LMFDB - Embedded newform 33.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,6,Mod(1,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("33.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 33 = 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	33.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:	$5.29266605383$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	33.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +9.00000 q^{3} -31.0000 q^{4} -92.0000 q^{5} +9.00000 q^{6} -26.0000 q^{7} -63.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +9.00000 q^{3} -31.0000 q^{4} -92.0000 q^{5} +9.00000 q^{6} -26.0000 q^{7} -63.0000 q^{8} +81.0000 q^{9} -92.0000 q^{10} +121.000 q^{11} -279.000 q^{12} -692.000 q^{13} -26.0000 q^{14} -828.000 q^{15} +929.000 q^{16} -1442.00 q^{17} +81.0000 q^{18} +2160.00 q^{19} +2852.00 q^{20} -234.000 q^{21} +121.000 q^{22} -1582.00 q^{23} -567.000 q^{24} +5339.00 q^{25} -692.000 q^{26} +729.000 q^{27} +806.000 q^{28} -5526.00 q^{29} -828.000 q^{30} +4792.00 q^{31} +2945.00 q^{32} +1089.00 q^{33} -1442.00 q^{34} +2392.00 q^{35} -2511.00 q^{36} -10194.0 q^{37} +2160.00 q^{38} -6228.00 q^{39} +5796.00 q^{40} -10622.0 q^{41} -234.000 q^{42} +8580.00 q^{43} -3751.00 q^{44} -7452.00 q^{45} -1582.00 q^{46} -2362.00 q^{47} +8361.00 q^{48} -16131.0 q^{49} +5339.00 q^{50} -12978.0 q^{51} +21452.0 q^{52} -30804.0 q^{53} +729.000 q^{54} -11132.0 q^{55} +1638.00 q^{56} +19440.0 q^{57} -5526.00 q^{58} +6416.00 q^{59} +25668.0 q^{60} +42096.0 q^{61} +4792.00 q^{62} -2106.00 q^{63} -26783.0 q^{64} +63664.0 q^{65} +1089.00 q^{66} -28444.0 q^{67} +44702.0 q^{68} -14238.0 q^{69} +2392.00 q^{70} +45690.0 q^{71} -5103.00 q^{72} -18374.0 q^{73} -10194.0 q^{74} +48051.0 q^{75} -66960.0 q^{76} -3146.00 q^{77} -6228.00 q^{78} -105214. q^{79} -85468.0 q^{80} +6561.00 q^{81} -10622.0 q^{82} +62292.0 q^{83} +7254.00 q^{84} +132664. q^{85} +8580.00 q^{86} -49734.0 q^{87} -7623.00 q^{88} -72246.0 q^{89} -7452.00 q^{90} +17992.0 q^{91} +49042.0 q^{92} +43128.0 q^{93} -2362.00 q^{94} -198720. q^{95} +26505.0 q^{96} +79262.0 q^{97} -16131.0 q^{98} +9801.00 q^{99} +O(q^{100})\)

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	$-0.471828\pi$
$2$	1.00000	0.176777	0.0883883	+	0.996086i	$0.471828\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−31.0000	−0.968750
$5$	−92.0000	−1.64575	−0.822873	−	0.568225i	$-0.807630\pi$
$5$	−92.0000	−1.64575	−0.822873	+	0.568225i	$0.807630\pi$
$6$	9.00000	0.102062
$7$	−26.0000	−0.200553	−0.100276	−	0.994960i	$-0.531973\pi$
$7$	−26.0000	−0.200553	−0.100276	+	0.994960i	$0.531973\pi$
$8$	−63.0000	−0.348029
$9$	81.0000	0.333333
$10$	−92.0000	−0.290930
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−279.000	−0.559308
$13$	−692.000	−1.13566	−0.567829	−	0.823146i	$-0.692217\pi$
$13$	−692.000	−1.13566	−0.567829	+	0.823146i	$0.692217\pi$
$14$	−26.0000	−0.0354530
$15$	−828.000	−0.950172
$16$	929.000	0.907227
$17$	−1442.00	−1.21016	−0.605080	−	0.796165i	$-0.706859\pi$
$17$	−1442.00	−1.21016	−0.605080	+	0.796165i	$0.706859\pi$
$18$	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	$0.259216\pi$
$20$	2852.00	1.59432
$21$	−234.000	−0.115789
$22$	121.000	0.0533002
$23$	−1582.00	−0.623572	−0.311786	−	0.950152i	$-0.600927\pi$
$23$	−1582.00	−0.623572	−0.311786	+	0.950152i	$0.600927\pi$
$24$	−567.000	−0.200935
$25$	5339.00	1.70848
$26$	−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	$0.708862\pi$
$30$	−828.000	−0.167968
$31$	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
$35$	2392.00	0.330059
$36$	−2511.00	−0.322917
$37$	−10194.0	−1.22417	−0.612083	−	0.790794i	$-0.709668\pi$
$37$	−10194.0	−1.22417	−0.612083	+	0.790794i	$0.709668\pi$
$38$	2160.00	0.242658
$39$	−6228.00	−0.655673
$40$	5796.00	0.572768
$41$	−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	$-0.384881\pi$
$43$	8580.00	0.707646	0.353823	+	0.935312i	$0.384881\pi$
$44$	−3751.00	−0.292089
$45$	−7452.00	−0.548582
$46$	−1582.00	−0.110233
$47$	−2362.00	−0.155968	−0.0779840	−	0.996955i	$-0.524848\pi$
$47$	−2362.00	−0.155968	−0.0779840	+	0.996955i	$0.524848\pi$
$48$	8361.00	0.523788
$49$	−16131.0	−0.959779
$50$	5339.00	0.302019
$51$	−12978.0	−0.698686
$52$	21452.0	1.10017
$53$	−30804.0	−1.50632	−0.753160	−	0.657837i	$-0.771472\pi$
$53$	−30804.0	−1.50632	−0.753160	+	0.657837i	$0.771472\pi$
$54$	729.000	0.0340207
$55$	−11132.0	−0.496211
