−0.848818	−0.424409	+	0.905471i	$0.639518\pi$
$770$	0	0
$771$	−3.57856e6	−0.216807
$772$	3.26015e6	0.196877
$773$	4.17883e6	0.251539	0.125770	−	0.992059i	$-0.459860\pi$
$773$	4.17883e6	0.251539	0.125770	+	0.992059i	$0.459860\pi$
$774$	−1.34525e6	−0.0807142
$775$	0	0
$776$	8.47896e6	0.505462
$777$	1.08132e7	0.642541
$778$	2.55558e6	0.151370
$779$	−1.10981e7	−0.655246
$780$	0	0
$781$	−8.42741e6	−0.494386
$782$	3.66198e6	0.214141
$783$	3.38985e6	0.197595
$784$	3.34363e6	0.194280
$785$	0	0
$786$	965736.	0.0557573
$787$	−9.66705e6	−0.556361	−0.278181	−	0.960529i	$-0.589731\pi$
$787$	−9.66705e6	−0.556361	−0.278181	+	0.960529i	$0.589731\pi$
$788$	−6.93678e6	−0.397962
$789$	−1.91850e7	−1.09716
$790$	0	0
$791$	−6.32345e6	−0.359346
$792$	−1.17612e6	−0.0666252
$793$	3.07997e7	1.73926
$794$	−1.09080e7	−0.614036
$795$	0	0
$796$	1.43741e7	0.804077
$797$	5.79884e6	0.323367	0.161683	−	0.986843i	$-0.448308\pi$
$797$	5.79884e6	0.323367	0.161683	+	0.986843i	$0.448308\pi$
$798$	−5.75424e6	−0.319875
$799$	−1.78536e7	−0.989371
$800$	0	0
$801$	1.07382e7	0.591356
$802$	−2.97960e6	−0.163577
$803$	2.23197e6	0.122151
$804$	7.78882e6	0.424944
$805$	0	0
$806$	−5.77674e6	−0.313217
$807$	−6.53229e6	−0.353087
$808$	1.22398e7	0.659545
$809$	−1.92543e7	−1.03433	−0.517163	−	0.855887i	$-0.673012\pi$
$809$	−1.92543e7	−1.03433	−0.517163	+	0.855887i	$0.673012\pi$
$810$	0	0
$811$	−1.31938e7	−0.704396	−0.352198	−	0.935926i	$-0.614566\pi$
$811$	−1.31938e7	−0.704396	−0.352198	+	0.935926i	$0.614566\pi$
$812$	1.92696e7	1.02561
$813$	−1.32108e7	−0.700976
$814$	−1.96456e6	−0.103921
$815$	0	0
$816$	4.26269e6	0.224108
$817$	−1.79366e7	−0.940126
$818$	−8.79798e6	−0.459727
$819$	6.88111e6	0.358467
$820$	0	0
$821$	1.33779e7	0.692677	0.346338	−	0.938110i	$-0.387425\pi$
$821$	1.33779e7	0.692677	0.346338	+	0.938110i	$0.387425\pi$
$822$	4.11952e6	0.212651
$823$	1.88613e7	0.970673	0.485336	−	0.874327i	$-0.338697\pi$
$823$	1.88613e7	0.970673	0.485336	+	0.874327i	$0.338697\pi$
$824$	1.57085e7	0.805965
$825$	0	0
$826$	1.73456e6	0.0884584
$827$	−1.62680e7	−0.827123	−0.413561	−	0.910476i	$-0.635715\pi$
$827$	−1.62680e7	−0.827123	−0.413561	+	0.910476i	$0.635715\pi$
$828$	−5.75165e6	−0.291552
$829$	−2.18098e7	−1.10221	−0.551107	−	0.834435i	$-0.685794\pi$
$829$	−2.18098e7	−1.10221	−0.551107	+	0.834435i	$0.685794\pi$
$830$	0	0
$831$	−1.36890e7	−0.687652
$832$	6.13491e6	0.307256
$833$	3.68003e6	0.183755
$834$	6.74856e6	0.335967
$835$	0	0
$836$	−7.31808e6	−0.362144
$837$	3.66833e6	0.180990
$838$	−560840.	−0.0275886
$839$	1.17771e7	0.577607	0.288804	−	0.957388i	$-0.406743\pi$
$839$	1.17771e7	0.577607	0.288804	+	0.957388i	$0.406743\pi$
$840$	0	0
$841$	1.11135e6	0.0541828
$842$	1.63492e6	0.0794726
$843$	4.17944e6	0.202558
$844$	1.73793e7	0.839799
$845$	0	0
$846$	−4.00594e6	−0.192432
$847$	−2.16687e6	−0.103782
$848$	1.88574e7	0.900516
$849$	3.73622e6	0.177895
$850$	0	0
$851$	−2.05872e7	−0.974483
$852$	1.75513e7	0.828343
$853$	1.43993e7	0.677591	0.338796	−	0.940860i	$-0.389980\pi$
$853$	1.43993e7	0.677591	0.338796	+	0.940860i	$0.389980\pi$
