LMFDB - Embedded newform 825.6.a.a.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:	$0$
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	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -9.00000 q^{3} -31.0000 q^{4} +9.00000 q^{6} +26.0000 q^{7} +63.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -9.00000 q^{3} -31.0000 q^{4} +9.00000 q^{6} +26.0000 q^{7} +63.0000 q^{8} +81.0000 q^{9} +121.000 q^{11} +279.000 q^{12} +692.000 q^{13} -26.0000 q^{14} +929.000 q^{16} +1442.00 q^{17} -81.0000 q^{18} +2160.00 q^{19} -234.000 q^{21} -121.000 q^{22} +1582.00 q^{23} -567.000 q^{24} -692.000 q^{26} -729.000 q^{27} -806.000 q^{28} -5526.00 q^{29} +4792.00 q^{31} -2945.00 q^{32} -1089.00 q^{33} -1442.00 q^{34} -2511.00 q^{36} +10194.0 q^{37} -2160.00 q^{38} -6228.00 q^{39} -10622.0 q^{41} +234.000 q^{42} -8580.00 q^{43} -3751.00 q^{44} -1582.00 q^{46} +2362.00 q^{47} -8361.00 q^{48} -16131.0 q^{49} -12978.0 q^{51} -21452.0 q^{52} +30804.0 q^{53} +729.000 q^{54} +1638.00 q^{56} -19440.0 q^{57} +5526.00 q^{58} +6416.00 q^{59} +42096.0 q^{61} -4792.00 q^{62} +2106.00 q^{63} -26783.0 q^{64} +1089.00 q^{66} +28444.0 q^{67} -44702.0 q^{68} -14238.0 q^{69} +45690.0 q^{71} +5103.00 q^{72} +18374.0 q^{73} -10194.0 q^{74} -66960.0 q^{76} +3146.00 q^{77} +6228.00 q^{78} -105214. q^{79} +6561.00 q^{81} +10622.0 q^{82} -62292.0 q^{83} +7254.00 q^{84} +8580.00 q^{86} +49734.0 q^{87} +7623.00 q^{88} -72246.0 q^{89} +17992.0 q^{91} -49042.0 q^{92} -43128.0 q^{93} -2362.00 q^{94} +26505.0 q^{96} -79262.0 q^{97} +16131.0 q^{98} +9801.00 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.176777	−0.0883883	−	0.996086i	$-0.528172\pi$
$2$	−1.00000	−0.176777	−0.0883883	+	0.996086i	$0.528172\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−31.0000	−0.968750
$5$	0	0
$5$	0	0
$6$	9.00000	0.102062
$7$	26.0000	0.200553	0.100276	−	0.994960i	$-0.468027\pi$
$7$	26.0000	0.200553	0.100276	+	0.994960i	$0.468027\pi$
$8$	63.0000	0.348029
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	279.000	0.559308
$13$	692.000	1.13566	0.567829	−	0.823146i	$-0.307783\pi$
$13$	692.000	1.13566	0.567829	+	0.823146i	$0.307783\pi$
$14$	−26.0000	−0.0354530
$15$	0	0
$16$	929.000	0.907227
$17$	1442.00	1.21016	0.605080	−	0.796165i	$-0.293141\pi$
$17$	1442.00	1.21016	0.605080	+	0.796165i	$0.293141\pi$
$18$	−81.0000	−0.0589256
$19$	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
$21$	−234.000	−0.115789
$22$	−121.000	−0.0533002
$23$	1582.00	0.623572	0.311786	−	0.950152i	$-0.399073\pi$
$23$	1582.00	0.623572	0.311786	+	0.950152i	$0.399073\pi$
$24$	−567.000	−0.200935
$25$	0	0
$26$	−692.000	−0.200758
$27$	−729.000	−0.192450
$28$	−806.000	−0.194285
$29$	−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	$-0.352208\pi$
$31$	4792.00	0.895597	0.447798	+	0.894135i	$0.352208\pi$
$32$	−2945.00	−0.508406
$33$	−1089.00	−0.174078
$34$	−1442.00	−0.213928
$35$	0	0
$36$	−2511.00	−0.322917
$37$	10194.0	1.22417	0.612083	−	0.790794i	$-0.290332\pi$
$37$	10194.0	1.22417	0.612083	+	0.790794i	$0.290332\pi$
$38$	−2160.00	−0.242658
$39$	−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	$0.664253\pi$
$42$	234.000	0.0204688
$43$	−8580.00	−0.707646	−0.353823	−	0.935312i	$-0.615119\pi$
$43$	−8580.00	−0.707646	−0.353823	+	0.935312i	$0.615119\pi$
$44$	−3751.00	−0.292089
$45$	0	0
$46$	−1582.00	−0.110233
$47$	2362.00	0.155968	0.0779840	−	0.996955i	$-0.475152\pi$
$47$	2362.00	0.155968	0.0779840	+	0.996955i	$0.475152\pi$
$48$	−8361.00	−0.523788
$49$	−16131.0	−0.959779
$50$	0	0
$51$	−12978.0	−0.698686
$52$	−21452.0	−1.10017
$53$	30804.0	1.50632	0.753160	−	0.657837i	$-0.228528\pi$
$53$	30804.0	1.50632	0.753160	+	0.657837i	$0.228528\pi$
$54$	729.000	0.0340207
$55$	0	0
$56$	1638.00	0.0697981
$57$	−19440.0	−0.792518
$58$	5526.00	0.215695
$59$	6416.00	0.239957	0.119979	−	0.992776i	$-0.461717\pi$
$59$	6416.00	0.239957	0.119979	+	0.992776i	$0.461717\pi$
$60$	0	0
$61$	42096.0	1.44849	0.724246	−	0.689541i	$-0.242188\pi$
$61$	42096.0	1.44849	0.724246	+	0.689541i	$0.242188\pi$
$62$	−4792.00	−0.158321
$63$	2106.00	0.0668509
$64$	−26783.0	−0.817352
$65$	0	0
$66$	1089.00	0.0307729
$67$	28444.0	0.774112	0.387056	−	0.922056i	$-0.373492\pi$
$67$	28444.0	0.774112	0.387056	+	0.922056i	$0.373492\pi$
$68$	−44702.0	−1.17234
$69$	−14238.0	−0.360020
$70$	0	0
$71$	45690.0	1.07566	0.537830	−	0.843053i	$-0.319244\pi$
$71$	45690.0	1.07566	0.537830	+	0.843053i	$0.319244\pi$
$72$	5103.00	0.116010
$73$	18374.0	0.403549	0.201775	−	0.979432i	$-0.435329\pi$
$73$	18374.0	0.403549	0.201775	+	0.979432i	$0.435329\pi$
$74$	−10194.0	−0.216404
$75$	0	0
$76$	−66960.0	−1.32979
$77$	3146.00	0.0604689