$56$	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	$0.461717\pi$
$60$	25668.0	0.920479
$61$	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
$65$	63664.0	1.86901
$66$	1089.00	0.0307729
$67$	−28444.0	−0.774112	−0.387056	−	0.922056i	$-0.626508\pi$
$67$	−28444.0	−0.774112	−0.387056	+	0.922056i	$0.626508\pi$
$68$	44702.0	1.17234
$69$	−14238.0	−0.360020
$70$	2392.00	0.0583467
$71$	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	$-0.564671\pi$
$73$	−18374.0	−0.403549	−0.201775	+	0.979432i	$0.564671\pi$
$74$	−10194.0	−0.216404
$75$	48051.0	0.986391
$76$	−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	$0.897264\pi$
$80$	−85468.0	−1.49306
$81$	6561.00	0.111111
$82$	−10622.0	−0.174450
$83$	62292.0	0.992515	0.496257	−	0.868175i	$-0.334707\pi$
$83$	62292.0	0.992515	0.496257	+	0.868175i	$0.334707\pi$
$84$	7254.00	0.112171
$85$	132664.	1.99162
$86$	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	$0.660599\pi$
$90$	−7452.00	−0.0969765
$91$	17992.0	0.227759
$92$	49042.0	0.604086
$93$	43128.0	0.517073
$94$	−2362.00	−0.0275715
$95$	−198720.	−2.25908
$96$	26505.0	0.293528
$97$	79262.0	0.855334	0.427667	−	0.903936i	$-0.359336\pi$
$97$	79262.0	0.855334	0.427667	+	0.903936i	$0.359336\pi$
$98$	−16131.0	−0.169667
$99$	9801.00	0.100504
$100$	−165509.	−1.65509
$101$	−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	$-0.584984\pi$
$103$	−56812.0	−0.527651	−0.263826	+	0.964570i	$0.584984\pi$
$104$	43596.0	0.395242
$105$	21528.0	0.190559
$106$	−30804.0	−0.266282
$107$	−12492.0	−0.105481	−0.0527403	−	0.998608i	$-0.516796\pi$
$107$	−12492.0	−0.105481	−0.0527403	+	0.998608i	$0.516796\pi$
$108$	−22599.0	−0.186436
$109$	198748.	1.60227	0.801137	−	0.598482i	$-0.204229\pi$
$109$	198748.	1.60227	0.801137	+	0.598482i	$0.204229\pi$
$110$	−11132.0	−0.0877186
$111$	−91746.0	−0.706773
$112$	−24154.0	−0.181947
$113$	166554.	1.22704	0.613520	−	0.789679i	$-0.289753\pi$
$113$	166554.	1.22704	0.613520	+	0.789679i	$0.289753\pi$
$114$	19440.0	0.140099
$115$	145544.	1.02624
$116$	171306.	1.18203
$117$	−56052.0	−0.378553
$118$	6416.00	0.0424189
$119$	37492.0	0.242701
$120$	52164.0	0.330687
$121$	14641.0	0.0909091
$122$	42096.0	0.256060
$123$	−95598.0	−0.569752
$124$	−148552.	−0.867609
$125$	−203688.	−1.16598
$126$	−2106.00	−0.0118177
$127$	304226.	1.67374	0.836868	−	0.547405i	$-0.184384\pi$
$127$	304226.	1.67374	0.836868	+	0.547405i	$0.184384\pi$
$128$	−121023.	−0.652894
$129$	77220.0	0.408560
$130$	63664.0	0.330397
$131$	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
$135$	−67068.0	−0.316724
$136$	90846.0	0.421171
$137$	−245458.	−1.11732	−0.558658	−	0.829398i	$-0.688683\pi$
$137$	−245458.	−1.11732	−0.558658	+	0.829398i	$0.688683\pi$
$138$	−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	$0.541964\pi$
$140$	−74152.0	−0.319744
$141$	−21258.0	−0.0900481
$142$	45690.0	0.190152
$143$	−83732.0	−0.342414
$144$	75249.0	0.302409
$145$	508392.	2.00807
$146$	−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	$0.457567\pi$
$150$	48051.0	0.174371
$151$	−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
$155$	−440864.	−1.47393
$156$	193068.	0.635183
$157$	−318766.	−1.03210	−0.516051	−	0.856558i	$-0.672599\pi$
$157$	−318766.	−1.03210	−0.516051	+	0.856558i	$0.672599\pi$
$158$	−105214.	−0.335298
$159$	−277236.	−0.869675
$160$	−270940.	−0.836707
$161$	41132.0	0.125059
$162$	6561.00	0.0196419
$163$	−431996.	−1.27353	−0.636767	−	0.771056i	$-0.719729\pi$
$163$	−431996.	−1.27353	−0.636767	+	0.771056i	$0.719729\pi$
$164$	329282.	0.956001
$165$	−100188.	−0.286488
$166$	62292.0	0.175454
$167$	−251580.	−0.698047	−0.349024	−	0.937114i	$-0.613487\pi$
$167$	−251580.	−0.698047	−0.349024	+	0.937114i	$0.613487\pi$
$168$	14742.0	0.0402980
$169$	107571.	0.289720
$170$	132664.	0.352071
$171$	174960.	0.457560
$172$	−265980.	−0.685532
$173$	476634.	1.21079	0.605396	−	0.795924i	$-0.293015\pi$
$173$	476634.	1.21079	0.605396	+	0.795924i	$0.293015\pi$
$174$	−49734.0	−0.124532
$175$	−138814.	−0.342640
$176$	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	$0.466453\pi$
$180$	231012.	0.531439
$181$	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
$185$	937848.	2.01467
$186$	43128.0	0.0914065
$187$	−174482.	−0.364877
$188$	73222.0	0.151094
$189$	−18954.0	−0.0385964
$190$	−198720.	−0.399354
$191$	−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	$-0.636802\pi$
$193$	−431234.	−0.833335	−0.416668	+	0.909059i	$0.636802\pi$