$854$	1.58828e7	0.745215
$855$	0	0
$856$	−1.69934e7	−0.792678
$857$	−6.27604e6	−0.291900	−0.145950	−	0.989292i	$-0.546624\pi$
$857$	−6.27604e6	−0.291900	−0.145950	+	0.989292i	$0.546624\pi$
$858$	−1.25017e6	−0.0579764
$859$	−4.71738e6	−0.218131	−0.109066	−	0.994035i	$-0.534786\pi$
$859$	−4.71738e6	−0.218131	−0.109066	+	0.994035i	$0.534786\pi$
$860$	0	0
$861$	6.84382e6	0.314623
$862$	3.77198e6	0.172903
$863$	−7.53926e6	−0.344589	−0.172295	−	0.985045i	$-0.555118\pi$
$863$	−7.53926e6	−0.344589	−0.172295	+	0.985045i	$0.555118\pi$
$864$	3.75581e6	0.171167
$865$	0	0
$866$	−1.16813e7	−0.529296
$867$	−8.08716e6	−0.365383
$868$	2.08526e7	0.939423
$869$	−3.06130e6	−0.137517
$870$	0	0
$871$	1.77412e7	0.792387
$872$	2.87772e7	1.28161
$873$	−5.72330e6	−0.254162
$874$	1.09555e7	0.485126
$875$	0	0
$876$	−4.64839e6	−0.204664
$877$	1.04331e7	0.458051	0.229025	−	0.973420i	$-0.426446\pi$
$877$	1.04331e7	0.458051	0.229025	+	0.973420i	$0.426446\pi$
$878$	−1.01908e6	−0.0446141
$879$	2.33389e7	1.01884
$880$	0	0
$881$	3.91076e7	1.69755	0.848774	−	0.528756i	$-0.177342\pi$
$881$	3.91076e7	1.69755	0.848774	+	0.528756i	$0.177342\pi$
$882$	825714.	0.0357403
$883$	−1.29282e7	−0.558003	−0.279001	−	0.960291i	$-0.590003\pi$
$883$	−1.29282e7	−0.558003	−0.279001	+	0.960291i	$0.590003\pi$
$884$	1.16040e7	0.499432
$885$	0	0
$886$	−8.20537e6	−0.351167
$887$	−3.36466e7	−1.43592	−0.717962	−	0.696082i	$-0.754925\pi$
$887$	−3.36466e7	−1.43592	−0.717962	+	0.696082i	$0.754925\pi$
$888$	8.76744e6	0.373113
$889$	7.66462e6	0.325264
$890$	0	0
$891$	793881.	0.0335013
$892$	−3.69044e7	−1.55298
$893$	−5.34125e7	−2.24137
$894$	−1.18494e6	−0.0495853
$895$	0	0
$896$	2.75635e7	1.14700
$897$	−1.31010e7	−0.543654
$898$	1.02682e6	0.0424916
$899$	2.33988e7	0.965594
$900$	0	0
$901$	2.07546e7	0.851731
$902$	−1.24340e6	−0.0508854
$903$	1.10609e7	0.451411
$904$	−5.12712e6	−0.208666
$905$	0	0
$906$	2.77294e6	0.112233
$907$	4.19629e7	1.69374	0.846872	−	0.531797i	$-0.178483\pi$
$907$	4.19629e7	1.69374	0.846872	+	0.531797i	$0.178483\pi$
$908$	−2.48475e7	−1.00016
$909$	−8.26184e6	−0.331640
$910$	0	0
$911$	−1.92521e6	−0.0768567	−0.0384283	−	0.999261i	$-0.512235\pi$
$911$	−1.92521e6	−0.0768567	−0.0384283	+	0.999261i	$0.512235\pi$
$912$	1.27526e7	0.507707
$913$	2.12428e6	0.0843401
$914$	−2.45476e6	−0.0971948
$915$	0	0
$916$	6.64860e6	0.261813
$917$	−7.94050e6	−0.311835
$918$	1.05268e6	0.0412276
$919$	−1.72481e7	−0.673678	−0.336839	−	0.941562i	$-0.609358\pi$
$919$	−1.72481e7	−0.673678	−0.336839	+	0.941562i	$0.609358\pi$
$920$	0	0
$921$	8.37749e6	0.325435
$922$	−1.28200e7	−0.496662
$923$	3.99780e7	1.54460
$924$	4.51282e6	0.173887
$925$	0	0
$926$	−1.32606e7	−0.508202
$927$	−1.06032e7	−0.405265
$928$	2.39568e7	0.913185
$929$	2.51145e6	0.0954740	0.0477370	−	0.998860i	$-0.484799\pi$
$929$	2.51145e6	0.0954740	0.0477370	+	0.998860i	$0.484799\pi$
$930$	0	0
$931$	1.10095e7	0.416288
$932$	−2.56118e7	−0.965828
$933$	2.23674e7	0.841225
$934$	8.29538e6	0.311150
$935$	0	0
$936$	5.57928e6	0.208156