$78$	6228.00	0.115908
$79$	−105214.	−1.89673	−0.948366	−	0.317179i	$-0.897264\pi$
$79$	−105214.	−1.89673	−0.948366	+	0.317179i	$0.897264\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	10622.0	0.174450
$83$	−62292.0	−0.992515	−0.496257	−	0.868175i	$-0.665293\pi$
$83$	−62292.0	−0.992515	−0.496257	+	0.868175i	$0.665293\pi$
$84$	7254.00	0.112171
$85$	0	0
$86$	8580.00	0.125095
$87$	49734.0	0.704458
$88$	7623.00	0.104935
$89$	−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
$92$	−49042.0	−0.604086
$93$	−43128.0	−0.517073
$94$	−2362.00	−0.0275715
$95$	0	0
$96$	26505.0	0.293528
$97$	−79262.0	−0.855334	−0.427667	−	0.903936i	$-0.640664\pi$
$97$	−79262.0	−0.855334	−0.427667	+	0.903936i	$0.640664\pi$
$98$	16131.0	0.169667
$99$	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	$0.538842\pi$
$102$	12978.0	0.123511
$103$	56812.0	0.527651	0.263826	−	0.964570i	$-0.415016\pi$
$103$	56812.0	0.527651	0.263826	+	0.964570i	$0.415016\pi$
$104$	43596.0	0.395242
$105$	0	0
$106$	−30804.0	−0.266282
$107$	12492.0	0.105481	0.0527403	−	0.998608i	$-0.483204\pi$
$107$	12492.0	0.105481	0.0527403	+	0.998608i	$0.483204\pi$
$108$	22599.0	0.186436
$109$	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
$112$	24154.0	0.181947
$113$	−166554.	−1.22704	−0.613520	−	0.789679i	$-0.710247\pi$
$113$	−166554.	−1.22704	−0.613520	+	0.789679i	$0.710247\pi$
$114$	19440.0	0.140099
$115$	0	0
$116$	171306.	1.18203
$117$	56052.0	0.378553
$118$	−6416.00	−0.0424189
$119$	37492.0	0.242701
$120$	0	0
$121$	14641.0	0.0909091
$122$	−42096.0	−0.256060
$123$	95598.0	0.569752
$124$	−148552.	−0.867609
$125$	0	0
$126$	−2106.00	−0.0118177
$127$	−304226.	−1.67374	−0.836868	−	0.547405i	$-0.815616\pi$
$127$	−304226.	−1.67374	−0.836868	+	0.547405i	$0.815616\pi$
$128$	121023.	0.652894
$129$	77220.0	0.408560
$130$	0	0
$131$	274428.	1.39717	0.698586	−	0.715526i	$-0.253813\pi$
$131$	274428.	1.39717	0.698586	+	0.715526i	$0.253813\pi$
$132$	33759.0	0.168638
$133$	56160.0	0.275295
$134$	−28444.0	−0.136845
$135$	0	0
$136$	90846.0	0.421171
$137$	245458.	1.11732	0.558658	−	0.829398i	$-0.311317\pi$
$137$	245458.	1.11732	0.558658	+	0.829398i	$0.311317\pi$
$138$	14238.0	0.0636431
$139$	−59888.0	−0.262907	−0.131454	−	0.991322i	$-0.541964\pi$
$139$	−59888.0	−0.262907	−0.131454	+	0.991322i	$0.541964\pi$
$140$	0	0
$141$	−21258.0	−0.0900481
$142$	−45690.0	−0.190152
$143$	83732.0	0.342414
$144$	75249.0	0.302409
$145$	0	0
$146$	−18374.0	−0.0713381
$147$	145179.	0.554128
$148$	−316014.	−1.18591
$149$	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	$-0.695623\pi$
$151$	−323110.	−1.15321	−0.576605	+	0.817023i	$0.695623\pi$
$152$	136080.	0.477733
$153$	116802.	0.403387
$154$	−3146.00	−0.0106895
$155$	0	0
$156$	193068.	0.635183
$157$	318766.	1.03210	0.516051	−	0.856558i	$-0.327401\pi$
$157$	318766.	1.03210	0.516051	+	0.856558i	$0.327401\pi$
$158$	105214.	0.335298
$159$	−277236.	−0.869675
$160$	0	0
$161$	41132.0	0.125059
$162$	−6561.00	−0.0196419
$163$	431996.	1.27353	0.636767	−	0.771056i	$-0.280271\pi$
$163$	431996.	1.27353	0.636767	+	0.771056i	$0.280271\pi$
$164$	329282.	0.956001
$165$	0	0
$166$	62292.0	0.175454
$167$	251580.	0.698047	0.349024	−	0.937114i	$-0.386513\pi$
$167$	251580.	0.698047	0.349024	+	0.937114i	$0.386513\pi$
$168$	−14742.0	−0.0402980
$169$	107571.	0.289720
$170$	0	0
$171$	174960.	0.457560
$172$	265980.	0.685532
$173$	−476634.	−1.21079	−0.605396	−	0.795924i	$-0.706985\pi$
$173$	−476634.	−1.21079	−0.605396	+	0.795924i	$0.706985\pi$
$174$	−49734.0	−0.124532
$175$	0	0
$176$	112409.	0.273539
$177$	−57744.0	−0.138540
$178$	72246.0	0.170909
$179$	90192.0	0.210395	0.105198	−	0.994451i	$-0.466453\pi$
$179$	90192.0	0.210395	0.105198	+	0.994451i	$0.466453\pi$
$180$	0	0
$181$	248002.	0.562676	0.281338	−	0.959609i	$-0.409222\pi$
$181$	248002.	0.562676	0.281338	+	0.959609i	$0.409222\pi$
$182$	−17992.0	−0.0402625
$183$	−378864.	−0.836288
$184$	99666.0	0.217021
$185$	0	0
$186$	43128.0	0.0914065
$187$	174482.	0.364877
$188$	−73222.0	−0.151094
$189$	−18954.0	−0.0385964
$190$	0	0
$191$	−156802.	−0.311006	−0.155503	−	0.987835i	$-0.549700\pi$
$191$	−156802.	−0.311006	−0.155503	+	0.987835i	$0.549700\pi$
$192$	241047.	0.471899
$193$	431234.	0.833335	0.416668	−	0.909059i	$-0.363198\pi$
$193$	431234.	0.833335	0.416668	+	0.909059i	$0.363198\pi$
$194$	79262.0	0.151203
$195$	0	0
$196$	500061.	0.929786
$197$	864974.	1.58795	0.793976	−	0.607949i	$-0.208007\pi$
$197$	864974.	1.58795	0.793976	+	0.607949i	$0.208007\pi$
$198$	−9801.00	−0.0177667
$199$	−480060.	−0.859336	−0.429668	−	0.902987i	$-0.641369\pi$
$199$	−480060.	−0.859336	−0.429668	+	0.902987i	$0.641369\pi$
$200$	0	0