$194$	79262.0	0.151203
$195$	572976.	1.07907
$196$	500061.	0.929786
$197$	−864974.	−1.58795	−0.793976	−	0.607949i	$-0.791993\pi$
$197$	−864974.	−1.58795	−0.793976	+	0.607949i	$0.791993\pi$
$198$	9801.00	0.0177667
$199$	−480060.	−0.859336	−0.429668	−	0.902987i	$-0.641369\pi$
$199$	−480060.	−0.859336	−0.429668	+	0.902987i	$0.641369\pi$
$200$	−336357.	−0.594601
$201$	−255996.	−0.446934
$202$	−24958.0	−0.0430359
$203$	143676.	0.244706
$204$	402318.	0.676853
$205$	977224.	1.62409
$206$	−56812.0	−0.0932765
$207$	−128142.	−0.207857
$208$	−642868.	−1.03030
$209$	261360.	0.413879
$210$	21528.0	0.0336865
$211$	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
$215$	−789360.	−1.16461
$216$	−45927.0	−0.0669782
$217$	−124592.	−0.179614
$218$	198748.	0.283245
$219$	−165366.	−0.232989
$220$	345092.	0.480705
$221$	997864.	1.37433
$222$	−91746.0	−0.124941
$223$	−245264.	−0.330272	−0.165136	−	0.986271i	$-0.552806\pi$
$223$	−245264.	−0.330272	−0.165136	+	0.986271i	$0.552806\pi$
$224$	−76570.0	−0.101962
$225$	432459.	0.569493
$226$	166554.	0.216912
$227$	−799308.	−1.02955	−0.514777	−	0.857324i	$-0.672125\pi$
$227$	−799308.	−1.02955	−0.514777	+	0.857324i	$0.672125\pi$
$228$	−602640.	−0.767752
$229$	−1.53989e6	−1.94045	−0.970224	−	0.242208i	$-0.922128\pi$
$229$	−1.53989e6	−1.94045	−0.970224	+	0.242208i	$0.922128\pi$
$230$	145544.	0.181416
$231$	−28314.0	−0.0349117
$232$	348138.	0.424650
$233$	−721830.	−0.871054	−0.435527	−	0.900176i	$-0.643438\pi$
$233$	−721830.	−0.871054	−0.435527	+	0.900176i	$0.643438\pi$
$234$	−56052.0	−0.0669193
$235$	217304.	0.256684
$236$	−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	$0.617731\pi$
$240$	−769212.	−0.862021
$241$	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
$245$	1.48405e6	1.57955
$246$	−95598.0	−0.100719
$247$	−1.49472e6	−1.55890
$248$	−301896.	−0.311694
$249$	560628.	0.573029
$250$	−203688.	−0.206118
$251$	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
$255$	1.19398e6	1.14986
$256$	736033.	0.701936
$257$	−468014.	−0.442004	−0.221002	−	0.975273i	$-0.570933\pi$
$257$	−468014.	−0.442004	−0.221002	+	0.975273i	$0.570933\pi$
$258$	77220.0	0.0722238
$259$	265044.	0.245510
$260$	−1.97358e6	−1.81060
$261$	−447606.	−0.406719
$262$	274428.	0.246988
$263$	1.54510e6	1.37743	0.688713	−	0.725034i	$-0.258176\pi$
$263$	1.54510e6	1.37743	0.688713	+	0.725034i	$0.258176\pi$
$264$	−68607.0	−0.0605841
$265$	2.83397e6	2.47902
$266$	−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	$0.649544\pi$
$270$	−67068.0	−0.0559894
$271$	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
$275$	646019.	0.515126
$276$	441378.	0.348769
$277$	692704.	0.542436	0.271218	−	0.962518i	$-0.412574\pi$
$277$	692704.	0.542436	0.271218	+	0.962518i	$0.412574\pi$
$278$	−59888.0	−0.0464759
$279$	388152.	0.298532
$280$	−150696.	−0.114870
$281$	−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	$-0.414525\pi$
$283$	714916.	0.530626	0.265313	+	0.964162i	$0.414525\pi$
$284$	−1.41639e6	−1.04205
$285$	−1.78848e6	−1.30428
$286$	−83732.0	−0.0605308
$287$	276172.	0.197913
$288$	238545.	0.169469
$289$	659507.	0.464488
$290$	508392.	0.354980
$291$	713358.	0.493827
$292$	569594.	0.390938
$293$	2.14409e6	1.45907	0.729533	−	0.683946i	$-0.239738\pi$
$293$	2.14409e6	1.45907	0.729533	+	0.683946i	$0.239738\pi$
$294$	−145179.	−0.0979570
$295$	−590272.	−0.394909
$296$	642222.	0.426045
$297$	88209.0	0.0580259
$298$	72038.0	0.0469917
$299$	1.09474e6	0.708165
$300$	−1.48958e6	−0.955567
$301$	−223080.	−0.141920
$302$	−323110.	−0.203860
$303$	−224622.	−0.140555
$304$	2.00664e6	1.24533
$305$	−3.87283e6	−2.38385
$306$	−116802.	−0.0713094
$307$	−588808.	−0.356556	−0.178278	−	0.983980i	$-0.557053\pi$
$307$	−588808.	−0.356556	−0.178278	+	0.983980i	$0.557053\pi$
$308$	97526.0	0.0585792
$309$	−511308.	−0.304640
$310$	−440864.	−0.260556
$311$	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	$-0.723110\pi$
$313$	−2.23562e6	−1.28985	−0.644923	+	0.764248i	$0.723110\pi$
$314$	−318766.	−0.182452
$315$	193752.	0.110020
$316$	3.26163e6	1.83746
$317$	1.06079e6	0.592901	0.296450	−	0.955048i	$-0.404197\pi$
$317$	1.06079e6	0.592901	0.296450	+	0.955048i	$0.404197\pi$
$318$	−277236.	−0.153738
$319$	−668646.	−0.367891
$320$	2.46404e6	1.34515
$321$	−112428.	−0.0608992
$322$	41132.0	0.0221075
$323$	−3.11472e6	−1.66116
$324$	−203391.	−0.107639
$325$	−3.69459e6	−1.94025
