LMFDB - Embedded newform 528.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,6,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("528.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	528.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:	$84.6826568613$
Analytic rank:	$0$
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	$\chi$	$=$	528.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-9.00000 q^{3} -92.0000 q^{5} +26.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q-9.00000 q^{3} -92.0000 q^{5} +26.0000 q^{7} +81.0000 q^{9} -121.000 q^{11} -692.000 q^{13} +828.000 q^{15} -1442.00 q^{17} -2160.00 q^{19} -234.000 q^{21} +1582.00 q^{23} +5339.00 q^{25} -729.000 q^{27} -5526.00 q^{29} -4792.00 q^{31} +1089.00 q^{33} -2392.00 q^{35} -10194.0 q^{37} +6228.00 q^{39} -10622.0 q^{41} -8580.00 q^{43} -7452.00 q^{45} +2362.00 q^{47} -16131.0 q^{49} +12978.0 q^{51} -30804.0 q^{53} +11132.0 q^{55} +19440.0 q^{57} -6416.00 q^{59} +42096.0 q^{61} +2106.00 q^{63} +63664.0 q^{65} +28444.0 q^{67} -14238.0 q^{69} -45690.0 q^{71} -18374.0 q^{73} -48051.0 q^{75} -3146.00 q^{77} +105214. q^{79} +6561.00 q^{81} -62292.0 q^{83} +132664. q^{85} +49734.0 q^{87} -72246.0 q^{89} -17992.0 q^{91} +43128.0 q^{93} +198720. q^{95} +79262.0 q^{97} -9801.00 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−92.0000	−1.64575	−0.822873	−	0.568225i	$-0.807630\pi$
$5$	−92.0000	−1.64575	−0.822873	+	0.568225i	$0.807630\pi$
$6$	0	0
$7$	26.0000	0.200553	0.100276	−	0.994960i	$-0.468027\pi$
$7$	26.0000	0.200553	0.100276	+	0.994960i	$0.468027\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−692.000	−1.13566	−0.567829	−	0.823146i	$-0.692217\pi$
$13$	−692.000	−1.13566	−0.567829	+	0.823146i	$0.692217\pi$
$14$	0	0
$15$	828.000	0.950172
$16$	0	0
$17$	−1442.00	−1.21016	−0.605080	−	0.796165i	$-0.706859\pi$
$17$	−1442.00	−1.21016	−0.605080	+	0.796165i	$0.706859\pi$
$18$	0	0
$19$	−2160.00	−1.37268	−0.686341	−	0.727280i	$-0.740784\pi$
$19$	−2160.00	−1.37268	−0.686341	+	0.727280i	$0.740784\pi$
$20$	0	0
$21$	−234.000	−0.115789
$22$	0	0
$23$	1582.00	0.623572	0.311786	−	0.950152i	$-0.399073\pi$
$23$	1582.00	0.623572	0.311786	+	0.950152i	$0.399073\pi$
$24$	0	0
$25$	5339.00	1.70848
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−5526.00	−1.22016	−0.610079	−	0.792341i	$-0.708862\pi$
$29$	−5526.00	−1.22016	−0.610079	+	0.792341i	$0.708862\pi$
$30$	0	0
$31$	−4792.00	−0.895597	−0.447798	−	0.894135i	$-0.647792\pi$
$31$	−4792.00	−0.895597	−0.447798	+	0.894135i	$0.647792\pi$
$32$	0	0
$33$	1089.00	0.174078
$34$	0	0
$35$	−2392.00	−0.330059
$36$	0	0
$37$	−10194.0	−1.22417	−0.612083	−	0.790794i	$-0.709668\pi$
$37$	−10194.0	−1.22417	−0.612083	+	0.790794i	$0.709668\pi$
$38$	0	0
$39$	6228.00	0.655673
$40$	0	0
$41$	−10622.0	−0.986840	−0.493420	−	0.869791i	$-0.664253\pi$
$41$	−10622.0	−0.986840	−0.493420	+	0.869791i	$0.664253\pi$
$42$	0	0
$43$	−8580.00	−0.707646	−0.353823	−	0.935312i	$-0.615119\pi$
$43$	−8580.00	−0.707646	−0.353823	+	0.935312i	$0.615119\pi$
$44$	0	0
$45$	−7452.00	−0.548582
$46$	0	0
$47$	2362.00	0.155968	0.0779840	−	0.996955i	$-0.475152\pi$
$47$	2362.00	0.155968	0.0779840	+	0.996955i	$0.475152\pi$
$48$	0	0
$49$	−16131.0	−0.959779
$50$	0	0
$51$	12978.0	0.698686
$52$	0	0
$53$	−30804.0	−1.50632	−0.753160	−	0.657837i	$-0.771472\pi$
$53$	−30804.0	−1.50632	−0.753160	+	0.657837i	$0.771472\pi$
$54$	0	0
$55$	11132.0	0.496211
$56$	0	0
$57$	19440.0	0.792518
$58$	0	0
$59$	−6416.00	−0.239957	−0.119979	−	0.992776i	$-0.538283\pi$
$59$	−6416.00	−0.239957	−0.119979	+	0.992776i	$0.538283\pi$
$60$	0	0
$61$	42096.0	1.44849	0.724246	−	0.689541i	$-0.242188\pi$
$61$	42096.0	1.44849	0.724246	+	0.689541i	$0.242188\pi$
$62$	0	0
$63$	2106.00	0.0668509
$64$	0	0
$65$	63664.0	1.86901
$66$	0	0
$67$	28444.0	0.774112	0.387056	−	0.922056i	$-0.373492\pi$
$67$	28444.0	0.774112	0.387056	+	0.922056i	$0.373492\pi$
$68$	0	0
$69$	−14238.0	−0.360020
$70$	0	0
$71$	−45690.0	−1.07566	−0.537830	−	0.843053i	$-0.680756\pi$
$71$	−45690.0	−1.07566	−0.537830	+	0.843053i	$0.680756\pi$
$72$	0	0
$73$	−18374.0	−0.403549	−0.201775	−	0.979432i	$-0.564671\pi$
$73$	−18374.0	−0.403549	−0.201775	+	0.979432i	$0.564671\pi$
$74$	0	0
$75$	−48051.0	−0.986391
$76$	0	0
$77$	−3146.00	−0.0604689
$78$	0	0
$79$	105214.	1.89673	0.948366	−	0.317179i	$-0.102736\pi$
$79$	105214.	1.89673	0.948366	+	0.317179i	$0.102736\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−62292.0	−0.992515	−0.496257	−	0.868175i	$-0.665293\pi$