$201$	−255996.	−0.446934
$202$	24958.0	0.0430359
$203$	−143676.	−0.244706
$204$	402318.	0.676853
$205$	0	0
$206$	−56812.0	−0.0932765
$207$	128142.	0.207857
$208$	642868.	1.03030
$209$	261360.	0.413879
$210$	0	0
$211$	525900.	0.813199	0.406600	−	0.913606i	$-0.366714\pi$
$211$	525900.	0.813199	0.406600	+	0.913606i	$0.366714\pi$
$212$	−954924.	−1.45925
$213$	−411210.	−0.621033
$214$	−12492.0	−0.0186465
$215$	0	0
$216$	−45927.0	−0.0669782
$217$	124592.	0.179614
$218$	−198748.	−0.283245
$219$	−165366.	−0.232989
$220$	0	0
$221$	997864.	1.37433
$222$	91746.0	0.124941
$223$	245264.	0.330272	0.165136	−	0.986271i	$-0.447194\pi$
$223$	245264.	0.330272	0.165136	+	0.986271i	$0.447194\pi$
$224$	−76570.0	−0.101962
$225$	0	0
$226$	166554.	0.216912
$227$	799308.	1.02955	0.514777	−	0.857324i	$-0.327875\pi$
$227$	799308.	1.02955	0.514777	+	0.857324i	$0.327875\pi$
$228$	602640.	0.767752
$229$	−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
$232$	−348138.	−0.424650
$233$	721830.	0.871054	0.435527	−	0.900176i	$-0.356562\pi$
$233$	721830.	0.871054	0.435527	+	0.900176i	$0.356562\pi$
$234$	−56052.0	−0.0669193
$235$	0	0
$236$	−198896.	−0.232459
$237$	946926.	1.09508
$238$	−37492.0	−0.0429038
$239$	−638436.	−0.722974	−0.361487	−	0.932377i	$-0.617731\pi$
$239$	−638436.	−0.722974	−0.361487	+	0.932377i	$0.617731\pi$
$240$	0	0
$241$	220990.	0.245092	0.122546	−	0.992463i	$-0.460894\pi$
$241$	220990.	0.245092	0.122546	+	0.992463i	$0.460894\pi$
$242$	−14641.0	−0.0160706
$243$	−59049.0	−0.0641500
$244$	−1.30498e6	−1.40323
$245$	0	0
$246$	−95598.0	−0.100719
$247$	1.49472e6	1.55890
$248$	301896.	0.311694
$249$	560628.	0.573029
$250$	0	0
$251$	627304.	0.628483	0.314242	−	0.949343i	$-0.398250\pi$
$251$	627304.	0.628483	0.314242	+	0.949343i	$0.398250\pi$
$252$	−65286.0	−0.0647618
$253$	191422.	0.188014
$254$	304226.	0.295878
$255$	0	0
$256$	736033.	0.701936
$257$	468014.	0.442004	0.221002	−	0.975273i	$-0.429067\pi$
$257$	468014.	0.442004	0.221002	+	0.975273i	$0.429067\pi$
$258$	−77220.0	−0.0722238
$259$	265044.	0.245510
$260$	0	0
$261$	−447606.	−0.406719
$262$	−274428.	−0.246988
$263$	−1.54510e6	−1.37743	−0.688713	−	0.725034i	$-0.741824\pi$
$263$	−1.54510e6	−1.37743	−0.688713	+	0.725034i	$0.741824\pi$
$264$	−68607.0	−0.0605841
$265$	0	0
$266$	−56160.0	−0.0486657
$267$	650214.	0.558185
$268$	−881764.	−0.749921
$269$	−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	$-0.272067\pi$
$271$	1.58723e6	1.31285	0.656427	+	0.754389i	$0.272067\pi$
$272$	1.33962e6	1.09789
$273$	−161928.	−0.131497
$274$	−245458.	−0.197515
$275$	0	0
$276$	441378.	0.348769
$277$	−692704.	−0.542436	−0.271218	−	0.962518i	$-0.587426\pi$
$277$	−692704.	−0.542436	−0.271218	+	0.962518i	$0.587426\pi$
$278$	59888.0	0.0464759
$279$	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	$0.568711\pi$
$282$	21258.0	0.0159184
$283$	−714916.	−0.530626	−0.265313	−	0.964162i	$-0.585475\pi$
$283$	−714916.	−0.530626	−0.265313	+	0.964162i	$0.585475\pi$
$284$	−1.41639e6	−1.04205
$285$	0	0
$286$	−83732.0	−0.0605308
$287$	−276172.	−0.197913
$288$	−238545.	−0.169469
$289$	659507.	0.464488
$290$	0	0
$291$	713358.	0.493827
$292$	−569594.	−0.390938
$293$	−2.14409e6	−1.45907	−0.729533	−	0.683946i	$-0.760262\pi$
$293$	−2.14409e6	−1.45907	−0.729533	+	0.683946i	$0.760262\pi$
$294$	−145179.	−0.0979570
$295$	0	0
$296$	642222.	0.426045
$297$	−88209.0	−0.0580259
$298$	−72038.0	−0.0469917
$299$	1.09474e6	0.708165
$300$	0	0
$301$	−223080.	−0.141920
$302$	323110.	0.203860
$303$	224622.	0.140555
$304$	2.00664e6	1.24533
$305$	0	0
$306$	−116802.	−0.0713094
$307$	588808.	0.356556	0.178278	−	0.983980i	$-0.442947\pi$
$307$	588808.	0.356556	0.178278	+	0.983980i	$0.442947\pi$
$308$	−97526.0	−0.0585792
$309$	−511308.	−0.304640
$310$	0	0
$311$	2.51827e6	1.47639	0.738194	−	0.674588i	$-0.235679\pi$
$311$	2.51827e6	1.47639	0.738194	+	0.674588i	$0.235679\pi$
$312$	−392364.	−0.228193
$313$	2.23562e6	1.28985	0.644923	−	0.764248i	$-0.276890\pi$
$313$	2.23562e6	1.28985	0.644923	+	0.764248i	$0.276890\pi$
$314$	−318766.	−0.182452
$315$	0	0
$316$	3.26163e6	1.83746
$317$	−1.06079e6	−0.592901	−0.296450	−	0.955048i	$-0.595803\pi$
$317$	−1.06079e6	−0.592901	−0.296450	+	0.955048i	$0.595803\pi$
$318$	277236.	0.153738
$319$	−668646.	−0.367891
$320$	0	0
$321$	−112428.	−0.0608992
$322$	−41132.0	−0.0221075
$323$	3.11472e6	1.66116
$324$	−203391.	−0.107639
$325$	0	0
$326$	−431996.	−0.225131
$327$	−1.78873e6	−0.925073
$328$	−669186.	−0.343449
$329$	61412.0	0.0312798
$330$	0	0
$331$	−2.34566e6	−1.17678	−0.588390	−	0.808577i	$-0.700238\pi$
$331$	−2.34566e6	−1.17678	−0.588390	+	0.808577i	$0.700238\pi$
$332$	1.93105e6	0.961499
$333$	825714.	0.408055