$326$	−431996.	−0.225131
$327$	1.78873e6	0.925073
$328$	669186.	0.343449
$329$	61412.0	0.0312798
$330$	−100188.	−0.0506443
$331$	−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
$335$	2.61685e6	1.27399
$336$	−217386.	−0.105047
$337$	839978.	0.402896	0.201448	−	0.979499i	$-0.435435\pi$
$337$	839978.	0.402896	0.201448	+	0.979499i	$0.435435\pi$
$338$	107571.	0.0512157
$339$	1.49899e6	0.708432
$340$	−4.11258e6	−1.92938
$341$	579832.	0.270033
$342$	174960.	0.0808860
$343$	856388.	0.393039
$344$	−540540.	−0.246281
$345$	1.30990e6	0.592501
$346$	476634.	0.214040
$347$	−2.02560e6	−0.903086	−0.451543	−	0.892249i	$-0.649126\pi$
$347$	−2.02560e6	−0.903086	−0.451543	+	0.892249i	$0.649126\pi$
$348$	1.54175e6	0.682444
$349$	−378924.	−0.166528	−0.0832642	−	0.996528i	$-0.526535\pi$
$349$	−378924.	−0.166528	−0.0832642	+	0.996528i	$0.526535\pi$
$350$	−138814.	−0.0605708
$351$	−504468.	−0.218558
$352$	356345.	0.153290
$353$	−1.98730e6	−0.848842	−0.424421	−	0.905465i	$-0.639522\pi$
$353$	−1.98730e6	−0.848842	−0.424421	+	0.905465i	$0.639522\pi$
$354$	57744.0	0.0244906
$355$	−4.20348e6	−1.77026
$356$	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	$0.748741\pi$
$360$	469476.	0.190923
$361$	2.18950e6	0.884254
$362$	248002.	0.0994681
$363$	131769.	0.0524864
$364$	−557752.	−0.220642
$365$	1.69041e6	0.664140
$366$	378864.	0.147836
$367$	−1.79679e6	−0.696358	−0.348179	−	0.937428i	$-0.613200\pi$
$367$	−1.79679e6	−0.696358	−0.348179	+	0.937428i	$0.613200\pi$
$368$	−1.46968e6	−0.565721
$369$	−860382.	−0.328947
$370$	937848.	0.356146
$371$	800904.	0.302096
$372$	−1.33697e6	−0.500915
$373$	−1.43541e6	−0.534201	−0.267100	−	0.963669i	$-0.586066\pi$
$373$	−1.43541e6	−0.534201	−0.267100	+	0.963669i	$0.586066\pi$
$374$	−174482.	−0.0645018
$375$	−1.83319e6	−0.673178
$376$	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	$0.342074\pi$
$380$	6.16032e6	2.18849
$381$	2.73803e6	0.966332
$382$	−156802.	−0.0549785
$383$	2.04091e6	0.710932	0.355466	−	0.934689i	$-0.384322\pi$
$383$	2.04091e6	0.710932	0.355466	+	0.934689i	$0.384322\pi$
$384$	−1.08921e6	−0.376949
$385$	289432.	0.0995164
$386$	−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	$0.755994\pi$
$390$	572976.	0.190755
$391$	2.28124e6	0.754623
$392$	1.01625e6	0.334031
$393$	2.46985e6	0.806658
$394$	−864974.	−0.280713
$395$	9.67969e6	3.12154
$396$	−303831.	−0.0973630
$397$	728818.	0.232083	0.116041	−	0.993244i	$-0.462979\pi$
$397$	728818.	0.232083	0.116041	+	0.993244i	$0.462979\pi$
$398$	−480060.	−0.151911
$399$	−505440.	−0.158942
$400$	4.95993e6	1.54998
$401$	−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
$405$	−603612.	−0.182861
$406$	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	$0.434518\pi$
$410$	977224.	0.287101
$411$	−2.20912e6	−0.645082
$412$	1.76117e6	0.511162
$413$	−166816.	−0.0481241
$414$	−128142.	−0.0367444
$415$	−5.73086e6	−1.63343
$416$	−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	$0.224134\pi$
$420$	−667368.	−0.184604
$421$	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
$425$	−7.69884e6	−2.06753
$426$	411210.	0.109784
$427$	−1.09450e6	−0.290499
$428$	387252.	0.102184
$429$	−753588.	−0.197693
$430$	−789360.	−0.205875
$431$	−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	$-0.483146\pi$
$433$	412954.	0.105848	0.0529239	+	0.998599i	$0.483146\pi$
$434$	−124592.	−0.0317516
$435$	4.57553e6	1.15936
$436$	−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	$0.235558\pi$
$440$	701316.	0.172696
$441$	−1.30661e6	−0.319926
$442$	997864.	0.242949
$443$	2.18433e6	0.528821	0.264410	−	0.964410i	$-0.414823\pi$
$443$	2.18433e6	0.528821	0.264410	+	0.964410i	$0.414823\pi$
$444$	2.84413e6	0.684686
$445$	6.64663e6	1.59112
$446$	−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	$0.500293\pi$
$450$	432459.	0.100673
$451$	−1.28526e6	−0.297543
$452$	−5.16317e6	−1.18870
$453$	−2.90799e6	−0.665806
$454$	−799308.	−0.182001
$455$	−1.65526e6	−0.374834
$456$	−1.22472e6	−0.275819
$457$	−899922.	−0.201565	−0.100782	−	0.994908i	$-0.532135\pi$
$457$	−899922.	−0.201565	−0.100782	+	0.994908i	$0.532135\pi$
$458$	−1.53989e6	−0.343026
$459$	−1.05122e6	−0.232895
$460$	−4.51186e6	−0.994172
$461$	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	$-0.795707\pi$
$463$	−7.38964e6	−1.60203	−0.801016	+	0.598643i	$0.795707\pi$
$464$	−5.13365e6	−1.10696
$465$	−3.96778e6	−0.850971
$466$	−721830.	−0.153982
