LMFDB - Embedded newform 693.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,6,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	693.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:	$111.145987130$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -28.0000 q^{4} +74.0000 q^{5} -49.0000 q^{7} -120.000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} -28.0000 q^{4} +74.0000 q^{5} -49.0000 q^{7} -120.000 q^{8} +148.000 q^{10} -121.000 q^{11} +364.000 q^{13} -98.0000 q^{14} +656.000 q^{16} -148.000 q^{17} -1320.00 q^{19} -2072.00 q^{20} -242.000 q^{22} +436.000 q^{23} +2351.00 q^{25} +728.000 q^{26} +1372.00 q^{28} -2970.00 q^{29} +8842.00 q^{31} +5152.00 q^{32} -296.000 q^{34} -3626.00 q^{35} +138.000 q^{37} -2640.00 q^{38} -8880.00 q^{40} -532.000 q^{41} -20676.0 q^{43} +3388.00 q^{44} +872.000 q^{46} +11722.0 q^{47} +2401.00 q^{49} +4702.00 q^{50} -10192.0 q^{52} -5274.00 q^{53} -8954.00 q^{55} +5880.00 q^{56} -5940.00 q^{58} +27670.0 q^{59} +19512.0 q^{61} +17684.0 q^{62} -10688.0 q^{64} +26936.0 q^{65} +64088.0 q^{67} +4144.00 q^{68} -7252.00 q^{70} +3708.00 q^{71} -24296.0 q^{73} +276.000 q^{74} +36960.0 q^{76} +5929.00 q^{77} -2200.00 q^{79} +48544.0 q^{80} -1064.00 q^{82} -74424.0 q^{83} -10952.0 q^{85} -41352.0 q^{86} +14520.0 q^{88} -34170.0 q^{89} -17836.0 q^{91} -12208.0 q^{92} +23444.0 q^{94} -97680.0 q^{95} +151718. q^{97} +4802.00 q^{98} +O(q^{100})\)

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 $

$2$	2.00000	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	2.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	0	0
$3$	0	0
$4$	−28.0000	−0.875000
$5$	74.0000	1.32375	0.661876	−	0.749613i	$-0.269760\pi$
$5$	74.0000	1.32375	0.661876	+	0.749613i	$0.269760\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−120.000	−0.662913
$9$	0	0
$10$	148.000	0.468017
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	364.000	0.597369	0.298685	−	0.954352i	$-0.403452\pi$
$13$	364.000	0.597369	0.298685	+	0.954352i	$0.403452\pi$
$14$	−98.0000	−0.133631
$15$	0	0
$16$	656.000	0.640625
$17$	−148.000	−0.124205	−0.0621025	−	0.998070i	$-0.519781\pi$
$17$	−148.000	−0.124205	−0.0621025	+	0.998070i	$0.519781\pi$
$18$	0	0
$19$	−1320.00	−0.838861	−0.419430	−	0.907787i	$-0.637770\pi$
$19$	−1320.00	−0.838861	−0.419430	+	0.907787i	$0.637770\pi$
$20$	−2072.00	−1.15828
$21$	0	0
$22$	−242.000	−0.106600
$23$	436.000	0.171857	0.0859284	−	0.996301i	$-0.472614\pi$
$23$	436.000	0.171857	0.0859284	+	0.996301i	$0.472614\pi$
$24$	0	0
$25$	2351.00	0.752320
$26$	728.000	0.211202
$27$	0	0
$28$	1372.00	0.330719
$29$	−2970.00	−0.655785	−0.327892	−	0.944715i	$-0.606338\pi$
$29$	−2970.00	−0.655785	−0.327892	+	0.944715i	$0.606338\pi$
$30$	0	0
$31$	8842.00	1.65252	0.826259	−	0.563290i	$-0.190465\pi$
$31$	8842.00	1.65252	0.826259	+	0.563290i	$0.190465\pi$
$32$	5152.00	0.889408
$33$	0	0
$34$	−296.000	−0.0439131
$35$	−3626.00	−0.500331
$36$	0	0
$37$	138.000	0.0165720	0.00828600	−	0.999966i	$-0.497362\pi$
$37$	138.000	0.0165720	0.00828600	+	0.999966i	$0.497362\pi$
$38$	−2640.00	−0.296582
$39$	0	0