$334$	−251580.	−0.123399
$335$	0	0
$336$	−217386.	−0.105047
$337$	−839978.	−0.402896	−0.201448	−	0.979499i	$-0.564565\pi$
$337$	−839978.	−0.402896	−0.201448	+	0.979499i	$0.564565\pi$
$338$	−107571.	−0.0512157
$339$	1.49899e6	0.708432
$340$	0	0
$341$	579832.	0.270033
$342$	−174960.	−0.0808860
$343$	−856388.	−0.393039
$344$	−540540.	−0.246281
$345$	0	0
$346$	476634.	0.214040
$347$	2.02560e6	0.903086	0.451543	−	0.892249i	$-0.350874\pi$
$347$	2.02560e6	0.903086	0.451543	+	0.892249i	$0.350874\pi$
$348$	−1.54175e6	−0.682444
$349$	−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
$352$	−356345.	−0.153290
$353$	1.98730e6	0.848842	0.424421	−	0.905465i	$-0.360478\pi$
$353$	1.98730e6	0.848842	0.424421	+	0.905465i	$0.360478\pi$
$354$	57744.0	0.0244906
$355$	0	0
$356$	2.23963e6	0.936592
$357$	−337428.	−0.140123
$358$	−90192.0	−0.0371929
$359$	−3.43975e6	−1.40861	−0.704305	−	0.709898i	$-0.748741\pi$
$359$	−3.43975e6	−1.40861	−0.704305	+	0.709898i	$0.748741\pi$
$360$	0	0
$361$	2.18950e6	0.884254
$362$	−248002.	−0.0994681
$363$	−131769.	−0.0524864
$364$	−557752.	−0.220642
$365$	0	0
$366$	378864.	0.147836
$367$	1.79679e6	0.696358	0.348179	−	0.937428i	$-0.386800\pi$
$367$	1.79679e6	0.696358	0.348179	+	0.937428i	$0.386800\pi$
$368$	1.46968e6	0.565721
$369$	−860382.	−0.328947
$370$	0	0
$371$	800904.	0.302096
$372$	1.33697e6	0.500915
$373$	1.43541e6	0.534201	0.267100	−	0.963669i	$-0.413934\pi$
$373$	1.43541e6	0.534201	0.267100	+	0.963669i	$0.413934\pi$
$374$	−174482.	−0.0645018
$375$	0	0
$376$	148806.	0.0542814
$377$	−3.82399e6	−1.38568
$378$	18954.0	0.00682294
$379$	2.66235e6	0.952065	0.476033	−	0.879428i	$-0.342074\pi$
$379$	2.66235e6	0.952065	0.476033	+	0.879428i	$0.342074\pi$
$380$	0	0
$381$	2.73803e6	0.966332
$382$	156802.	0.0549785
$383$	−2.04091e6	−0.710932	−0.355466	−	0.934689i	$-0.615678\pi$
$383$	−2.04091e6	−0.710932	−0.355466	+	0.934689i	$0.615678\pi$
$384$	−1.08921e6	−0.376949
$385$	0	0
$386$	−431234.	−0.147314
$387$	−694980.	−0.235882
$388$	2.45712e6	0.828605
$389$	−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
$392$	−1.01625e6	−0.334031
$393$	−2.46985e6	−0.806658
$394$	−864974.	−0.280713
$395$	0	0
$396$	−303831.	−0.0973630
$397$	−728818.	−0.232083	−0.116041	−	0.993244i	$-0.537021\pi$
$397$	−728818.	−0.232083	−0.116041	+	0.993244i	$0.537021\pi$
$398$	480060.	0.151911
$399$	−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	$0.871848\pi$
$402$	255996.	0.0790075
$403$	3.31606e6	1.01709
$404$	773698.	0.235840
$405$	0	0
$406$	143676.	0.0432583
$407$	1.23347e6	0.369100
$408$	−817614.	−0.243163
$409$	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
$412$	−1.76117e6	−0.511162
$413$	166816.	0.0481241
$414$	−128142.	−0.0367444
$415$	0	0
$416$	−2.03794e6	−0.577375
$417$	538992.	0.151790
$418$	−261360.	−0.0731642
$419$	5.47794e6	1.52434	0.762170	−	0.647377i	$-0.224134\pi$
$419$	5.47794e6	1.52434	0.762170	+	0.647377i	$0.224134\pi$
$420$	0	0
$421$	1.02873e6	0.282877	0.141439	−	0.989947i	$-0.454827\pi$
$421$	1.02873e6	0.282877	0.141439	+	0.989947i	$0.454827\pi$
$422$	−525900.	−0.143755
$423$	191322.	0.0519893
$424$	1.94065e6	0.524243
$425$	0	0
$426$	411210.	0.109784
$427$	1.09450e6	0.290499
$428$	−387252.	−0.102184
$429$	−753588.	−0.197693
$430$	0	0
$431$	−5.14310e6	−1.33362	−0.666810	−	0.745228i	$-0.732341\pi$
$431$	−5.14310e6	−1.33362	−0.666810	+	0.745228i	$0.732341\pi$
$432$	−677241.	−0.174596
$433$	−412954.	−0.105848	−0.0529239	−	0.998599i	$-0.516854\pi$
$433$	−412954.	−0.105848	−0.0529239	+	0.998599i	$0.516854\pi$
$434$	−124592.	−0.0317516
$435$	0	0
$436$	−6.16119e6	−1.55220
$437$	3.41712e6	0.855966
$438$	165366.	0.0411871
$439$	5.96365e6	1.47690	0.738450	−	0.674309i	$-0.235558\pi$
$439$	5.96365e6	1.47690	0.738450	+	0.674309i	$0.235558\pi$
$440$	0	0
$441$	−1.30661e6	−0.319926
$442$	−997864.	−0.242949
$443$	−2.18433e6	−0.528821	−0.264410	−	0.964410i	$-0.585177\pi$
$443$	−2.18433e6	−0.528821	−0.264410	+	0.964410i	$0.585177\pi$
$444$	2.84413e6	0.684686
$445$	0	0
$446$	−245264.	−0.0583844
$447$	−648342.	−0.153474
$448$	−696358.	−0.163922
$449$	−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
$452$	5.16317e6	1.18870
$453$	2.90799e6	0.665806
$454$	−799308.	−0.182001
$455$	0	0
$456$	−1.22472e6	−0.275819
$457$	899922.	0.201565	0.100782	−	0.994908i	$-0.467865\pi$
$457$	899922.	0.201565	0.100782	+	0.994908i	$0.467865\pi$
$458$	1.53989e6	0.343026
$459$	−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	$0.460267\pi$
$462$	28314.0	0.00617158
$463$	7.38964e6	1.60203	0.801016	−	0.598643i	$-0.204293\pi$
$463$	7.38964e6	1.60203	0.801016	+	0.598643i	$0.204293\pi$
$464$	−5.13365e6	−1.10696
$465$	0	0