$467$	4.20851e6	0.892968	0.446484	−	0.894792i	$-0.352676\pi$
$467$	4.20851e6	0.892968	0.446484	+	0.894792i	$0.352676\pi$
$468$	1.73761e6	0.366723
$469$	739544.	0.155250
$470$	217304.	0.0453757
$471$	−2.86889e6	−0.595885
$472$	−404208.	−0.0835122
$473$	1.03818e6	0.213363
$474$	−946926.	−0.193584
$475$	1.15322e7	2.34520
$476$	−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	$0.236587\pi$
$480$	−2.43846e6	−0.483073
$481$	7.05425e6	1.39023
$482$	220990.	0.0433266
$483$	370188.	0.0722029
$484$	−453871.	−0.0880682
$485$	−7.29210e6	−1.40766
$486$	59049.0	0.0113402
$487$	−3.81644e6	−0.729181	−0.364591	−	0.931168i	$-0.618791\pi$
$487$	−3.81644e6	−0.729181	−0.364591	+	0.931168i	$0.618791\pi$
$488$	−2.65205e6	−0.504118
$489$	−3.88796e6	−0.735275
$490$	1.48405e6	0.279228
$491$	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
$495$	−901692.	−0.165404
$496$	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	$0.285144\pi$
$500$	6.31433e6	1.12954
$501$	−2.26422e6	−0.403018
$502$	627304.	0.111101
$503$	6.01023e6	1.05918	0.529591	−	0.848253i	$-0.322345\pi$
$503$	6.01023e6	1.05918	0.529591	+	0.848253i	$0.322345\pi$
$504$	132678.	0.0232660
$505$	2.29614e6	0.400654
$506$	−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	$0.482983\pi$
$510$	1.19398e6	0.203269
$511$	477724.	0.0809328
$512$	4.60877e6	0.776980
$513$	1.57464e6	0.264173
$514$	−468014.	−0.0781360
$515$	5.22670e6	0.868380
$516$	−2.39382e6	−0.395792
$517$	−285802.	−0.0470261
$518$	265044.	0.0434004
$519$	4.28971e6	0.699051
$520$	−4.01083e6	−0.650468
$521$	−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	$-0.502918\pi$
$523$	−114676.	−0.0183324	−0.00916618	+	0.999958i	$0.502918\pi$
$524$	−8.50727e6	−1.35351
$525$	−1.24933e6	−0.197823
$526$	1.54510e6	0.243497
$527$	−6.91006e6	−1.08382
$528$	1.01168e6	0.157928
$529$	−3.93362e6	−0.611157
$530$	2.83397e6	0.438233
$531$	519696.	0.0799858
$532$	1.74096e6	0.266692
$533$	7.35042e6	1.12071
$534$	−650214.	−0.0986741
$535$	1.14926e6	0.173594
$536$	1.79197e6	0.269413
$537$	811728.	0.121472
$538$	−1.07457e6	−0.160059
$539$	−1.95185e6	−0.289384
$540$	2.07911e6	0.306826
$541$	−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
$545$	−1.82848e7	−2.63693
$546$	161928.	0.0232456
$547$	1.22672e7	1.75299	0.876494	−	0.481413i	$-0.159876\pi$
$547$	1.22672e7	1.75299	0.876494	+	0.481413i	$0.159876\pi$
$548$	7.60920e6	1.08240
$549$	3.40978e6	0.482831
$550$	646019.	0.0910623
$551$	−1.19362e7	−1.67489
$552$	896994.	0.125297
$553$	2.73556e6	0.380394
$554$	692704.	0.0958900
$555$	8.44063e6	1.16317
$556$	1.85653e6	0.254692
$557$	1.10980e7	1.51568	0.757839	−	0.652442i	$-0.226255\pi$
$557$	1.10980e7	1.51568	0.757839	+	0.652442i	$0.226255\pi$
$558$	388152.	0.0527736
$559$	−5.93736e6	−0.803644
$560$	2.22217e6	0.299438
$561$	−1.57034e6	−0.210662
$562$	−567018.	−0.0757279
$563$	4.61984e6	0.614265	0.307132	−	0.951667i	$-0.400631\pi$
$563$	4.61984e6	0.614265	0.307132	+	0.951667i	$0.400631\pi$
$564$	658998.	0.0872341
$565$	−1.53230e7	−2.01940
$566$	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	$0.271176\pi$
$570$	−1.78848e6	−0.230567
$571$	−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
$575$	−8.44630e6	−1.06536
$576$	−2.16942e6	−0.272451
$577$	−6.14973e6	−0.768983	−0.384491	−	0.923129i	$-0.625623\pi$
$577$	−6.14973e6	−0.768983	−0.384491	+	0.923129i	$0.625623\pi$
$578$	659507.	0.0821107
$579$	−3.88111e6	−0.481126
$580$	−1.57602e7	−1.94532
$581$	−1.61959e6	−0.199051
$582$	713358.	0.0872972
$583$	−3.72728e6	−0.454173
$584$	1.15756e6	0.140447
$585$	5.15678e6	0.623002
$586$	2.14409e6	0.257929
$587$	1.04649e6	0.125354	0.0626771	−	0.998034i	$-0.480036\pi$
$587$	1.04649e6	0.125354	0.0626771	+	0.998034i	$0.480036\pi$
$588$	4.50055e6	0.536812
$589$	1.03507e7	1.22937
$590$	−590272.	−0.0698107
$591$	−7.78477e6	−0.916805
$592$	−9.47023e6	−1.11060
$593$	−3.31784e6	−0.387453	−0.193726	−	0.981056i	$-0.562057\pi$
$593$	−3.31784e6	−0.387453	−0.193726	+	0.981056i	$0.562057\pi$
$594$	88209.0	0.0102576
$595$	−3.44926e6	−0.399424
$596$	−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	$0.956485\pi$
$600$	−3.02721e6	−0.343293
$601$	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
$605$	−1.34697e6	−0.149613
$606$	−224622.	−0.0248468
$607$	−9.64617e6	−1.06263	−0.531317	−	0.847173i	$-0.678303\pi$
$607$	−9.64617e6	−1.06263	−0.531317	+	0.847173i	$0.678303\pi$
$608$	6.36120e6	0.697879
$609$	1.29308e6	0.141281