$40$	−8880.00	−0.877532
$41$	−532.000	−0.0494256	−0.0247128	−	0.999695i	$-0.507867\pi$
$41$	−532.000	−0.0494256	−0.0247128	+	0.999695i	$0.507867\pi$
$42$	0	0
$43$	−20676.0	−1.70528	−0.852639	−	0.522500i	$-0.825000\pi$
$43$	−20676.0	−1.70528	−0.852639	+	0.522500i	$0.825000\pi$
$44$	3388.00	0.263822
$45$	0	0
$46$	872.000	0.0607606
$47$	11722.0	0.774029	0.387014	−	0.922074i	$-0.373506\pi$
$47$	11722.0	0.774029	0.387014	+	0.922074i	$0.373506\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	4702.00	0.265985
$51$	0	0
$52$	−10192.0	−0.522698
$53$	−5274.00	−0.257899	−0.128950	−	0.991651i	$-0.541161\pi$
$53$	−5274.00	−0.257899	−0.128950	+	0.991651i	$0.541161\pi$
$54$	0	0
$55$	−8954.00	−0.399126
$56$	5880.00	0.250557
$57$	0	0
$58$	−5940.00	−0.231855
$59$	27670.0	1.03485	0.517427	−	0.855727i	$-0.326890\pi$
$59$	27670.0	1.03485	0.517427	+	0.855727i	$0.326890\pi$
$60$	0	0
$61$	19512.0	0.671394	0.335697	−	0.941970i	$-0.391028\pi$
$61$	19512.0	0.671394	0.335697	+	0.941970i	$0.391028\pi$
$62$	17684.0	0.584253
$63$	0	0
$64$	−10688.0	−0.326172
$65$	26936.0	0.790769
$66$	0	0
$67$	64088.0	1.74417	0.872087	−	0.489351i	$-0.162766\pi$
$67$	64088.0	1.74417	0.872087	+	0.489351i	$0.162766\pi$
$68$	4144.00	0.108679
$69$	0	0
$70$	−7252.00	−0.176894
$71$	3708.00	0.0872959	0.0436480	−	0.999047i	$-0.486102\pi$
$71$	3708.00	0.0872959	0.0436480	+	0.999047i	$0.486102\pi$
$72$	0	0
$73$	−24296.0	−0.533615	−0.266807	−	0.963750i	$-0.585969\pi$
$73$	−24296.0	−0.533615	−0.266807	+	0.963750i	$0.585969\pi$
$74$	276.000	0.00585908
$75$	0	0
$76$	36960.0	0.734003
$77$	5929.00	0.113961
$78$	0	0
$79$	−2200.00	−0.0396602	−0.0198301	−	0.999803i	$-0.506313\pi$
$79$	−2200.00	−0.0396602	−0.0198301	+	0.999803i	$0.506313\pi$
$80$	48544.0	0.848029
$81$	0	0
$82$	−1064.00	−0.0174746
$83$	−74424.0	−1.18582	−0.592909	−	0.805270i	$-0.702020\pi$
$83$	−74424.0	−1.18582	−0.592909	+	0.805270i	$0.702020\pi$
$84$	0	0
$85$	−10952.0	−0.164417
$86$	−41352.0	−0.602907
$87$	0	0
$88$	14520.0	0.199876
$89$	−34170.0	−0.457267	−0.228634	−	0.973513i	$-0.573426\pi$
$89$	−34170.0	−0.457267	−0.228634	+	0.973513i	$0.573426\pi$
$90$	0	0
$91$	−17836.0	−0.225784
$92$	−12208.0	−0.150375
$93$	0	0
$94$	23444.0	0.273660
$95$	−97680.0	−1.11044
$96$	0	0
$97$	151718.	1.63722	0.818611	−	0.574348i	$-0.194744\pi$
$97$	151718.	1.63722	0.818611	+	0.574348i	$0.194744\pi$
$98$	4802.00	0.0505076
$99$	0	0
$100$	−65828.0	−0.658280
$101$	−116852.	−1.13981	−0.569905	−	0.821710i	$-0.693020\pi$
$101$	−116852.	−1.13981	−0.569905	+	0.821710i	$0.693020\pi$
$102$	0	0
$103$	103694.	0.963076	0.481538	−	0.876425i	$-0.340078\pi$
$103$	103694.	0.963076	0.481538	+	0.876425i	$0.340078\pi$
$104$	−43680.0	−0.396004
$105$	0	0
$106$	−10548.0	−0.0911812
$107$	97092.0	0.819830	0.409915	−	0.912124i	$-0.365558\pi$
$107$	97092.0	0.819830	0.409915	+	0.912124i	$0.365558\pi$
$108$	0	0
$109$	52930.0	0.426713	0.213356	−	0.976974i	$-0.431560\pi$
$109$	52930.0	0.426713	0.213356	+	0.976974i	$0.431560\pi$
$110$	−17908.0	−0.141112
$111$	0	0
$112$	−32144.0	−0.242133
$113$	80526.0	0.593253	0.296627	−	0.954994i	$-0.404138\pi$