$937$	−1.79853e7	−0.669221	−0.334611	−	0.942357i	$-0.608605\pi$
$937$	−1.79853e7	−0.669221	−0.334611	+	0.942357i	$0.608605\pi$
$938$	9.14877e6	0.339512
$939$	−1.18547e7	−0.438761
$940$	0	0
$941$	−3.22586e7	−1.18760	−0.593802	−	0.804611i	$-0.702374\pi$
$941$	−3.22586e7	−1.18760	−0.593802	+	0.804611i	$0.702374\pi$
$942$	−5.17820e6	−0.190131
$943$	−1.30300e7	−0.477160
$944$	−3.84416e6	−0.140401
$945$	0	0
$946$	−2.00957e6	−0.0730087
$947$	−4.41659e7	−1.60034	−0.800169	−	0.599774i	$-0.795257\pi$
$947$	−4.41659e7	−1.60034	−0.800169	+	0.599774i	$0.795257\pi$
$948$	6.37560e6	0.230409
$949$	−1.05880e7	−0.381635
$950$	0	0
$951$	−2.02986e7	−0.727804
$952$	1.28227e7	0.458551
$953$	−1.87488e7	−0.668714	−0.334357	−	0.942446i	$-0.608519\pi$
$953$	−1.87488e7	−0.668714	−0.334357	+	0.942446i	$0.608519\pi$
$954$	4.65685e6	0.165661
$955$	0	0
$956$	−3.94498e7	−1.39605
$957$	5.06385e6	0.178731
$958$	−1.01026e7	−0.355649
$959$	−3.38716e7	−1.18929
$960$	0	0
$961$	−3.30813e6	−0.115551
$962$	9.31946e6	0.324678
$963$	1.14706e7	0.398584
$964$	2.31380e7	0.801925
$965$	0	0
$966$	−6.75590e6	−0.232938
$967$	−1.08673e7	−0.373730	−0.186865	−	0.982386i	$-0.559833\pi$
$967$	−1.08673e7	−0.373730	−0.186865	+	0.982386i	$0.559833\pi$
$968$	−1.75692e6	−0.0602648
$969$	1.40357e7	0.480202
$970$	0	0
$971$	−4.79123e7	−1.63079	−0.815397	−	0.578902i	$-0.803481\pi$
$971$	−4.79123e7	−1.63079	−0.815397	+	0.578902i	$0.803481\pi$
$972$	−1.65337e6	−0.0561313
$973$	−5.54882e7	−1.87896
$974$	−5.32442e6	−0.179835
$975$	0	0
$976$	−3.51996e7	−1.18281
$977$	−4.01385e7	−1.34532	−0.672658	−	0.739954i	$-0.734847\pi$
$977$	−4.01385e7	−1.34532	−0.672658	+	0.739954i	$0.734847\pi$
$978$	−1.89223e6	−0.0632597
$979$	1.60410e7	0.534902
$980$	0	0
$981$	−1.94246e7	−0.644436
$982$	−1.10932e7	−0.367094
$983$	−3.22682e6	−0.106510	−0.0532551	−	0.998581i	$-0.516960\pi$
$983$	−3.22682e6	−0.106510	−0.0532551	+	0.998581i	$0.516960\pi$
$984$	5.54904e6	0.182696
$985$	0	0
$986$	6.71460e6	0.219952
$987$	3.29377e7	1.07622
$988$	3.47155e7	1.13144
$989$	−2.10589e7	−0.684614
$990$	0	0
$991$	−5.95345e6	−0.192568	−0.0962841	−	0.995354i	$-0.530696\pi$
$991$	−5.95345e6	−0.192568	−0.0962841	+	0.995354i	$0.530696\pi$
$992$	2.59249e7	0.836445
$993$	−2.85364e7	−0.918387
$994$	2.06158e7	0.661812
$995$	0	0
$996$	−4.42411e6	−0.141312
$997$	−3.20783e7	−1.02205	−0.511027	−	0.859565i	$-0.670735\pi$
$997$	−3.20783e7	−1.02205	−0.511027	+	0.859565i	$0.670735\pi$
$998$	−13640.0	−0.000433499 0
$999$	−5.91802e6	−0.187613

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.b.1.1		1
5.4	even	2		33.6.a.a.1.1	✓	1
15.14	odd	2		99.6.a.b.1.1		1
20.19	odd	2		528.6.a.i.1.1		1
55.54	odd	2		363.6.a.c.1.1		1
165.164	even	2		1089.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.a.1.1	✓	1	5.4	even	2
99.6.a.b.1.1		1	15.14	odd	2
363.6.a.c.1.1		1	55.54	odd	2
528.6.a.i.1.1		1	20.19	odd	2
825.6.a.b.1.1		1	1.1	even	1	trivial
1089.6.a.d.1.1		1	165.164	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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