$610$	−3.87283e6	−0.421409
$611$	1.63450e6	0.177126
$612$	3.62086e6	0.390781
$613$	3.68170e6	0.395729	0.197864	−	0.980229i	$-0.436599\pi$
$613$	3.68170e6	0.395729	0.197864	+	0.980229i	$0.436599\pi$
$614$	−588808.	−0.0630308
$615$	8.79502e6	0.937667
$616$	198198.	0.0210449
$617$	1.83190e7	1.93727	0.968635	−	0.248489i	$-0.0799340\pi$
$617$	1.83190e7	1.93727	0.968635	+	0.248489i	$0.0799340\pi$
$618$	−511308.	−0.0538532
$619$	1.09660e6	0.115033	0.0575166	−	0.998345i	$-0.481682\pi$
$619$	1.09660e6	0.115033	0.0575166	+	0.998345i	$0.481682\pi$
$620$	1.36668e7	1.42786
$621$	−1.15328e6	−0.120007
$622$	2.51827e6	0.260991
$623$	1.87840e6	0.193895
$624$	−5.78581e6	−0.594844
$625$	2.05492e6	0.210424
$626$	−2.23562e6	−0.228015
$627$	2.35224e6	0.238953
$628$	9.88175e6	0.999849
$629$	1.46997e7	1.48144
$630$	193752.	0.0194489
$631$	−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
$635$	−2.79888e7	−2.75454
$636$	8.59432e6	0.842497
$637$	1.11627e7	1.08998
$638$	−668646.	−0.0650346
$639$	3.70089e6	0.358554
$640$	1.11341e7	1.07450
$641$	−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	$-0.590523\pi$
$643$	−5.88298e6	−0.561138	−0.280569	+	0.959834i	$0.590523\pi$
$644$	−1.27509e6	−0.121151
$645$	−7.10424e6	−0.672385
$646$	−3.11472e6	−0.293655
$647$	−3.62822e6	−0.340748	−0.170374	−	0.985379i	$-0.554498\pi$
$647$	−3.62822e6	−0.340748	−0.170374	+	0.985379i	$0.554498\pi$
$648$	−413343.	−0.0386699
$649$	776336.	0.0723499
$650$	−3.69459e6	−0.342991
$651$	−1.12133e6	−0.103700
$652$	1.33919e7	1.23374
$653$	−5.70795e6	−0.523838	−0.261919	−	0.965090i	$-0.584355\pi$
$653$	−5.70795e6	−0.523838	−0.261919	+	0.965090i	$0.584355\pi$
$654$	1.78873e6	0.163531
$655$	−2.52474e7	−2.29939
$656$	−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	$0.338713\pi$
$660$	3.10583e6	0.277535
$661$	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
$665$	5.16672e6	0.453065
$666$	−825714.	−0.0721347
$667$	8.74213e6	0.760857
$668$	7.79898e6	0.676233
$669$	−2.20738e6	−0.190683
$670$	2.61685e6	0.225212
$671$	5.09362e6	0.436737
$672$	−689130.	−0.0588678
$673$	−2.03858e7	−1.73496	−0.867482	−	0.497468i	$-0.834263\pi$
$673$	−2.03858e7	−1.73496	−0.867482	+	0.497468i	$0.834263\pi$
$674$	839978.	0.0712227
$675$	3.89213e6	0.328797
$676$	−3.33470e6	−0.280666
$677$	6.09278e6	0.510909	0.255455	−	0.966821i	$-0.417775\pi$
$677$	6.09278e6	0.510909	0.255455	+	0.966821i	$0.417775\pi$
$678$	1.49899e6	0.125234
$679$	−2.06081e6	−0.171539
$680$	−8.35783e6	−0.693141
$681$	−7.19377e6	−0.594414
$682$	579832.	0.0477355
$683$	1.44978e7	1.18918	0.594592	−	0.804027i	$-0.297314\pi$
$683$	1.44978e7	1.18918	0.594592	+	0.804027i	$0.297314\pi$
$684$	−5.42376e6	−0.443262
$685$	2.25821e7	1.83882
$686$	856388.	0.0694801
$687$	−1.38590e7	−1.12032
$688$	7.97082e6	0.641995
$689$	2.13164e7	1.71067
$690$	1.30990e6	0.104740
$691$	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
$695$	5.50970e6	0.432679
$696$	3.13324e6	0.245172
$697$	1.53169e7	1.19423
$698$	−378924.	−0.0294384
$699$	−6.49647e6	−0.502903
$700$	4.30323e6	0.331933
$701$	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
$705$	1.95574e6	0.148196
$706$	−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	$0.504892\pi$
$710$	−4.20348e6	−0.312941
$711$	−8.52233e6	−0.632244
$712$	4.55150e6	0.336476
$713$	−7.58094e6	−0.558470
$714$	337428.	0.0247705
$715$	7.70334e6	0.563526
$716$	−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	$0.427053\pi$
$720$	−6.92291e6	−0.497688
$721$	1.47711e6	0.105822
$722$	2.18950e6	0.156316
$723$	1.98891e6	0.141504
$724$	−7.68806e6	−0.545093
$725$	−2.95033e7	−2.08461
$726$	131769.	0.00927837
$727$	1.14699e7	0.804866	0.402433	−	0.915449i	$-0.368165\pi$
$727$	1.14699e7	0.804866	0.402433	+	0.915449i	$0.368165\pi$
$728$	−1.13350e6	−0.0792668
$729$	531441.	0.0370370
$730$	1.69041e6	0.117404
$731$	−1.23724e7	−0.856365
$732$	−1.17448e7	−0.810154
$733$	−1.87547e7	−1.28929	−0.644646	−	0.764481i	$-0.722995\pi$
$733$	−1.87547e7	−1.28929	−0.644646	+	0.764481i	$0.722995\pi$
$734$	−1.79679e6	−0.123100
$735$	1.33565e7	0.911955
$736$	−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	$0.469968\pi$
$740$	−2.90733e7	−1.95171
$741$	−1.34525e7	−0.900030
$742$	800904.	0.0534036
$743$	−2.25651e7	−1.49956	−0.749781	−	0.661686i	$-0.769841\pi$
$743$	−2.25651e7	−1.49956	−0.749781	+	0.661686i	$0.769841\pi$
$744$	−2.71706e6	−0.179956
$745$	−6.62750e6	−0.437481