$113$	80526.0	0.593253	0.296627	+	0.954994i	$0.404138\pi$
$114$	0	0
$115$	32264.0	0.227496
$116$	83160.0	0.573812
$117$	0	0
$118$	55340.0	0.365876
$119$	7252.00	0.0469451
$120$	0	0
$121$	14641.0	0.0909091
$122$	39024.0	0.237373
$123$	0	0
$124$	−247576.	−1.44595
$125$	−57276.0	−0.327867
$126$	0	0
$127$	221048.	1.21612	0.608061	−	0.793890i	$-0.291948\pi$
$127$	221048.	1.21612	0.608061	+	0.793890i	$0.291948\pi$
$128$	−186240.	−1.00473
$129$	0	0
$130$	53872.0	0.279579
$131$	−37572.0	−0.191287	−0.0956436	−	0.995416i	$-0.530491\pi$
$131$	−37572.0	−0.191287	−0.0956436	+	0.995416i	$0.530491\pi$
$132$	0	0
$133$	64680.0	0.317060
$134$	128176.	0.616658
$135$	0	0
$136$	17760.0	0.0823371
$137$	290602.	1.32281	0.661405	−	0.750029i	$-0.269961\pi$
$137$	290602.	1.32281	0.661405	+	0.750029i	$0.269961\pi$
$138$	0	0
$139$	367360.	1.61270	0.806352	−	0.591435i	$-0.201439\pi$
$139$	367360.	1.61270	0.806352	+	0.591435i	$0.201439\pi$
$140$	101528.	0.437790
$141$	0	0
$142$	7416.00	0.0308638
$143$	−44044.0	−0.180114
$144$	0	0
$145$	−219780.	−0.868097
$146$	−48592.0	−0.188661
$147$	0	0
$148$	−3864.00	−0.0145005
$149$	462730.	1.70751	0.853753	−	0.520679i	$-0.174321\pi$
$149$	462730.	1.70751	0.853753	+	0.520679i	$0.174321\pi$
$150$	0	0
$151$	−7648.00	−0.0272964	−0.0136482	−	0.999907i	$-0.504344\pi$
$151$	−7648.00	−0.0272964	−0.0136482	+	0.999907i	$0.504344\pi$
$152$	158400.	0.556091
$153$	0	0
$154$	11858.0	0.0402911
$155$	654308.	2.18752
$156$	0	0
$157$	−161482.	−0.522847	−0.261424	−	0.965224i	$-0.584192\pi$
$157$	−161482.	−0.522847	−0.261424	+	0.965224i	$0.584192\pi$
$158$	−4400.00	−0.0140220
$159$	0	0
$160$	381248.	1.17736
$161$	−21364.0	−0.0649558
$162$	0	0
$163$	179464.	0.529064	0.264532	−	0.964377i	$-0.414782\pi$
$163$	179464.	0.529064	0.264532	+	0.964377i	$0.414782\pi$
$164$	14896.0	0.0432474
$165$	0	0
$166$	−148848.	−0.419250
$167$	−316848.	−0.879144	−0.439572	−	0.898207i	$-0.644870\pi$
$167$	−316848.	−0.879144	−0.439572	+	0.898207i	$0.644870\pi$
$168$	0	0
$169$	−238797.	−0.643150
$170$	−21904.0	−0.0581301
$171$	0	0
$172$	578928.	1.49212
$173$	175116.	0.444847	0.222423	−	0.974950i	$-0.428603\pi$
$173$	175116.	0.444847	0.222423	+	0.974950i	$0.428603\pi$
$174$	0	0
$175$	−115199.	−0.284350
$176$	−79376.0	−0.193156
$177$	0	0
$178$	−68340.0	−0.161668
$179$	69780.0	0.162779	0.0813895	−	0.996682i	$-0.474064\pi$
$179$	69780.0	0.162779	0.0813895	+	0.996682i	$0.474064\pi$
$180$	0	0
$181$	−78638.0	−0.178417	−0.0892085	−	0.996013i	$-0.528434\pi$
$181$	−78638.0	−0.178417	−0.0892085	+	0.996013i	$0.528434\pi$
$182$	−35672.0	−0.0798268
$183$	0	0
$184$	−52320.0	−0.113926
$185$	10212.0	0.0219372
$186$	0	0
$187$	17908.0	0.0374492
$188$	−328216.	−0.677275
$189$	0	0
$190$	−195360.	−0.392601
$191$	927208.	1.83905	0.919525	−	0.393030i	$-0.128573\pi$
$191$	927208.	1.83905	0.919525	+	0.393030i	$0.128573\pi$
$192$	0	0
$193$	877474.	1.69567	0.847834	−	0.530261i	$-0.177906\pi$
$193$	877474.	1.69567	0.847834	+	0.530261i	$0.177906\pi$
$194$	303436.	0.578846
$195$	0	0
$196$	−67228.0	−0.125000
$197$	744602.	1.36697	0.683484	−	0.729965i	$-0.260464\pi$