$466$	−721830.	−0.153982
$467$	−4.20851e6	−0.892968	−0.446484	−	0.894792i	$-0.647324\pi$
$467$	−4.20851e6	−0.892968	−0.446484	+	0.894792i	$0.647324\pi$
$468$	−1.73761e6	−0.366723
$469$	739544.	0.155250
$470$	0	0
$471$	−2.86889e6	−0.595885
$472$	404208.	0.0835122
$473$	−1.03818e6	−0.213363
$474$	−946926.	−0.193584
$475$	0	0
$476$	−1.16225e6	−0.235116
$477$	2.49512e6	0.502107
$478$	638436.	0.127805
$479$	7.39441e6	1.47253	0.736266	−	0.676692i	$-0.236587\pi$
$479$	7.39441e6	1.47253	0.736266	+	0.676692i	$0.236587\pi$
$480$	0	0
$481$	7.05425e6	1.39023
$482$	−220990.	−0.0433266
$483$	−370188.	−0.0722029
$484$	−453871.	−0.0880682
$485$	0	0
$486$	59049.0	0.0113402
$487$	3.81644e6	0.729181	0.364591	−	0.931168i	$-0.381209\pi$
$487$	3.81644e6	0.729181	0.364591	+	0.931168i	$0.381209\pi$
$488$	2.65205e6	0.504118
$489$	−3.88796e6	−0.735275
$490$	0	0
$491$	1.69716e6	0.317702	0.158851	−	0.987303i	$-0.449221\pi$
$491$	1.69716e6	0.317702	0.158851	+	0.987303i	$0.449221\pi$
$492$	−2.96354e6	−0.551947
$493$	−7.96849e6	−1.47659
$494$	−1.49472e6	−0.275577
$495$	0	0
$496$	4.45177e6	0.812509
$497$	1.18794e6	0.215727
$498$	−560628.	−0.101298
$499$	6.95160e6	1.24978	0.624889	−	0.780713i	$-0.285144\pi$
$499$	6.95160e6	1.24978	0.624889	+	0.780713i	$0.285144\pi$
$500$	0	0
$501$	−2.26422e6	−0.403018
$502$	−627304.	−0.111101
$503$	−6.01023e6	−1.05918	−0.529591	−	0.848253i	$-0.677655\pi$
$503$	−6.01023e6	−1.05918	−0.529591	+	0.848253i	$0.677655\pi$
$504$	132678.	0.0232660
$505$	0	0
$506$	−191422.	−0.0332365
$507$	−968139.	−0.167270
$508$	9.43101e6	1.62143
$509$	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
$512$	−4.60877e6	−0.776980
$513$	−1.57464e6	−0.264173
$514$	−468014.	−0.0781360
$515$	0	0
$516$	−2.39382e6	−0.395792
$517$	285802.	0.0470261
$518$	−265044.	−0.0434004
$519$	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	$0.516640\pi$
$522$	447606.	0.0718985
$523$	114676.	0.0183324	0.00916618	−	0.999958i	$-0.497082\pi$
$523$	114676.	0.0183324	0.00916618	+	0.999958i	$0.497082\pi$
$524$	−8.50727e6	−1.35351
$525$	0	0
$526$	1.54510e6	0.243497
$527$	6.91006e6	1.08382
$528$	−1.01168e6	−0.157928
$529$	−3.93362e6	−0.611157
$530$	0	0
$531$	519696.	0.0799858
$532$	−1.74096e6	−0.266692
$533$	−7.35042e6	−1.12071
$534$	−650214.	−0.0986741
$535$	0	0
$536$	1.79197e6	0.269413
$537$	−811728.	−0.121472
$538$	1.07457e6	0.160059
$539$	−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	$0.549862\pi$
$542$	−1.58723e6	−0.232082
$543$	−2.23202e6	−0.324861
$544$	−4.24669e6	−0.615252
$545$	0	0
$546$	161928.	0.0232456
$547$	−1.22672e7	−1.75299	−0.876494	−	0.481413i	$-0.840124\pi$
$547$	−1.22672e7	−1.75299	−0.876494	+	0.481413i	$0.840124\pi$
$548$	−7.60920e6	−1.08240
$549$	3.40978e6	0.482831
$550$	0	0
$551$	−1.19362e7	−1.67489
$552$	−896994.	−0.125297
$553$	−2.73556e6	−0.380394
$554$	692704.	0.0958900
$555$	0	0
$556$	1.85653e6	0.254692
$557$	−1.10980e7	−1.51568	−0.757839	−	0.652442i	$-0.773745\pi$
$557$	−1.10980e7	−1.51568	−0.757839	+	0.652442i	$0.773745\pi$
$558$	−388152.	−0.0527736
$559$	−5.93736e6	−0.803644
$560$	0	0
$561$	−1.57034e6	−0.210662
$562$	567018.	0.0757279
$563$	−4.61984e6	−0.614265	−0.307132	−	0.951667i	$-0.599369\pi$
$563$	−4.61984e6	−0.614265	−0.307132	+	0.951667i	$0.599369\pi$
$564$	658998.	0.0872341
$565$	0	0
$566$	714916.	0.0938024
$567$	170586.	0.0222836
$568$	2.87847e6	0.374361
$569$	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	$-0.705331\pi$
$571$	−9.36866e6	−1.20251	−0.601253	+	0.799059i	$0.705331\pi$
$572$	−2.59569e6	−0.331713
$573$	1.41122e6	0.179559
$574$	276172.	0.0349865
$575$	0	0
$576$	−2.16942e6	−0.272451
$577$	6.14973e6	0.768983	0.384491	−	0.923129i	$-0.374377\pi$
$577$	6.14973e6	0.768983	0.384491	+	0.923129i	$0.374377\pi$
$578$	−659507.	−0.0821107
$579$	−3.88111e6	−0.481126
$580$	0	0
$581$	−1.61959e6	−0.199051
$582$	−713358.	−0.0872972
$583$	3.72728e6	0.454173
$584$	1.15756e6	0.140447
$585$	0	0
$586$	2.14409e6	0.257929
$587$	−1.04649e6	−0.125354	−0.0626771	−	0.998034i	$-0.519964\pi$
$587$	−1.04649e6	−0.125354	−0.0626771	+	0.998034i	$0.519964\pi$
$588$	−4.50055e6	−0.536812
$589$	1.03507e7	1.22937
$590$	0	0
$591$	−7.78477e6	−0.916805
$592$	9.47023e6	1.11060
$593$	3.31784e6	0.387453	0.193726	−	0.981056i	$-0.437943\pi$
$593$	3.31784e6	0.387453	0.193726	+	0.981056i	$0.437943\pi$
$594$	88209.0	0.0102576
$595$	0	0
$596$	−2.23318e6	−0.257518
$597$	4.32054e6	0.496138
$598$	−1.09474e6	−0.125187
$599$	−1.73991e7	−1.98134	−0.990670	−	0.136280i	$-0.956485\pi$
$599$	−1.73991e7	−1.98134	−0.990670	+	0.136280i	$0.956485\pi$
$600$	0	0
$601$	7.13163e6	0.805383	0.402691	−	0.915336i	$-0.368075\pi$