$83$	−62292.0	−0.992515	−0.496257	+	0.868175i	$0.665293\pi$
$84$	0	0
$85$	132664.	1.99162
$86$	0	0
$87$	49734.0	0.704458
$88$	0	0
$89$	−72246.0	−0.966805	−0.483402	−	0.875398i	$-0.660599\pi$
$89$	−72246.0	−0.966805	−0.483402	+	0.875398i	$0.660599\pi$
$90$	0	0
$91$	−17992.0	−0.227759
$92$	0	0
$93$	43128.0	0.517073
$94$	0	0
$95$	198720.	2.25908
$96$	0	0
$97$	79262.0	0.855334	0.427667	−	0.903936i	$-0.359336\pi$
$97$	79262.0	0.855334	0.427667	+	0.903936i	$0.359336\pi$
$98$	0	0
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−24958.0	−0.243448	−0.121724	−	0.992564i	$-0.538842\pi$
$101$	−24958.0	−0.243448	−0.121724	+	0.992564i	$0.538842\pi$
$102$	0	0
$103$	56812.0	0.527651	0.263826	−	0.964570i	$-0.415016\pi$
$103$	56812.0	0.527651	0.263826	+	0.964570i	$0.415016\pi$
$104$	0	0
$105$	21528.0	0.190559
$106$	0	0
$107$	12492.0	0.105481	0.0527403	−	0.998608i	$-0.483204\pi$
$107$	12492.0	0.105481	0.0527403	+	0.998608i	$0.483204\pi$
$108$	0	0
$109$	198748.	1.60227	0.801137	−	0.598482i	$-0.204229\pi$
$109$	198748.	1.60227	0.801137	+	0.598482i	$0.204229\pi$
$110$	0	0
$111$	91746.0	0.706773
$112$	0	0
$113$	166554.	1.22704	0.613520	−	0.789679i	$-0.289753\pi$
$113$	166554.	1.22704	0.613520	+	0.789679i	$0.289753\pi$
$114$	0	0
$115$	−145544.	−1.02624
$116$	0	0
$117$	−56052.0	−0.378553
$118$	0	0
$119$	−37492.0	−0.242701
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	95598.0	0.569752
$124$	0	0
$125$	−203688.	−1.16598
$126$	0	0
$127$	−304226.	−1.67374	−0.836868	−	0.547405i	$-0.815616\pi$
$127$	−304226.	−1.67374	−0.836868	+	0.547405i	$0.815616\pi$
$128$	0	0
$129$	77220.0	0.408560
$130$	0	0
$131$	−274428.	−1.39717	−0.698586	−	0.715526i	$-0.746187\pi$
$131$	−274428.	−1.39717	−0.698586	+	0.715526i	$0.746187\pi$
$132$	0	0
$133$	−56160.0	−0.275295
$134$	0	0
$135$	67068.0	0.316724
$136$	0	0
$137$	−245458.	−1.11732	−0.558658	−	0.829398i	$-0.688683\pi$
$137$	−245458.	−1.11732	−0.558658	+	0.829398i	$0.688683\pi$
$138$	0	0
$139$	59888.0	0.262907	0.131454	−	0.991322i	$-0.458036\pi$
$139$	59888.0	0.262907	0.131454	+	0.991322i	$0.458036\pi$
$140$	0	0
$141$	−21258.0	−0.0900481
$142$	0	0
$143$	83732.0	0.342414
$144$	0	0
$145$	508392.	2.00807
$146$	0	0
$147$	145179.	0.554128
$148$	0	0
$149$	72038.0	0.265825	0.132913	−	0.991128i	$-0.457567\pi$
$149$	72038.0	0.265825	0.132913	+	0.991128i	$0.457567\pi$
$150$	0	0
$151$	323110.	1.15321	0.576605	−	0.817023i	$-0.304377\pi$
$151$	323110.	1.15321	0.576605	+	0.817023i	$0.304377\pi$
$152$	0	0
$153$	−116802.	−0.403387
$154$	0	0
$155$	440864.	1.47393
$156$	0	0
$157$	−318766.	−1.03210	−0.516051	−	0.856558i	$-0.672599\pi$
$157$	−318766.	−1.03210	−0.516051	+	0.856558i	$0.672599\pi$
$158$	0	0
$159$	277236.	0.869675
$160$	0	0
$161$	41132.0	0.125059
$162$	0	0
$163$	431996.	1.27353	0.636767	−	0.771056i	$-0.280271\pi$
$163$	431996.	1.27353	0.636767	+	0.771056i	$0.280271\pi$
$164$	0	0
$165$	−100188.	−0.286488
$166$	0	0
$167$	251580.	0.698047	0.349024	−	0.937114i	$-0.386513\pi$
$167$	251580.	0.698047	0.349024	+	0.937114i	$0.386513\pi$
$168$	0	0
$169$	107571.	0.289720
$170$	0	0
$171$	−174960.	−0.457560
$172$	0	0
$173$	476634.	1.21079	0.605396	−	0.795924i	$-0.293015\pi$
$173$	476634.	1.21079	0.605396	+	0.795924i	$0.293015\pi$
$174$	0	0
$175$	138814.	0.342640
$176$	0	0
$177$	57744.0	0.138540
$178$	0	0
$179$	−90192.0	−0.210395	−0.105198	−	0.994451i	$-0.533547\pi$
$179$	−90192.0	−0.210395	−0.105198	+	0.994451i	$0.533547\pi$
$180$	0	0
$181$	248002.	0.562676	0.281338	−	0.959609i	$-0.409222\pi$
$181$	248002.	0.562676	0.281338	+	0.959609i	$0.409222\pi$
$182$	0	0
$183$	−378864.	−0.836288
$184$	0	0
$185$	937848.	2.01467
$186$	0	0
$187$	174482.	0.364877
$188$	0	0
$189$	−18954.0	−0.0385964
$190$	0	0
$191$	156802.	0.311006	0.155503	−	0.987835i	$-0.450300\pi$
$191$	156802.	0.311006	0.155503	+	0.987835i	$0.450300\pi$
$192$	0	0
$193$	−431234.	−0.833335	−0.416668	−	0.909059i	$-0.636802\pi$
$193$	−431234.	−0.833335	−0.416668	+	0.909059i	$0.636802\pi$
$194$	0	0
$195$	−572976.	−1.07907
$196$	0	0
$197$	−864974.	−1.58795	−0.793976	−	0.607949i	$-0.791993\pi$
$197$	−864974.	−1.58795	−0.793976	+	0.607949i	$0.791993\pi$
$198$	0	0
$199$	480060.	0.859336	0.429668	−	0.902987i	$-0.358631\pi$
$199$	480060.	0.859336	0.429668	+	0.902987i	$0.358631\pi$
$200$	0	0
$201$	−255996.	−0.446934
$202$	0	0
$203$	−143676.	−0.244706
$204$	0	0
$205$	977224.	1.62409
$206$	0	0
$207$	128142.	0.207857
$208$	0	0
$209$	261360.	0.413879
$210$	0	0
$211$	−525900.	−0.813199	−0.406600	−	0.913606i	$-0.633286\pi$