$746$	−1.43541e6	−0.0944342
$747$	5.04565e6	0.330838
$748$	5.40894e6	0.353475
$749$	324792.	0.0211544
$750$	−1.83319e6	−0.119002
$751$	−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
$755$	2.97261e7	1.89789
$756$	587574.	0.0373902
$757$	2.88492e7	1.82976	0.914880	−	0.403727i	$-0.132285\pi$
$757$	2.88492e7	1.82976	0.914880	+	0.403727i	$0.132285\pi$
$758$	2.66235e6	0.168303
$759$	−1.72280e6	−0.108550
$760$	1.25194e7	0.786227
$761$	−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
$765$	1.07458e7	0.663872
$766$	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	$0.507230\pi$
$770$	289432.	0.0175922
$771$	−4.21213e6	−0.255191
$772$	1.33683e7	0.807293
$773$	6.07336e6	0.365578	0.182789	−	0.983152i	$-0.441487\pi$
$773$	6.07336e6	0.365578	0.182789	+	0.983152i	$0.441487\pi$
$774$	694980.	0.0416984
$775$	2.55845e7	1.53011
$776$	−4.99351e6	−0.297681
$777$	2.38540e6	0.141745
$778$	−4.29947e6	−0.254663
$779$	−2.29435e7	−1.35462
$780$	−1.77623e7	−1.04535
$781$	5.52849e6	0.324324
$782$	2.28124e6	0.133400
$783$	−4.02845e6	−0.234819
$784$	−1.49857e7	−0.870737
$785$	2.93265e7	1.69858
$786$	2.46985e6	0.142598
$787$	−1.47512e7	−0.848966	−0.424483	−	0.905436i	$-0.639544\pi$
$787$	−1.47512e7	−0.848966	−0.424483	+	0.905436i	$0.639544\pi$
$788$	2.68142e7	1.53833
$789$	1.39059e7	0.795257
$790$	9.67969e6	0.551815
$791$	−4.33040e6	−0.246086
$792$	−617463.	−0.0349782
$793$	−2.91304e7	−1.64499
$794$	728818.	0.0410268
$795$	2.55057e7	1.43126
$796$	1.48819e7	0.832481
$797$	−2.78359e7	−1.55224	−0.776121	−	0.630584i	$-0.782815\pi$
$797$	−2.78359e7	−1.55224	−0.776121	+	0.630584i	$0.782815\pi$
$798$	−505440.	−0.0280972
$799$	3.40600e6	0.188746
$800$	1.57234e7	0.868601
$801$	−5.85193e6	−0.322268
$802$	−5.92515e6	−0.325285
$803$	−2.22325e6	−0.121675
$804$	7.93588e6	0.432967
$805$	−3.78414e6	−0.205815
$806$	−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	$0.260111\pi$
$810$	−603612.	−0.0323255
$811$	−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
$815$	3.97436e7	2.09591
$816$	−1.20566e7	−0.633867
$817$	1.85328e7	0.971373
$818$	1.38212e6	0.0722207
$819$	1.45735e6	0.0759197
$820$	−3.02939e7	−1.57333
$821$	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	$-0.806777\pi$
$823$	−3.19195e7	−1.64269	−0.821347	+	0.570430i	$0.806777\pi$
$824$	3.57916e6	0.183638
$825$	5.81417e6	0.297408
$826$	−166816.	−0.00850722
$827$	2.45556e7	1.24850	0.624248	−	0.781226i	$-0.285405\pi$
$827$	2.45556e7	1.24850	0.624248	+	0.781226i	$0.285405\pi$
$828$	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	$0.615931\pi$
$830$	−5.73086e6	−0.288752
$831$	6.23434e6	0.313175
$832$	1.85338e7	0.928233
$833$	2.32609e7	1.16149
$834$	−538992.	−0.0268329
$835$	2.31454e7	1.14881
$836$	−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	$0.523551\pi$
$840$	−1.35626e6	−0.0663202
$841$	1.00255e7	0.488784
$842$	1.02873e6	0.0500061
$843$	−5.10316e6	−0.247326
$844$	−1.63029e7	−0.787787
$845$	−9.89653e6	−0.476806
$846$	−191322.	−0.00919050
$847$	−380666.	−0.0182321
$848$	−2.86169e7	−1.36657
$849$	6.43424e6	0.306357
$850$	−7.69884e6	−0.365492
$851$	1.61269e7	0.763356
$852$	−1.27475e7	−0.601626
$853$	−1.67201e7	−0.786806	−0.393403	−	0.919366i	$-0.628702\pi$
$853$	−1.67201e7	−0.786806	−0.393403	+	0.919366i	$0.628702\pi$
$854$	−1.09450e6	−0.0513534
$855$	−1.60963e7	−0.753028
$856$	786996.	0.0367103
$857$	−9.15871e6	−0.425973	−0.212987	−	0.977055i	$-0.568319\pi$
$857$	−9.15871e6	−0.425973	−0.212987	+	0.977055i	$0.568319\pi$
$858$	−753588.	−0.0349475
$859$	1.51068e7	0.698536	0.349268	−	0.937023i	$-0.386430\pi$
$859$	1.51068e7	0.698536	0.349268	+	0.937023i	$0.386430\pi$
$860$	2.44702e7	1.12821
$861$	2.48555e6	0.114265
$862$	−5.14310e6	−0.235753
$863$	5.11568e6	0.233817	0.116909	−	0.993143i	$-0.462702\pi$
$863$	5.11568e6	0.233817	0.116909	+	0.993143i	$0.462702\pi$
$864$	2.14690e6	0.0978427
$865$	−4.38503e7	−1.99266
$866$	412954.	0.0187114
$867$	5.93556e6	0.268172
$868$	3.86235e6	0.174001
$869$	−1.27309e7	−0.571886
$870$	4.57553e6	0.204948
$871$	1.96832e7	0.879127
$872$	−1.25211e7	−0.557638
$873$	6.42022e6	0.285111
$874$	−3.41712e6	−0.151315
$875$	5.29589e6	0.233840
$876$	5.12635e6	0.225708
$877$	−1.26998e7	−0.557568	−0.278784	−	0.960354i	$-0.589931\pi$
$877$	−1.26998e7	−0.557568	−0.278784	+	0.960354i	$0.589931\pi$
$878$	5.96365e6	0.261081
$879$	1.92968e7	0.842392
$880$	−1.03416e7	−0.450176