$601$	7.13163e6	0.805383	0.402691	+	0.915336i	$0.368075\pi$
$602$	223080.	0.0250882
$603$	2.30396e6	0.258037
$604$	1.00164e7	1.11717
$605$	0	0
$606$	−224622.	−0.0248468
$607$	9.64617e6	1.06263	0.531317	−	0.847173i	$-0.321697\pi$
$607$	9.64617e6	1.06263	0.531317	+	0.847173i	$0.321697\pi$
$608$	−6.36120e6	−0.697879
$609$	1.29308e6	0.141281
$610$	0	0
$611$	1.63450e6	0.177126
$612$	−3.62086e6	−0.390781
$613$	−3.68170e6	−0.395729	−0.197864	−	0.980229i	$-0.563401\pi$
$613$	−3.68170e6	−0.395729	−0.197864	+	0.980229i	$0.563401\pi$
$614$	−588808.	−0.0630308
$615$	0	0
$616$	198198.	0.0210449
$617$	−1.83190e7	−1.93727	−0.968635	−	0.248489i	$-0.920066\pi$
$617$	−1.83190e7	−1.93727	−0.968635	+	0.248489i	$0.920066\pi$
$618$	511308.	0.0538532
$619$	1.09660e6	0.115033	0.0575166	−	0.998345i	$-0.481682\pi$
$619$	1.09660e6	0.115033	0.0575166	+	0.998345i	$0.481682\pi$
$620$	0	0
$621$	−1.15328e6	−0.120007
$622$	−2.51827e6	−0.260991
$623$	−1.87840e6	−0.193895
$624$	−5.78581e6	−0.594844
$625$	0	0
$626$	−2.23562e6	−0.228015
$627$	−2.35224e6	−0.238953
$628$	−9.88175e6	−0.999849
$629$	1.46997e7	1.48144
$630$	0	0
$631$	−9.58030e6	−0.957869	−0.478934	−	0.877851i	$-0.658977\pi$
$631$	−9.58030e6	−0.957869	−0.478934	+	0.877851i	$0.658977\pi$
$632$	−6.62848e6	−0.660118
$633$	−4.73310e6	−0.469501
$634$	1.06079e6	0.104811
$635$	0	0
$636$	8.59432e6	0.842497
$637$	−1.11627e7	−1.08998
$638$	668646.	0.0650346
$639$	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	$0.692075\pi$
$642$	112428.	0.0107656
$643$	5.88298e6	0.561138	0.280569	−	0.959834i	$-0.409477\pi$
$643$	5.88298e6	0.561138	0.280569	+	0.959834i	$0.409477\pi$
$644$	−1.27509e6	−0.121151
$645$	0	0
$646$	−3.11472e6	−0.293655
$647$	3.62822e6	0.340748	0.170374	−	0.985379i	$-0.445502\pi$
$647$	3.62822e6	0.340748	0.170374	+	0.985379i	$0.445502\pi$
$648$	413343.	0.0386699
$649$	776336.	0.0723499
$650$	0	0
$651$	−1.12133e6	−0.103700
$652$	−1.33919e7	−1.23374
$653$	5.70795e6	0.523838	0.261919	−	0.965090i	$-0.415645\pi$
$653$	5.70795e6	0.523838	0.261919	+	0.965090i	$0.415645\pi$
$654$	1.78873e6	0.163531
$655$	0	0
$656$	−9.86784e6	−0.895287
$657$	1.48829e6	0.134516
$658$	−61412.0	−0.00552953
$659$	1.08205e7	0.970588	0.485294	−	0.874351i	$-0.338713\pi$
$659$	1.08205e7	0.970588	0.485294	+	0.874351i	$0.338713\pi$
$660$	0	0
$661$	1.14311e7	1.01762	0.508809	−	0.860879i	$-0.330086\pi$
$661$	1.14311e7	1.01762	0.508809	+	0.860879i	$0.330086\pi$
$662$	2.34566e6	0.208027
$663$	−8.98078e6	−0.793469
$664$	−3.92440e6	−0.345424
$665$	0	0
$666$	−825714.	−0.0721347
$667$	−8.74213e6	−0.760857
$668$	−7.79898e6	−0.676233
$669$	−2.20738e6	−0.190683
$670$	0	0
$671$	5.09362e6	0.436737
$672$	689130.	0.0588678
$673$	2.03858e7	1.73496	0.867482	−	0.497468i	$-0.165737\pi$
$673$	2.03858e7	1.73496	0.867482	+	0.497468i	$0.165737\pi$
$674$	839978.	0.0712227
$675$	0	0
$676$	−3.33470e6	−0.280666
$677$	−6.09278e6	−0.510909	−0.255455	−	0.966821i	$-0.582225\pi$
$677$	−6.09278e6	−0.510909	−0.255455	+	0.966821i	$0.582225\pi$
$678$	−1.49899e6	−0.125234
$679$	−2.06081e6	−0.171539
$680$	0	0
$681$	−7.19377e6	−0.594414
$682$	−579832.	−0.0477355
$683$	−1.44978e7	−1.18918	−0.594592	−	0.804027i	$-0.702686\pi$
$683$	−1.44978e7	−1.18918	−0.594592	+	0.804027i	$0.702686\pi$
$684$	−5.42376e6	−0.443262
$685$	0	0
$686$	856388.	0.0694801
$687$	1.38590e7	1.12032
$688$	−7.97082e6	−0.641995
$689$	2.13164e7	1.71067
$690$	0	0
$691$	9.87069e6	0.786416	0.393208	−	0.919449i	$-0.371365\pi$
$691$	9.87069e6	0.786416	0.393208	+	0.919449i	$0.371365\pi$
$692$	1.47757e7	1.17296
$693$	254826.	0.0201563
$694$	−2.02560e6	−0.159645
$695$	0	0
$696$	3.13324e6	0.245172
$697$	−1.53169e7	−1.19423
$698$	378924.	0.0294384
$699$	−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	$0.421478\pi$
$702$	504468.	0.0386359
$703$	2.20190e7	1.68039
$704$	−3.24074e6	−0.246441
$705$	0	0
$706$	−1.98730e6	−0.150056
$707$	−648908.	−0.0488241
$708$	1.79006e6	0.134210
$709$	−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
$712$	−4.55150e6	−0.336476
$713$	7.58094e6	0.558470
$714$	337428.	0.0247705
$715$	0	0
$716$	−2.79595e6	−0.203820
$717$	5.74592e6	0.417409
$718$	3.43975e6	0.249009
$719$	6.29795e6	0.454336	0.227168	−	0.973856i	$-0.427053\pi$
$719$	6.29795e6	0.454336	0.227168	+	0.973856i	$0.427053\pi$
$720$	0	0
$721$	1.47711e6	0.105822
$722$	−2.18950e6	−0.156316
$723$	−1.98891e6	−0.141504
$724$	−7.68806e6	−0.545093
$725$	0	0
$726$	131769.	0.00927837
$727$	−1.14699e7	−0.804866	−0.402433	−	0.915449i	$-0.631835\pi$
$727$	−1.14699e7	−0.804866	−0.402433	+	0.915449i	$0.631835\pi$
$728$	1.13350e6	0.0792668
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.23724e7	−0.856365
$732$	1.17448e7	0.810154