$211$	−525900.	−0.813199	−0.406600	+	0.913606i	$0.633286\pi$
$212$	0	0
$213$	411210.	0.621033
$214$	0	0
$215$	789360.	1.16461
$216$	0	0
$217$	−124592.	−0.179614
$218$	0	0
$219$	165366.	0.232989
$220$	0	0
$221$	997864.	1.37433
$222$	0	0
$223$	245264.	0.330272	0.165136	−	0.986271i	$-0.447194\pi$
$223$	245264.	0.330272	0.165136	+	0.986271i	$0.447194\pi$
$224$	0	0
$225$	432459.	0.569493
$226$	0	0
$227$	799308.	1.02955	0.514777	−	0.857324i	$-0.327875\pi$
$227$	799308.	1.02955	0.514777	+	0.857324i	$0.327875\pi$
$228$	0	0
$229$	−1.53989e6	−1.94045	−0.970224	−	0.242208i	$-0.922128\pi$
$229$	−1.53989e6	−1.94045	−0.970224	+	0.242208i	$0.922128\pi$
$230$	0	0
$231$	28314.0	0.0349117
$232$	0	0
$233$	−721830.	−0.871054	−0.435527	−	0.900176i	$-0.643438\pi$
$233$	−721830.	−0.871054	−0.435527	+	0.900176i	$0.643438\pi$
$234$	0	0
$235$	−217304.	−0.256684
$236$	0	0
$237$	−946926.	−1.09508
$238$	0	0
$239$	638436.	0.722974	0.361487	−	0.932377i	$-0.382269\pi$
$239$	638436.	0.722974	0.361487	+	0.932377i	$0.382269\pi$
$240$	0	0
$241$	220990.	0.245092	0.122546	−	0.992463i	$-0.460894\pi$
$241$	220990.	0.245092	0.122546	+	0.992463i	$0.460894\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	1.48405e6	1.57955
$246$	0	0
$247$	1.49472e6	1.55890
$248$	0	0
$249$	560628.	0.573029
$250$	0	0
$251$	−627304.	−0.628483	−0.314242	−	0.949343i	$-0.601750\pi$
$251$	−627304.	−0.628483	−0.314242	+	0.949343i	$0.601750\pi$
$252$	0	0
$253$	−191422.	−0.188014
$254$	0	0
$255$	−1.19398e6	−1.14986
$256$	0	0
$257$	−468014.	−0.442004	−0.221002	−	0.975273i	$-0.570933\pi$
$257$	−468014.	−0.442004	−0.221002	+	0.975273i	$0.570933\pi$
$258$	0	0
$259$	−265044.	−0.245510
$260$	0	0
$261$	−447606.	−0.406719
$262$	0	0
$263$	−1.54510e6	−1.37743	−0.688713	−	0.725034i	$-0.741824\pi$
$263$	−1.54510e6	−1.37743	−0.688713	+	0.725034i	$0.741824\pi$
$264$	0	0
$265$	2.83397e6	2.47902
$266$	0	0
$267$	650214.	0.558185
$268$	0	0
$269$	−1.07457e6	−0.905430	−0.452715	−	0.891655i	$-0.649544\pi$
$269$	−1.07457e6	−0.905430	−0.452715	+	0.891655i	$0.649544\pi$
$270$	0	0
$271$	−1.58723e6	−1.31285	−0.656427	−	0.754389i	$-0.727933\pi$
$271$	−1.58723e6	−1.31285	−0.656427	+	0.754389i	$0.727933\pi$
$272$	0	0
$273$	161928.	0.131497
$274$	0	0
$275$	−646019.	−0.515126
$276$	0	0
$277$	692704.	0.542436	0.271218	−	0.962518i	$-0.412574\pi$
$277$	692704.	0.542436	0.271218	+	0.962518i	$0.412574\pi$
$278$	0	0
$279$	−388152.	−0.298532
$280$	0	0
$281$	−567018.	−0.428382	−0.214191	−	0.976792i	$-0.568711\pi$
$281$	−567018.	−0.428382	−0.214191	+	0.976792i	$0.568711\pi$
$282$	0	0
$283$	−714916.	−0.530626	−0.265313	−	0.964162i	$-0.585475\pi$
$283$	−714916.	−0.530626	−0.265313	+	0.964162i	$0.585475\pi$
$284$	0	0
$285$	−1.78848e6	−1.30428
$286$	0	0
$287$	−276172.	−0.197913
$288$	0	0
$289$	659507.	0.464488
$290$	0	0
$291$	−713358.	−0.493827
$292$	0	0
$293$	2.14409e6	1.45907	0.729533	−	0.683946i	$-0.239738\pi$
$293$	2.14409e6	1.45907	0.729533	+	0.683946i	$0.239738\pi$
$294$	0	0
$295$	590272.	0.394909
$296$	0	0
$297$	88209.0	0.0580259
$298$	0	0
$299$	−1.09474e6	−0.708165
$300$	0	0
$301$	−223080.	−0.141920
$302$	0	0
$303$	224622.	0.140555
$304$	0	0
$305$	−3.87283e6	−2.38385
$306$	0	0
$307$	588808.	0.356556	0.178278	−	0.983980i	$-0.442947\pi$
$307$	588808.	0.356556	0.178278	+	0.983980i	$0.442947\pi$
$308$	0	0
$309$	−511308.	−0.304640
$310$	0	0
$311$	−2.51827e6	−1.47639	−0.738194	−	0.674588i	$-0.764321\pi$
$311$	−2.51827e6	−1.47639	−0.738194	+	0.674588i	$0.764321\pi$
$312$	0	0
$313$	−2.23562e6	−1.28985	−0.644923	−	0.764248i	$-0.723110\pi$
$313$	−2.23562e6	−1.28985	−0.644923	+	0.764248i	$0.723110\pi$
$314$	0	0
$315$	−193752.	−0.110020
$316$	0	0
$317$	1.06079e6	0.592901	0.296450	−	0.955048i	$-0.404197\pi$
$317$	1.06079e6	0.592901	0.296450	+	0.955048i	$0.404197\pi$
$318$	0	0
$319$	668646.	0.367891
$320$	0	0
$321$	−112428.	−0.0608992
$322$	0	0
$323$	3.11472e6	1.66116
$324$	0	0
$325$	−3.69459e6	−1.94025
$326$	0	0
$327$	−1.78873e6	−0.925073
$328$	0	0
$329$	61412.0	0.0312798
$330$	0	0
$331$	2.34566e6	1.17678	0.588390	−	0.808577i	$-0.299762\pi$
$331$	2.34566e6	1.17678	0.588390	+	0.808577i	$0.299762\pi$
$332$	0	0
$333$	−825714.	−0.408055
$334$	0	0
$335$	−2.61685e6	−1.27399
$336$	0	0
$337$	839978.	0.402896	0.201448	−	0.979499i	$-0.435435\pi$
$337$	839978.	0.402896	0.201448	+	0.979499i	$0.435435\pi$
$338$	0	0
$339$	−1.49899e6	−0.708432
$340$	0	0
$341$	579832.	0.270033
$342$	0	0
$343$	−856388.	−0.393039
$344$	0	0
$345$	1.30990e6	0.592501
$346$	0	0