$881$	−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	$-0.619224\pi$
$883$	−1.69529e7	−0.731715	−0.365858	+	0.930671i	$0.619224\pi$
$884$	−3.09338e7	−1.33138
$885$	−5.31245e6	−0.228001
$886$	2.18433e6	0.0934832
$887$	−1.05143e7	−0.448717	−0.224359	−	0.974507i	$-0.572029\pi$
$887$	−1.05143e7	−0.448717	−0.224359	+	0.974507i	$0.572029\pi$
$888$	5.78000e6	0.245977
$889$	−7.90988e6	−0.335672
$890$	6.64663e6	0.281272
$891$	793881.	0.0335013
$892$	7.60318e6	0.319951
$893$	−5.10192e6	−0.214094
$894$	648342.	0.0271307
$895$	−8.29766e6	−0.346257
$896$	3.14660e6	0.130940
$897$	9.85270e6	0.408859
$898$	−7858.00	−0.000325178 0
$899$	−2.64806e7	−1.09277
$900$	−1.34062e7	−0.551697
$901$	4.44194e7	1.82289
$902$	−1.28526e6	−0.0525987
$903$	−2.00772e6	−0.0819377
$904$	−1.04929e7	−0.427046
$905$	−2.28162e7	−0.926023
$906$	−2.90799e6	−0.117699
$907$	−1.53747e7	−0.620569	−0.310284	−	0.950644i	$-0.600424\pi$
$907$	−1.53747e7	−0.620569	−0.310284	+	0.950644i	$0.600424\pi$
$908$	2.47785e7	0.997381
$909$	−2.02160e6	−0.0811494
$910$	−1.65526e6	−0.0662619
$911$	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
$915$	−3.48555e7	−1.37632
$916$	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	$0.275916\pi$
$920$	−9.16927e6	−0.357162
$921$	−5.29927e6	−0.205858
$922$	1.13619e6	0.0440173
$923$	−3.16175e7	−1.22158
$924$	877734.	0.0338207
$925$	−5.44258e7	−2.09146
$926$	−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	$0.299096\pi$
$930$	−3.96778e6	−0.150432
$931$	−3.48430e7	−1.31747
$932$	2.23767e7	0.843834
$933$	2.26644e7	0.852393
$934$	4.20851e6	0.157856
$935$	1.60523e7	0.600495
$936$	3.53128e6	0.131747
$937$	−3.10737e7	−1.15623	−0.578115	−	0.815955i	$-0.696212\pi$
$937$	−3.10737e7	−1.15623	−0.578115	+	0.815955i	$0.696212\pi$
$938$	739544.	0.0274446
$939$	−2.01206e7	−0.744692
$940$	−6.73642e6	−0.248662
$941$	−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
$945$	1.74377e6	0.0635198
$946$	1.03818e6	0.0377177
$947$	−5.37383e6	−0.194719	−0.0973596	−	0.995249i	$-0.531040\pi$
$947$	−5.37383e6	−0.194719	−0.0973596	+	0.995249i	$0.531040\pi$
$948$	2.93547e7	1.06086
$949$	1.27148e7	0.458294
$950$	1.15322e7	0.414576
$951$	9.54713e6	0.342311
$952$	−2.36200e6	−0.0844669
$953$	7.26908e6	0.259267	0.129634	−	0.991562i	$-0.458620\pi$
$953$	7.26908e6	0.259267	0.129634	+	0.991562i	$0.458620\pi$
$954$	−2.49512e6	−0.0887608
$955$	1.44258e7	0.511836
$956$	1.97915e7	0.700381
$957$	−6.01781e6	−0.212402
$958$	7.39441e6	0.260309
$959$	6.38191e6	0.224080
$960$	2.21763e7	0.776625
$961$	−5.66589e6	−0.197906
$962$	7.05425e6	0.245761
$963$	−1.01185e6	−0.0351602
$964$	−6.85069e6	−0.237433
$965$	3.96735e7	1.37146
$966$	370188.	0.0127638
$967$	−2.54428e7	−0.874983	−0.437491	−	0.899223i	$-0.644133\pi$
$967$	−2.54428e7	−0.874983	−0.437491	+	0.899223i	$0.644133\pi$
$968$	−922383.	−0.0316390
$969$	−2.80325e7	−0.959074
$970$	−7.29210e6	−0.248842
$971$	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
$975$	−3.32513e7	−1.12020
$976$	3.91072e7	1.31411
$977$	2.22197e6	0.0744736	0.0372368	−	0.999306i	$-0.488144\pi$
$977$	2.22197e6	0.0744736	0.0372368	+	0.999306i	$0.488144\pi$
$978$	−3.88796e6	−0.129980
$979$	−8.74177e6	−0.291503
$980$	−4.60056e7	−1.53019
$981$	1.60986e7	0.534091
$982$	1.69716e6	0.0561623
$983$	2.53706e7	0.837428	0.418714	−	0.908118i	$-0.362481\pi$
$983$	2.53706e7	0.837428	0.418714	+	0.908118i	$0.362481\pi$
$984$	6.02267e6	0.198290
$985$	7.95776e7	2.61337
$986$	7.96849e6	0.261026
$987$	552708.	0.0180594
$988$	4.63363e7	1.51018
$989$	−1.35736e7	−0.441269
$990$	−901692.	−0.0292395
$991$	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
$995$	4.41655e7	1.41425
$996$	−1.73795e7	−0.555122
$997$	−1.55048e6	−0.0494000	−0.0247000	−	0.999695i	$-0.507863\pi$
$997$	−1.55048e6	−0.0494000	−0.0247000	+	0.999695i	$0.507863\pi$
$998$	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	33.6.a.b.1.1	✓	1
3.2	odd	2		99.6.a.a.1.1		1
4.3	odd	2		528.6.a.a.1.1		1
5.4	even	2		825.6.a.a.1.1		1
11.10	odd	2		363.6.a.b.1.1		1
33.32	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	1.1	even	1	trivial
99.6.a.a.1.1		1	3.2	odd	2
363.6.a.b.1.1		1	11.10	odd	2
528.6.a.a.1.1		1	4.3	odd	2
825.6.a.a.1.1		1	5.4	even	2
1089.6.a.h.1.1		1	33.32	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:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	33.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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