$733$	1.87547e7	1.28929	0.644646	−	0.764481i	$-0.277005\pi$
$733$	1.87547e7	1.28929	0.644646	+	0.764481i	$0.277005\pi$
$734$	−1.79679e6	−0.123100
$735$	0	0
$736$	−4.65899e6	−0.317028
$737$	3.44172e6	0.233403
$738$	860382.	0.0581501
$739$	2.79727e6	0.188418	0.0942091	−	0.995552i	$-0.469968\pi$
$739$	2.79727e6	0.188418	0.0942091	+	0.995552i	$0.469968\pi$
$740$	0	0
$741$	−1.34525e7	−0.900030
$742$	−800904.	−0.0534036
$743$	2.25651e7	1.49956	0.749781	−	0.661686i	$-0.230159\pi$
$743$	2.25651e7	1.49956	0.749781	+	0.661686i	$0.230159\pi$
$744$	−2.71706e6	−0.179956
$745$	0	0
$746$	−1.43541e6	−0.0944342
$747$	−5.04565e6	−0.330838
$748$	−5.40894e6	−0.353475
$749$	324792.	0.0211544
$750$	0	0
$751$	−7.49233e6	−0.484749	−0.242375	−	0.970183i	$-0.577926\pi$
$751$	−7.49233e6	−0.484749	−0.242375	+	0.970183i	$0.577926\pi$
$752$	2.19430e6	0.141498
$753$	−5.64574e6	−0.362855
$754$	3.82399e6	0.244956
$755$	0	0
$756$	587574.	0.0373902
$757$	−2.88492e7	−1.82976	−0.914880	−	0.403727i	$-0.867715\pi$
$757$	−2.88492e7	−1.82976	−0.914880	+	0.403727i	$0.867715\pi$
$758$	−2.66235e6	−0.168303
$759$	−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	$0.596750\pi$
$762$	−2.73803e6	−0.170825
$763$	5.16745e6	0.321340
$764$	4.86086e6	0.301287
$765$	0	0
$766$	2.04091e6	0.125676
$767$	4.43987e6	0.272510
$768$	−6.62430e6	−0.405263
$769$	−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
$772$	−1.33683e7	−0.807293
$773$	−6.07336e6	−0.365578	−0.182789	−	0.983152i	$-0.558513\pi$
$773$	−6.07336e6	−0.365578	−0.182789	+	0.983152i	$0.558513\pi$
$774$	694980.	0.0416984
$775$	0	0
$776$	−4.99351e6	−0.297681
$777$	−2.38540e6	−0.141745
$778$	4.29947e6	0.254663
$779$	−2.29435e7	−1.35462
$780$	0	0
$781$	5.52849e6	0.324324
$782$	−2.28124e6	−0.133400
$783$	4.02845e6	0.234819
$784$	−1.49857e7	−0.870737
$785$	0	0
$786$	2.46985e6	0.142598
$787$	1.47512e7	0.848966	0.424483	−	0.905436i	$-0.360456\pi$
$787$	1.47512e7	0.848966	0.424483	+	0.905436i	$0.360456\pi$
$788$	−2.68142e7	−1.53833
$789$	1.39059e7	0.795257
$790$	0	0
$791$	−4.33040e6	−0.246086
$792$	617463.	0.0349782
$793$	2.91304e7	1.64499
$794$	728818.	0.0410268
$795$	0	0
$796$	1.48819e7	0.832481
$797$	2.78359e7	1.55224	0.776121	−	0.630584i	$-0.217185\pi$
$797$	2.78359e7	1.55224	0.776121	+	0.630584i	$0.217185\pi$
$798$	505440.	0.0280972
$799$	3.40600e6	0.188746
$800$	0	0
$801$	−5.85193e6	−0.322268
$802$	5.92515e6	0.325285
$803$	2.22325e6	0.121675
$804$	7.93588e6	0.432967
$805$	0	0
$806$	−3.31606e6	−0.179798
$807$	9.67115e6	0.522750
$808$	−1.57235e6	−0.0847270
$809$	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	$-0.671207\pi$
$811$	−1.91915e7	−1.02460	−0.512302	+	0.858805i	$0.671207\pi$
$812$	4.45396e6	0.237059
$813$	−1.42851e7	−0.757977
$814$	−1.23347e6	−0.0652483
$815$	0	0
$816$	−1.20566e7	−0.633867
$817$	−1.85328e7	−0.971373
$818$	−1.38212e6	−0.0722207
$819$	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	$0.473012\pi$
$822$	2.20912e6	0.114036
$823$	3.19195e7	1.64269	0.821347	−	0.570430i	$-0.193223\pi$
$823$	3.19195e7	1.64269	0.821347	+	0.570430i	$0.193223\pi$
$824$	3.57916e6	0.183638
$825$	0	0
$826$	−166816.	−0.00850722
$827$	−2.45556e7	−1.24850	−0.624248	−	0.781226i	$-0.714595\pi$
$827$	−2.45556e7	−1.24850	−0.624248	+	0.781226i	$0.714595\pi$
$828$	−3.97240e6	−0.201362
$829$	−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
$832$	−1.85338e7	−0.928233
$833$	−2.32609e7	−1.16149
$834$	−538992.	−0.0268329
$835$	0	0
$836$	−8.10216e6	−0.400945
$837$	−3.49337e6	−0.172358
$838$	−5.47794e6	−0.269468
$839$	−3.01443e6	−0.147843	−0.0739213	−	0.997264i	$-0.523551\pi$
$839$	−3.01443e6	−0.147843	−0.0739213	+	0.997264i	$0.523551\pi$
$840$	0	0
$841$	1.00255e7	0.488784
$842$	−1.02873e6	−0.0500061
$843$	5.10316e6	0.247326
$844$	−1.63029e7	−0.787787
$845$	0	0
$846$	−191322.	−0.00919050
$847$	380666.	0.0182321
$848$	2.86169e7	1.36657
$849$	6.43424e6	0.306357
$850$	0	0
$851$	1.61269e7	0.763356
$852$	1.27475e7	0.601626
$853$	1.67201e7	0.786806	0.393403	−	0.919366i	$-0.371298\pi$
$853$	1.67201e7	0.786806	0.393403	+	0.919366i	$0.371298\pi$
$854$	−1.09450e6	−0.0513534
$855$	0	0
$856$	786996.	0.0367103
$857$	9.15871e6	0.425973	0.212987	−	0.977055i	$-0.431681\pi$
$857$	9.15871e6	0.425973	0.212987	+	0.977055i	$0.431681\pi$
$858$	753588.	0.0349475
$859$	1.51068e7	0.698536	0.349268	−	0.937023i	$-0.386430\pi$
$859$	1.51068e7	0.698536	0.349268	+	0.937023i	$0.386430\pi$
$860$	0	0
$861$	2.48555e6	0.114265
$862$	5.14310e6	0.235753
$863$	−5.11568e6	−0.233817	−0.116909	−	0.993143i	$-0.537298\pi$
$863$	−5.11568e6	−0.233817	−0.116909	+	0.993143i	$0.537298\pi$
$864$	2.14690e6	0.0978427
$865$	0	0
$866$	412954.	0.0187114