$347$	2.02560e6	0.903086	0.451543	−	0.892249i	$-0.350874\pi$
$347$	2.02560e6	0.903086	0.451543	+	0.892249i	$0.350874\pi$
$348$	0	0
$349$	−378924.	−0.166528	−0.0832642	−	0.996528i	$-0.526535\pi$
$349$	−378924.	−0.166528	−0.0832642	+	0.996528i	$0.526535\pi$
$350$	0	0
$351$	504468.	0.218558
$352$	0	0
$353$	−1.98730e6	−0.848842	−0.424421	−	0.905465i	$-0.639522\pi$
$353$	−1.98730e6	−0.848842	−0.424421	+	0.905465i	$0.639522\pi$
$354$	0	0
$355$	4.20348e6	1.77026
$356$	0	0
$357$	337428.	0.140123
$358$	0	0
$359$	3.43975e6	1.40861	0.704305	−	0.709898i	$-0.251259\pi$
$359$	3.43975e6	1.40861	0.704305	+	0.709898i	$0.251259\pi$
$360$	0	0
$361$	2.18950e6	0.884254
$362$	0	0
$363$	−131769.	−0.0524864
$364$	0	0
$365$	1.69041e6	0.664140
$366$	0	0
$367$	1.79679e6	0.696358	0.348179	−	0.937428i	$-0.386800\pi$
$367$	1.79679e6	0.696358	0.348179	+	0.937428i	$0.386800\pi$
$368$	0	0
$369$	−860382.	−0.328947
$370$	0	0
$371$	−800904.	−0.302096
$372$	0	0
$373$	−1.43541e6	−0.534201	−0.267100	−	0.963669i	$-0.586066\pi$
$373$	−1.43541e6	−0.534201	−0.267100	+	0.963669i	$0.586066\pi$
$374$	0	0
$375$	1.83319e6	0.673178
$376$	0	0
$377$	3.82399e6	1.38568
$378$	0	0
$379$	−2.66235e6	−0.952065	−0.476033	−	0.879428i	$-0.657926\pi$
$379$	−2.66235e6	−0.952065	−0.476033	+	0.879428i	$0.657926\pi$
$380$	0	0
$381$	2.73803e6	0.966332
$382$	0	0
$383$	−2.04091e6	−0.710932	−0.355466	−	0.934689i	$-0.615678\pi$
$383$	−2.04091e6	−0.710932	−0.355466	+	0.934689i	$0.615678\pi$
$384$	0	0
$385$	289432.	0.0995164
$386$	0	0
$387$	−694980.	−0.235882
$388$	0	0
$389$	−4.29947e6	−1.44059	−0.720296	−	0.693667i	$-0.755994\pi$
$389$	−4.29947e6	−1.44059	−0.720296	+	0.693667i	$0.755994\pi$
$390$	0	0
$391$	−2.28124e6	−0.754623
$392$	0	0
$393$	2.46985e6	0.806658
$394$	0	0
$395$	−9.67969e6	−3.12154
$396$	0	0
$397$	728818.	0.232083	0.116041	−	0.993244i	$-0.462979\pi$
$397$	728818.	0.232083	0.116041	+	0.993244i	$0.462979\pi$
$398$	0	0
$399$	505440.	0.158942
$400$	0	0
$401$	−5.92515e6	−1.84009	−0.920044	−	0.391814i	$-0.871848\pi$
$401$	−5.92515e6	−1.84009	−0.920044	+	0.391814i	$0.871848\pi$
$402$	0	0
$403$	3.31606e6	1.01709
$404$	0	0
$405$	−603612.	−0.182861
$406$	0	0
$407$	1.23347e6	0.369100
$408$	0	0
$409$	1.38212e6	0.408542	0.204271	−	0.978914i	$-0.434518\pi$
$409$	1.38212e6	0.408542	0.204271	+	0.978914i	$0.434518\pi$
$410$	0	0
$411$	2.20912e6	0.645082
$412$	0	0
$413$	−166816.	−0.0481241
$414$	0	0
$415$	5.73086e6	1.63343
$416$	0	0
$417$	−538992.	−0.151790
$418$	0	0
$419$	−5.47794e6	−1.52434	−0.762170	−	0.647377i	$-0.775866\pi$
$419$	−5.47794e6	−1.52434	−0.762170	+	0.647377i	$0.775866\pi$
$420$	0	0
$421$	1.02873e6	0.282877	0.141439	−	0.989947i	$-0.454827\pi$
$421$	1.02873e6	0.282877	0.141439	+	0.989947i	$0.454827\pi$
$422$	0	0
$423$	191322.	0.0519893
$424$	0	0
$425$	−7.69884e6	−2.06753
$426$	0	0
$427$	1.09450e6	0.290499
$428$	0	0
$429$	−753588.	−0.197693
$430$	0	0
$431$	5.14310e6	1.33362	0.666810	−	0.745228i	$-0.267659\pi$
$431$	5.14310e6	1.33362	0.666810	+	0.745228i	$0.267659\pi$
$432$	0	0
$433$	412954.	0.105848	0.0529239	−	0.998599i	$-0.483146\pi$
$433$	412954.	0.105848	0.0529239	+	0.998599i	$0.483146\pi$
$434$	0	0
$435$	−4.57553e6	−1.15936
$436$	0	0
$437$	−3.41712e6	−0.855966
$438$	0	0
$439$	−5.96365e6	−1.47690	−0.738450	−	0.674309i	$-0.764442\pi$
$439$	−5.96365e6	−1.47690	−0.738450	+	0.674309i	$0.764442\pi$
$440$	0	0
$441$	−1.30661e6	−0.319926
$442$	0	0
$443$	−2.18433e6	−0.528821	−0.264410	−	0.964410i	$-0.585177\pi$
$443$	−2.18433e6	−0.528821	−0.264410	+	0.964410i	$0.585177\pi$
$444$	0	0
$445$	6.64663e6	1.59112
$446$	0	0
$447$	−648342.	−0.153474
$448$	0	0
$449$	−7858.00	−0.00183948	−0.000919742	−	1.00000i	$-0.500293\pi$
$449$	−7858.00	−0.00183948	−0.000919742		1.00000i	$0.500293\pi$
$450$	0	0
$451$	1.28526e6	0.297543
$452$	0	0
$453$	−2.90799e6	−0.665806
$454$	0	0
$455$	1.65526e6	0.374834
$456$	0	0
$457$	−899922.	−0.201565	−0.100782	−	0.994908i	$-0.532135\pi$
$457$	−899922.	−0.201565	−0.100782	+	0.994908i	$0.532135\pi$
$458$	0	0
$459$	1.05122e6	0.232895
$460$	0	0
$461$	1.13619e6	0.249000	0.124500	−	0.992220i	$-0.460267\pi$
$461$	1.13619e6	0.249000	0.124500	+	0.992220i	$0.460267\pi$
$462$	0	0
$463$	7.38964e6	1.60203	0.801016	−	0.598643i	$-0.204293\pi$
$463$	7.38964e6	1.60203	0.801016	+	0.598643i	$0.204293\pi$
$464$	0	0
$465$	−3.96778e6	−0.850971
$466$	0	0
$467$	−4.20851e6	−0.892968	−0.446484	−	0.894792i	$-0.647324\pi$
$467$	−4.20851e6	−0.892968	−0.446484	+	0.894792i	$0.647324\pi$
$468$	0	0
$469$	739544.	0.155250
$470$	0	0
$471$	2.86889e6	0.595885
$472$	0	0
$473$	1.03818e6	0.213363
$474$	0	0
$475$	−1.15322e7	−2.34520
$476$	0	0
$477$	−2.49512e6	−0.502107
$478$	0	0
$479$	−7.39441e6	−1.47253	−0.736266	−	0.676692i	$-0.763413\pi$
$479$	−7.39441e6	−1.47253	−0.736266	+	0.676692i	$0.763413\pi$
$480$	0	0
$481$	7.05425e6	1.39023
$482$	0	0
$483$	−370188.	−0.0722029
$484$	0	0
$485$	−7.29210e6	−1.40766
$486$	0	0
$487$	3.81644e6	0.729181	0.364591	−	0.931168i	$-0.381209\pi$
$487$	3.81644e6	0.729181	0.364591	+	0.931168i	$0.381209\pi$
$488$	0	0
$489$	−3.88796e6	−0.735275
$490$	0	0
$491$	−1.69716e6	−0.317702	−0.158851	−	0.987303i	$-0.550779\pi$
$491$	−1.69716e6	−0.317702	−0.158851	+	0.987303i	$0.550779\pi$
$492$	0	0
$493$	7.96849e6	1.47659
$494$	0	0
$495$	901692.	0.165404
$496$	0	0
$497$	−1.18794e6	−0.215727
$498$	0	0
$499$	−6.95160e6	−1.24978	−0.624889	−	0.780713i	$-0.714856\pi$
$499$	−6.95160e6	−1.24978	−0.624889	+	0.780713i	$0.714856\pi$
$500$	0	0
$501$	−2.26422e6	−0.403018
$502$	0	0
$503$	−6.01023e6	−1.05918	−0.529591	−	0.848253i	$-0.677655\pi$
$503$	−6.01023e6	−1.05918	−0.529591	+	0.848253i	$0.677655\pi$
$504$	0	0
$505$	2.29614e6	0.400654
$506$	0	0
$507$	−968139.	−0.167270
$508$	0	0
$509$	624660.	0.106868	0.0534342	−	0.998571i	$-0.482983\pi$
$509$	624660.	0.106868	0.0534342	+	0.998571i	$0.482983\pi$
$510$	0	0
$511$	−477724.	−0.0809328
$512$	0	0
$513$	1.57464e6	0.264173
$514$	0	0
$515$	−5.22670e6	−0.868380
$516$	0	0
$517$	−285802.	−0.0470261
$518$	0	0
$519$	−4.28971e6	−0.699051
$520$	0	0
$521$	−647490.	−0.104505	−0.0522527	−	0.998634i	$-0.516640\pi$
$521$	−647490.	−0.104505	−0.0522527	+	0.998634i	$0.516640\pi$
$522$	0	0
$523$	114676.	0.0183324	0.00916618	−	0.999958i	$-0.497082\pi$
$523$	114676.	0.0183324	0.00916618	+	0.999958i	$0.497082\pi$
$524$	0	0
$525$	−1.24933e6	−0.197823
$526$	0	0
$527$	6.91006e6	1.08382
$528$	0	0
$529$	−3.93362e6	−0.611157
$530$	0	0
$531$	−519696.	−0.0799858
$532$	0	0
$533$	7.35042e6	1.12071
$534$	0	0
$535$	−1.14926e6	−0.173594
$536$	0	0
$537$	811728.	0.121472
$538$	0	0
$539$	1.95185e6	0.289384
$540$	0	0
$541$	−2.12404e6	−0.312011	−0.156006	−	0.987756i	$-0.549862\pi$
$541$	−2.12404e6	−0.312011	−0.156006	+	0.987756i	$0.549862\pi$
$542$	0	0
$543$	−2.23202e6	−0.324861
$544$	0	0
$545$	−1.82848e7	−2.63693
$546$	0	0
$547$	−1.22672e7	−1.75299	−0.876494	−	0.481413i	$-0.840124\pi$
$547$	−1.22672e7	−1.75299	−0.876494	+	0.481413i	$0.840124\pi$
$548$	0	0
$549$	3.40978e6	0.482831
$550$	0	0
$551$	1.19362e7	1.67489
$552$	0	0
$553$	2.73556e6	0.380394
$554$	0	0
$555$	−8.44063e6	−1.16317
$556$	0	0
$557$	1.10980e7	1.51568	0.757839	−	0.652442i	$-0.226255\pi$
$557$	1.10980e7	1.51568	0.757839	+	0.652442i	$0.226255\pi$
$558$	0	0
$559$	5.93736e6	0.803644
$560$	0	0
$561$	−1.57034e6	−0.210662
$562$	0	0
$563$	−4.61984e6	−0.614265	−0.307132	−	0.951667i	$-0.599369\pi$
$563$	−4.61984e6	−0.614265	−0.307132	+	0.951667i	$0.599369\pi$
$564$	0	0
$565$	−1.53230e7	−2.01940
$566$	0	0
$567$	170586.	0.0222836
$568$	0	0
$569$	1.01716e7	1.31707	0.658537	−	0.752548i	$-0.271176\pi$
$569$	1.01716e7	1.31707	0.658537	+	0.752548i	$0.271176\pi$
$570$	0	0
$571$	9.36866e6	1.20251	0.601253	−	0.799059i	$-0.294669\pi$
$571$	9.36866e6	1.20251	0.601253	+	0.799059i	$0.294669\pi$
$572$	0	0
$573$	−1.41122e6	−0.179559
$574$	0	0
$575$	8.44630e6	1.06536
$576$	0	0
$577$	−6.14973e6	−0.768983	−0.384491	−	0.923129i	$-0.625623\pi$
$577$	−6.14973e6	−0.768983	−0.384491	+	0.923129i	$0.625623\pi$
$578$	0	0
$579$	3.88111e6	0.481126
$580$	0	0
$581$	−1.61959e6	−0.199051
$582$	0	0
$583$	3.72728e6	0.454173
$584$	0	0
$585$	5.15678e6	0.623002
$586$	0	0
$587$	−1.04649e6	−0.125354	−0.0626771	−	0.998034i	$-0.519964\pi$
$587$	−1.04649e6	−0.125354	−0.0626771	+	0.998034i	$0.519964\pi$
$588$	0	0
$589$	1.03507e7	1.22937
$590$	0	0
$591$	7.78477e6	0.916805
$592$	0	0
$593$	−3.31784e6	−0.387453	−0.193726	−	0.981056i	$-0.562057\pi$
$593$	−3.31784e6	−0.387453	−0.193726	+	0.981056i	$0.562057\pi$
$594$	0	0
$595$	3.44926e6	0.399424
$596$	0	0
$597$	−4.32054e6	−0.496138
$598$	0	0
$599$	1.73991e7	1.98134	0.990670	−	0.136280i	$-0.0435146\pi$
$599$	1.73991e7	1.98134	0.990670	+	0.136280i	$0.0435146\pi$
$600$	0	0
$601$	7.13163e6	0.805383	0.402691	−	0.915336i	$-0.368075\pi$
$601$	7.13163e6	0.805383	0.402691	+	0.915336i	$0.368075\pi$
$602$	0	0
$603$	2.30396e6	0.258037
$604$	0	0
$605$	−1.34697e6	−0.149613
$606$	0	0
$607$	9.64617e6	1.06263	0.531317	−	0.847173i	$-0.321697\pi$
$607$	9.64617e6	1.06263	0.531317	+	0.847173i	$0.321697\pi$