$867$	−5.93556e6	−0.268172
$868$	−3.86235e6	−0.174001
$869$	−1.27309e7	−0.571886
$870$	0	0
$871$	1.96832e7	0.879127
$872$	1.25211e7	0.557638
$873$	−6.42022e6	−0.285111
$874$	−3.41712e6	−0.151315
$875$	0	0
$876$	5.12635e6	0.225708
$877$	1.26998e7	0.557568	0.278784	−	0.960354i	$-0.410069\pi$
$877$	1.26998e7	0.557568	0.278784	+	0.960354i	$0.410069\pi$
$878$	−5.96365e6	−0.261081
$879$	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	$0.558229\pi$
$882$	1.30661e6	0.0565555
$883$	1.69529e7	0.731715	0.365858	−	0.930671i	$-0.380776\pi$
$883$	1.69529e7	0.731715	0.365858	+	0.930671i	$0.380776\pi$
$884$	−3.09338e7	−1.33138
$885$	0	0
$886$	2.18433e6	0.0934832
$887$	1.05143e7	0.448717	0.224359	−	0.974507i	$-0.427971\pi$
$887$	1.05143e7	0.448717	0.224359	+	0.974507i	$0.427971\pi$
$888$	−5.78000e6	−0.245977
$889$	−7.90988e6	−0.335672
$890$	0	0
$891$	793881.	0.0335013
$892$	−7.60318e6	−0.319951
$893$	5.10192e6	0.214094
$894$	648342.	0.0271307
$895$	0	0
$896$	3.14660e6	0.130940
$897$	−9.85270e6	−0.408859
$898$	7858.00	0.000325178 0
$899$	−2.64806e7	−1.09277
$900$	0	0
$901$	4.44194e7	1.82289
$902$	1.28526e6	0.0525987
$903$	2.00772e6	0.0819377
$904$	−1.04929e7	−0.427046
$905$	0	0
$906$	−2.90799e6	−0.117699
$907$	1.53747e7	0.620569	0.310284	−	0.950644i	$-0.399576\pi$
$907$	1.53747e7	0.620569	0.310284	+	0.950644i	$0.399576\pi$
$908$	−2.47785e7	−0.997381
$909$	−2.02160e6	−0.0811494
$910$	0	0
$911$	1.25424e7	0.500708	0.250354	−	0.968154i	$-0.419453\pi$
$911$	1.25424e7	0.500708	0.250354	+	0.968154i	$0.419453\pi$
$912$	−1.80598e7	−0.718993
$913$	−7.53733e6	−0.299255
$914$	−899922.	−0.0356319
$915$	0	0
$916$	4.77367e7	1.87981
$917$	7.13513e6	0.280207
$918$	1.05122e6	0.0411705
$919$	3.31432e7	1.29451	0.647256	−	0.762273i	$-0.275916\pi$
$919$	3.31432e7	1.29451	0.647256	+	0.762273i	$0.275916\pi$
$920$	0	0
$921$	−5.29927e6	−0.205858
$922$	−1.13619e6	−0.0440173
$923$	3.16175e7	1.22158
$924$	877734.	0.0338207
$925$	0	0
$926$	−7.38964e6	−0.283202
$927$	4.60177e6	0.175884
$928$	1.62741e7	0.620335
$929$	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
$932$	−2.23767e7	−0.843834
$933$	−2.26644e7	−0.852393
$934$	4.20851e6	0.157856
$935$	0	0
$936$	3.53128e6	0.131747
$937$	3.10737e7	1.15623	0.578115	−	0.815955i	$-0.303788\pi$
$937$	3.10737e7	1.15623	0.578115	+	0.815955i	$0.303788\pi$
$938$	−739544.	−0.0274446
$939$	−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	$0.652449\pi$
$942$	2.86889e6	0.105339
$943$	−1.68040e7	−0.615366
$944$	5.96046e6	0.217696
$945$	0	0
$946$	1.03818e6	0.0377177
$947$	5.37383e6	0.194719	0.0973596	−	0.995249i	$-0.468960\pi$
$947$	5.37383e6	0.194719	0.0973596	+	0.995249i	$0.468960\pi$
$948$	−2.93547e7	−1.06086
$949$	1.27148e7	0.458294
$950$	0	0
$951$	9.54713e6	0.342311
$952$	2.36200e6	0.0844669
$953$	−7.26908e6	−0.259267	−0.129634	−	0.991562i	$-0.541380\pi$
$953$	−7.26908e6	−0.259267	−0.129634	+	0.991562i	$0.541380\pi$
$954$	−2.49512e6	−0.0887608
$955$	0	0
$956$	1.97915e7	0.700381
$957$	6.01781e6	0.212402
$958$	−7.39441e6	−0.260309
$959$	6.38191e6	0.224080
$960$	0	0
$961$	−5.66589e6	−0.197906
$962$	−7.05425e6	−0.245761
$963$	1.01185e6	0.0351602
$964$	−6.85069e6	−0.237433
$965$	0	0
$966$	370188.	0.0127638
$967$	2.54428e7	0.874983	0.437491	−	0.899223i	$-0.355867\pi$
$967$	2.54428e7	0.874983	0.437491	+	0.899223i	$0.355867\pi$
$968$	922383.	0.0316390
$969$	−2.80325e7	−0.959074
$970$	0	0
$971$	9.88213e6	0.336358	0.168179	−	0.985756i	$-0.446211\pi$
$971$	9.88213e6	0.336358	0.168179	+	0.985756i	$0.446211\pi$
$972$	1.83052e6	0.0621453
$973$	−1.55709e6	−0.0527268
$974$	−3.81644e6	−0.128902
$975$	0	0
$976$	3.91072e7	1.31411
$977$	−2.22197e6	−0.0744736	−0.0372368	−	0.999306i	$-0.511856\pi$
$977$	−2.22197e6	−0.0744736	−0.0372368	+	0.999306i	$0.511856\pi$
$978$	3.88796e6	0.129980
$979$	−8.74177e6	−0.291503
$980$	0	0
$981$	1.60986e7	0.534091
$982$	−1.69716e6	−0.0561623
$983$	−2.53706e7	−0.837428	−0.418714	−	0.908118i	$-0.637519\pi$
$983$	−2.53706e7	−0.837428	−0.418714	+	0.908118i	$0.637519\pi$
$984$	6.02267e6	0.198290
$985$	0	0
$986$	7.96849e6	0.261026
$987$	−552708.	−0.0180594
$988$	−4.63363e7	−1.51018
$989$	−1.35736e7	−0.441269
$990$	0	0
$991$	3.24132e7	1.04843	0.524214	−	0.851587i	$-0.324359\pi$
$991$	3.24132e7	1.04843	0.524214	+	0.851587i	$0.324359\pi$
$992$	−1.41124e7	−0.455327
$993$	2.11109e7	0.679414
$994$	−1.18794e6	−0.0381354
$995$	0	0
$996$	−1.73795e7	−0.555122
$997$	1.55048e6	0.0494000	0.0247000	−	0.999695i	$-0.492137\pi$
$997$	1.55048e6	0.0494000	0.0247000	+	0.999695i	$0.492137\pi$
$998$	−6.95160e6	−0.220932
$999$	−7.43143e6	−0.235591

Twists

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