$608$	0	0
$609$	1.29308e6	0.141281
$610$	0	0
$611$	−1.63450e6	−0.177126
$612$	0	0
$613$	3.68170e6	0.395729	0.197864	−	0.980229i	$-0.436599\pi$
$613$	3.68170e6	0.395729	0.197864	+	0.980229i	$0.436599\pi$
$614$	0	0
$615$	−8.79502e6	−0.937667
$616$	0	0
$617$	1.83190e7	1.93727	0.968635	−	0.248489i	$-0.0799340\pi$
$617$	1.83190e7	1.93727	0.968635	+	0.248489i	$0.0799340\pi$
$618$	0	0
$619$	−1.09660e6	−0.115033	−0.0575166	−	0.998345i	$-0.518318\pi$
$619$	−1.09660e6	−0.115033	−0.0575166	+	0.998345i	$0.518318\pi$
$620$	0	0
$621$	−1.15328e6	−0.120007
$622$	0	0
$623$	−1.87840e6	−0.193895
$624$	0	0
$625$	2.05492e6	0.210424
$626$	0	0
$627$	−2.35224e6	−0.238953
$628$	0	0
$629$	1.46997e7	1.48144
$630$	0	0
$631$	9.58030e6	0.957869	0.478934	−	0.877851i	$-0.341023\pi$
$631$	9.58030e6	0.957869	0.478934	+	0.877851i	$0.341023\pi$
$632$	0	0
$633$	4.73310e6	0.469501
$634$	0	0
$635$	2.79888e7	2.75454
$636$	0	0
$637$	1.11627e7	1.08998
$638$	0	0
$639$	−3.70089e6	−0.358554
$640$	0	0
$641$	−1.18062e7	−1.13492	−0.567462	−	0.823400i	$-0.692075\pi$
$641$	−1.18062e7	−1.13492	−0.567462	+	0.823400i	$0.692075\pi$
$642$	0	0
$643$	5.88298e6	0.561138	0.280569	−	0.959834i	$-0.409477\pi$
$643$	5.88298e6	0.561138	0.280569	+	0.959834i	$0.409477\pi$
$644$	0	0
$645$	−7.10424e6	−0.672385
$646$	0	0
$647$	3.62822e6	0.340748	0.170374	−	0.985379i	$-0.445502\pi$
$647$	3.62822e6	0.340748	0.170374	+	0.985379i	$0.445502\pi$
$648$	0	0
$649$	776336.	0.0723499
$650$	0	0
$651$	1.12133e6	0.103700
$652$	0	0
$653$	−5.70795e6	−0.523838	−0.261919	−	0.965090i	$-0.584355\pi$
$653$	−5.70795e6	−0.523838	−0.261919	+	0.965090i	$0.584355\pi$
$654$	0	0
$655$	2.52474e7	2.29939
$656$	0	0
$657$	−1.48829e6	−0.134516
$658$	0	0
$659$	−1.08205e7	−0.970588	−0.485294	−	0.874351i	$-0.661287\pi$
$659$	−1.08205e7	−0.970588	−0.485294	+	0.874351i	$0.661287\pi$
$660$	0	0
$661$	1.14311e7	1.01762	0.508809	−	0.860879i	$-0.330086\pi$
$661$	1.14311e7	1.01762	0.508809	+	0.860879i	$0.330086\pi$
$662$	0	0
$663$	−8.98078e6	−0.793469
$664$	0	0
$665$	5.16672e6	0.453065
$666$	0	0
$667$	−8.74213e6	−0.760857
$668$	0	0
$669$	−2.20738e6	−0.190683
$670$	0	0
$671$	−5.09362e6	−0.436737
$672$	0	0
$673$	−2.03858e7	−1.73496	−0.867482	−	0.497468i	$-0.834263\pi$
$673$	−2.03858e7	−1.73496	−0.867482	+	0.497468i	$0.834263\pi$
$674$	0	0
$675$	−3.89213e6	−0.328797
$676$	0	0
$677$	6.09278e6	0.510909	0.255455	−	0.966821i	$-0.417775\pi$
$677$	6.09278e6	0.510909	0.255455	+	0.966821i	$0.417775\pi$
$678$	0	0
$679$	2.06081e6	0.171539
$680$	0	0
$681$	−7.19377e6	−0.594414
$682$	0	0
$683$	−1.44978e7	−1.18918	−0.594592	−	0.804027i	$-0.702686\pi$
$683$	−1.44978e7	−1.18918	−0.594592	+	0.804027i	$0.702686\pi$
$684$	0	0
$685$	2.25821e7	1.83882
$686$	0	0
$687$	1.38590e7	1.12032
$688$	0	0
$689$	2.13164e7	1.71067
$690$	0	0
$691$	−9.87069e6	−0.786416	−0.393208	−	0.919449i	$-0.628635\pi$
$691$	−9.87069e6	−0.786416	−0.393208	+	0.919449i	$0.628635\pi$
$692$	0	0
$693$	−254826.	−0.0201563
$694$	0	0
$695$	−5.50970e6	−0.432679
$696$	0	0
$697$	1.53169e7	1.19423
$698$	0	0
$699$	6.49647e6	0.502903
$700$	0	0
$701$	6.35411e6	0.488382	0.244191	−	0.969727i	$-0.421478\pi$
$701$	6.35411e6	0.488382	0.244191	+	0.969727i	$0.421478\pi$
$702$	0	0
$703$	2.20190e7	1.68039
$704$	0	0
$705$	1.95574e6	0.148196
$706$	0	0
$707$	−648908.	−0.0488241
$708$	0	0
$709$	−411382.	−0.0307348	−0.0153674	−	0.999882i	$-0.504892\pi$
$709$	−411382.	−0.0307348	−0.0153674	+	0.999882i	$0.504892\pi$
$710$	0	0
$711$	8.52233e6	0.632244
$712$	0	0
$713$	−7.58094e6	−0.558470
$714$	0	0
$715$	−7.70334e6	−0.563526
$716$	0	0
$717$	−5.74592e6	−0.417409
$718$	0	0
$719$	−6.29795e6	−0.454336	−0.227168	−	0.973856i	$-0.572947\pi$
$719$	−6.29795e6	−0.454336	−0.227168	+	0.973856i	$0.572947\pi$
$720$	0	0
$721$	1.47711e6	0.105822
$722$	0	0
$723$	−1.98891e6	−0.141504
$724$	0	0
$725$	−2.95033e7	−2.08461
$726$	0	0
$727$	−1.14699e7	−0.804866	−0.402433	−	0.915449i	$-0.631835\pi$
$727$	−1.14699e7	−0.804866	−0.402433	+	0.915449i	$0.631835\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.23724e7	0.856365
$732$	0	0
$733$	−1.87547e7	−1.28929	−0.644646	−	0.764481i	$-0.722995\pi$
$733$	−1.87547e7	−1.28929	−0.644646	+	0.764481i	$0.722995\pi$
$734$	0	0
$735$	−1.33565e7	−0.911955
$736$	0	0
$737$	−3.44172e6	−0.233403
$738$	0	0
$739$	−2.79727e6	−0.188418	−0.0942091	−	0.995552i	$-0.530032\pi$
$739$	−2.79727e6	−0.188418	−0.0942091	+	0.995552i	$0.530032\pi$
$740$	0	0
$741$	−1.34525e7	−0.900030
$742$	0	0
$743$	2.25651e7	1.49956	0.749781	−	0.661686i	$-0.230159\pi$
$743$	2.25651e7	1.49956	0.749781	+	0.661686i	$0.230159\pi$
$744$	0	0
$745$	−6.62750e6	−0.437481
$746$	0	0
$747$	−5.04565e6	−0.330838
$748$	0	0
$749$	324792.	0.0211544
$750$	0	0
$751$	7.49233e6	0.484749	0.242375	−	0.970183i	$-0.422074\pi$
$751$	7.49233e6	0.484749	0.242375	+	0.970183i	$0.422074\pi$
$752$	0	0
$753$	5.64574e6	0.362855
$754$	0	0
$755$	−2.97261e7	−1.89789
$756$	0	0
$757$	2.88492e7	1.82976	0.914880	−	0.403727i	$-0.132285\pi$
$757$	2.88492e7	1.82976	0.914880	+	0.403727i	$0.132285\pi$
$758$	0	0
$759$	1.72280e6	0.108550
$760$	0	0
$761$	−9.56279e6	−0.598581	−0.299291	−	0.954162i	$-0.596750\pi$
$761$	−9.56279e6	−0.598581	−0.299291	+	0.954162i	$0.596750\pi$
$762$	0	0
$763$	5.16745e6	0.321340
$764$	0	0
$765$	1.07458e7	0.663872
$766$	0	0
$767$	4.43987e6	0.272510
$768$	0	0
$769$	−744898.	−0.0454235	−0.0227118	−	0.999742i	$-0.507230\pi$
$769$	−744898.	−0.0454235	−0.0227118	+	0.999742i	$0.507230\pi$
$770$	0	0
$771$	4.21213e6	0.255191
$772$	0	0
$773$	6.07336e6	0.365578	0.182789	−	0.983152i	$-0.441487\pi$
$773$	6.07336e6	0.365578	0.182789	+	0.983152i	$0.441487\pi$
$774$	0	0
$775$	−2.55845e7	−1.53011
$776$	0	0
$777$	2.38540e6	0.141745
$778$	0	0
$779$	2.29435e7	1.35462
$780$	0	0
$781$	5.52849e6	0.324324
$782$	0	0
$783$	4.02845e6	0.234819
$784$	0	0
$785$	2.93265e7	1.69858
$786$	0	0
$787$	1.47512e7	0.848966	0.424483	−	0.905436i	$-0.360456\pi$
$787$	1.47512e7	0.848966	0.424483	+	0.905436i	$0.360456\pi$
$788$	0	0
$789$	1.39059e7	0.795257
$790$	0	0
$791$	4.33040e6	0.246086
$792$	0	0
$793$	−2.91304e7	−1.64499
$794$	0	0
$795$	−2.55057e7	−1.43126
$796$	0	0
$797$	−2.78359e7	−1.55224	−0.776121	−	0.630584i	$-0.782815\pi$
$797$	−2.78359e7	−1.55224	−0.776121	+	0.630584i	$0.782815\pi$
$798$	0	0
$799$	−3.40600e6	−0.188746
$800$	0	0
$801$	−5.85193e6	−0.322268
$802$	0	0
$803$	2.22325e6	0.121675
$804$	0	0
$805$	−3.78414e6	−0.205815
$806$	0	0
$807$	9.67115e6	0.522750
$808$	0	0
$809$	2.54767e7	1.36859	0.684293	−	0.729207i	$-0.260111\pi$
$809$	2.54767e7	1.36859	0.684293	+	0.729207i	$0.260111\pi$
$810$	0	0
$811$	1.91915e7	1.02460	0.512302	−	0.858805i	$-0.328793\pi$
$811$	1.91915e7	1.02460	0.512302	+	0.858805i	$0.328793\pi$
$812$	0	0
$813$	1.42851e7	0.757977
$814$	0	0
$815$	−3.97436e7	−2.09591
$816$	0	0
$817$	1.85328e7	0.971373
$818$	0	0
$819$	−1.45735e6	−0.0759197
$820$	0	0
$821$	3.27107e6	0.169368	0.0846840	−	0.996408i	$-0.473012\pi$
$821$	3.27107e6	0.169368	0.0846840	+	0.996408i	$0.473012\pi$
$822$	0	0
$823$	3.19195e7	1.64269	0.821347	−	0.570430i	$-0.193223\pi$
$823$	3.19195e7	1.64269	0.821347	+	0.570430i	$0.193223\pi$
$824$	0	0
$825$	5.81417e6	0.297408
$826$	0	0
$827$	−2.45556e7	−1.24850	−0.624248	−	0.781226i	$-0.714595\pi$
$827$	−2.45556e7	−1.24850	−0.624248	+	0.781226i	$0.714595\pi$
$828$	0	0
$829$	−1.40969e7	−0.712421	−0.356211	−	0.934406i	$-0.615931\pi$
$829$	−1.40969e7	−0.712421	−0.356211	+	0.934406i	$0.615931\pi$
$830$	0	0
$831$	−6.23434e6	−0.313175
$832$	0	0
$833$	2.32609e7	1.16149
$834$	0	0
$835$	−2.31454e7	−1.14881
$836$	0	0
$837$	3.49337e6	0.172358
$838$	0	0
$839$	3.01443e6	0.147843	0.0739213	−	0.997264i	$-0.476449\pi$
$839$	3.01443e6	0.147843	0.0739213	+	0.997264i	$0.476449\pi$
$840$	0	0
$841$	1.00255e7	0.488784
$842$	0	0
$843$	5.10316e6	0.247326
$844$	0	0
$845$	−9.89653e6	−0.476806
$846$	0	0
$847$	380666.	0.0182321
$848$	0	0
$849$	6.43424e6	0.306357
$850$	0	0
$851$	−1.61269e7	−0.763356
$852$	0	0
$853$	−1.67201e7	−0.786806	−0.393403	−	0.919366i	$-0.628702\pi$
$853$	−1.67201e7	−0.786806	−0.393403	+	0.919366i	$0.628702\pi$
$854$	0	0
$855$	1.60963e7	0.753028
$856$	0	0
$857$	−9.15871e6	−0.425973	−0.212987	−	0.977055i	$-0.568319\pi$
$857$	−9.15871e6	−0.425973	−0.212987	+	0.977055i	$0.568319\pi$
$858$	0	0
$859$	−1.51068e7	−0.698536	−0.349268	−	0.937023i	$-0.613570\pi$
$859$	−1.51068e7	−0.698536	−0.349268	+	0.937023i	$0.613570\pi$
$860$	0	0
$861$	2.48555e6	0.114265
$862$	0	0
$863$	−5.11568e6	−0.233817	−0.116909	−	0.993143i	$-0.537298\pi$
$863$	−5.11568e6	−0.233817	−0.116909	+	0.993143i	$0.537298\pi$
$864$	0	0
$865$	−4.38503e7	−1.99266
$866$	0	0
$867$	−5.93556e6	−0.268172
$868$	0	0
$869$	−1.27309e7	−0.571886
$870$	0	0
$871$	−1.96832e7	−0.879127
$872$	0	0
$873$	6.42022e6	0.285111
$874$	0	0
$875$	−5.29589e6	−0.233840
$876$	0	0
$877$	−1.26998e7	−0.557568	−0.278784	−	0.960354i	$-0.589931\pi$
$877$	−1.26998e7	−0.557568	−0.278784	+	0.960354i	$0.589931\pi$
$878$	0	0
$879$	−1.92968e7	−0.842392
$880$	0	0
$881$	−8.38173e6	−0.363826	−0.181913	−	0.983315i	$-0.558229\pi$
$881$	−8.38173e6	−0.363826	−0.181913	+	0.983315i	$0.558229\pi$
$882$	0	0
$883$	1.69529e7	0.731715	0.365858	−	0.930671i	$-0.380776\pi$
$883$	1.69529e7	0.731715	0.365858	+	0.930671i	$0.380776\pi$
$884$	0	0
$885$	−5.31245e6	−0.228001
$886$	0	0
$887$	1.05143e7	0.448717	0.224359	−	0.974507i	$-0.427971\pi$
$887$	1.05143e7	0.448717	0.224359	+	0.974507i	$0.427971\pi$
$888$	0	0
$889$	−7.90988e6	−0.335672
$890$	0	0
$891$	−793881.	−0.0335013
$892$	0	0
$893$	−5.10192e6	−0.214094
$894$	0	0
$895$	8.29766e6	0.346257
$896$	0	0
$897$	9.85270e6	0.408859
$898$	0	0
$899$	2.64806e7	1.09277
$900$	0	0
$901$	4.44194e7	1.82289
$902$	0	0
$903$	2.00772e6	0.0819377
$904$	0	0
$905$	−2.28162e7	−0.926023
$906$	0	0
$907$	1.53747e7	0.620569	0.310284	−	0.950644i	$-0.399576\pi$
$907$	1.53747e7	0.620569	0.310284	+	0.950644i	$0.399576\pi$
$908$	0	0
$909$	−2.02160e6	−0.0811494
$910$	0	0
$911$	−1.25424e7	−0.500708	−0.250354	−	0.968154i	$-0.580547\pi$
$911$	−1.25424e7	−0.500708	−0.250354	+	0.968154i	$0.580547\pi$
$912$	0	0
$913$	7.53733e6	0.299255
$914$	0	0
$915$	3.48555e7	1.37632
$916$	0	0
$917$	−7.13513e6	−0.280207
$918$	0	0
$919$	−3.31432e7	−1.29451	−0.647256	−	0.762273i	$-0.724084\pi$
$919$	−3.31432e7	−1.29451	−0.647256	+	0.762273i	$0.724084\pi$
$920$	0	0
$921$	−5.29927e6	−0.205858
$922$	0	0
$923$	3.16175e7	1.22158
$924$	0	0
$925$	−5.44258e7	−2.09146
$926$	0	0
$927$	4.60177e6	0.175884
$928$	0	0
$929$	3.10442e7	1.18016	0.590080	−	0.807345i	$-0.299096\pi$
$929$	3.10442e7	1.18016	0.590080	+	0.807345i	$0.299096\pi$
$930$	0	0
$931$	3.48430e7	1.31747
$932$	0	0
$933$	2.26644e7	0.852393
$934$	0	0
$935$	−1.60523e7	−0.600495
$936$	0	0
$937$	−3.10737e7	−1.15623	−0.578115	−	0.815955i	$-0.696212\pi$
$937$	−3.10737e7	−1.15623	−0.578115	+	0.815955i	$0.696212\pi$
$938$	0	0
$939$	2.01206e7	0.744692
$940$	0	0
$941$	−2.50349e7	−0.921664	−0.460832	−	0.887488i	$-0.652449\pi$
$941$	−2.50349e7	−0.921664	−0.460832	+	0.887488i	$0.652449\pi$
$942$	0	0
$943$	−1.68040e7	−0.615366
$944$	0	0
$945$	1.74377e6	0.0635198
$946$	0	0
$947$	5.37383e6	0.194719	0.0973596	−	0.995249i	$-0.468960\pi$
$947$	5.37383e6	0.194719	0.0973596	+	0.995249i	$0.468960\pi$
$948$	0	0
$949$	1.27148e7	0.458294
$950$	0	0
$951$	−9.54713e6	−0.342311
$952$	0	0
$953$	7.26908e6	0.259267	0.129634	−	0.991562i	$-0.458620\pi$
$953$	7.26908e6	0.259267	0.129634	+	0.991562i	$0.458620\pi$
$954$	0	0
$955$	−1.44258e7	−0.511836
$956$	0	0
$957$	−6.01781e6	−0.212402
$958$	0	0
$959$	−6.38191e6	−0.224080
$960$	0	0
$961$	−5.66589e6	−0.197906
$962$	0	0
$963$	1.01185e6	0.0351602
$964$	0	0
$965$	3.96735e7	1.37146
$966$	0	0
$967$	2.54428e7	0.874983	0.437491	−	0.899223i	$-0.355867\pi$
$967$	2.54428e7	0.874983	0.437491	+	0.899223i	$0.355867\pi$
$968$	0	0
$969$	−2.80325e7	−0.959074
$970$	0	0
$971$	−9.88213e6	−0.336358	−0.168179	−	0.985756i	$-0.553789\pi$
$971$	−9.88213e6	−0.336358	−0.168179	+	0.985756i	$0.553789\pi$
$972$	0	0
$973$	1.55709e6	0.0527268
$974$	0	0
$975$	3.32513e7	1.12020
$976$	0	0
$977$	2.22197e6	0.0744736	0.0372368	−	0.999306i	$-0.488144\pi$
$977$	2.22197e6	0.0744736	0.0372368	+	0.999306i	$0.488144\pi$
$978$	0	0
$979$	8.74177e6	0.291503
$980$	0	0
$981$	1.60986e7	0.534091
$982$	0	0
$983$	−2.53706e7	−0.837428	−0.418714	−	0.908118i	$-0.637519\pi$
$983$	−2.53706e7	−0.837428	−0.418714	+	0.908118i	$0.637519\pi$
$984$	0	0
$985$	7.95776e7	2.61337
$986$	0	0
$987$	−552708.	−0.0180594
$988$	0	0
$989$	−1.35736e7	−0.441269
$990$	0	0
$991$	−3.24132e7	−1.04843	−0.524214	−	0.851587i	$-0.675641\pi$
$991$	−3.24132e7	−1.04843	−0.524214	+	0.851587i	$0.675641\pi$
$992$	0	0
$993$	−2.11109e7	−0.679414
$994$	0	0
$995$	−4.41655e7	−1.41425
$996$	0	0
$997$	−1.55048e6	−0.0494000	−0.0247000	−	0.999695i	$-0.507863\pi$
$997$	−1.55048e6	−0.0494000	−0.0247000	+	0.999695i	$0.507863\pi$
$998$	0	0
$999$	7.43143e6	0.235591

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.a.a.1.1		1
4.3	odd	2		33.6.a.b.1.1	✓	1
12.11	even	2		99.6.a.a.1.1		1
20.19	odd	2		825.6.a.a.1.1		1
44.43	even	2		363.6.a.b.1.1		1
132.131	odd	2		1089.6.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.b.1.1	✓	1	4.3	odd	2
99.6.a.a.1.1		1	12.11	even	2
363.6.a.b.1.1		1	44.43	even	2
528.6.a.a.1.1		1	1.1	even	1	trivial
825.6.a.a.1.1		1	20.19	odd	2
1089.6.a.h.1.1		1	132.131	odd	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:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	528.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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