$197$	744602.	1.36697	0.683484	+	0.729965i	$0.260464\pi$
$198$	0	0
$199$	1.07931e6	1.93203	0.966014	−	0.258489i	$-0.0832246\pi$
$199$	1.07931e6	1.93203	0.966014	+	0.258489i	$0.0832246\pi$
$200$	−282120.	−0.498722
$201$	0	0
$202$	−233704.	−0.402984
$203$	145530.	0.247863
$204$	0	0
$205$	−39368.0	−0.0654273
$206$	207388.	0.340499
$207$	0	0
$208$	238784.	0.382690
$209$	159720.	0.252926
$210$	0	0
$211$	728772.	1.12690	0.563450	−	0.826150i	$-0.309474\pi$
$211$	728772.	1.12690	0.563450	+	0.826150i	$0.309474\pi$
$212$	147672.	0.225662
$213$	0	0
$214$	194184.	0.289854
$215$	−1.53002e6	−2.25737
$216$	0	0
$217$	−433258.	−0.624593
$218$	105860.	0.150866
$219$	0	0
$220$	250712.	0.349236
$221$	−53872.0	−0.0741963
$222$	0	0
$223$	38374.0	0.0516743	0.0258372	−	0.999666i	$-0.491775\pi$
$223$	38374.0	0.0516743	0.0258372	+	0.999666i	$0.491775\pi$
$224$	−252448.	−0.336165
$225$	0	0
$226$	161052.	0.209747
$227$	−323268.	−0.416388	−0.208194	−	0.978088i	$-0.566759\pi$
$227$	−323268.	−0.416388	−0.208194	+	0.978088i	$0.566759\pi$
$228$	0	0
$229$	813690.	1.02535	0.512673	−	0.858584i	$-0.328655\pi$
$229$	813690.	1.02535	0.512673	+	0.858584i	$0.328655\pi$
$230$	64528.0	0.0804320
$231$	0	0
$232$	356400.	0.434728
$233$	−1.10801e6	−1.33707	−0.668537	−	0.743679i	$-0.733079\pi$
$233$	−1.10801e6	−1.33707	−0.668537	+	0.743679i	$0.733079\pi$
$234$	0	0
$235$	867428.	1.02462
$236$	−774760.	−0.905497
$237$	0	0
$238$	14504.0	0.0165976
$239$	1.31352e6	1.48745	0.743724	−	0.668487i	$-0.233058\pi$
$239$	1.31352e6	1.48745	0.743724	+	0.668487i	$0.233058\pi$
$240$	0	0
$241$	−1.05607e6	−1.17125	−0.585625	−	0.810582i	$-0.699151\pi$
$241$	−1.05607e6	−1.17125	−0.585625	+	0.810582i	$0.699151\pi$
$242$	29282.0	0.0321412
$243$	0	0
$244$	−546336.	−0.587469
$245$	177674.	0.189107
$246$	0	0
$247$	−480480.	−0.501110
$248$	−1.06104e6	−1.09548
$249$	0	0
$250$	−114552.	−0.115918
$251$	−1.68914e6	−1.69232	−0.846159	−	0.532931i	$-0.821091\pi$
$251$	−1.68914e6	−1.69232	−0.846159	+	0.532931i	$0.821091\pi$
$252$	0	0
$253$	−52756.0	−0.0518168
$254$	442096.	0.429964
$255$	0	0
$256$	−30464.0	−0.0290527
$257$	−641938.	−0.606262	−0.303131	−	0.952949i	$-0.598032\pi$
$257$	−641938.	−0.606262	−0.303131	+	0.952949i	$0.598032\pi$
$258$	0	0
$259$	−6762.00	−0.00626363
$260$	−754208.	−0.691923
$261$	0	0
$262$	−75144.0	−0.0676303
$263$	1.10150e6	0.981959	0.490980	−	0.871171i	$-0.336639\pi$
$263$	1.10150e6	0.981959	0.490980	+	0.871171i	$0.336639\pi$
$264$	0	0
$265$	−390276.	−0.341395
$266$	129360.	0.112097
$267$	0	0
$268$	−1.79446e6	−1.52615
$269$	2.15147e6	1.81282	0.906410	−	0.422399i	$-0.138812\pi$
$269$	2.15147e6	1.81282	0.906410	+	0.422399i	$0.138812\pi$
$270$	0	0
$271$	−1.08327e6	−0.896010	−0.448005	−	0.894031i	$-0.647865\pi$
$271$	−1.08327e6	−0.896010	−0.448005	+	0.894031i	$0.647865\pi$
$272$	−97088.0	−0.0795689
$273$	0	0
$274$	581204.	0.467684
$275$	−284471.	−0.226833
$276$	0	0
$277$	−2.22372e6	−1.74133	−0.870665	−	0.491877i	$-0.836311\pi$
$277$	−2.22372e6	−1.74133	−0.870665	+	0.491877i	$0.836311\pi$
$278$	734720.	0.570177
$279$	0	0
$280$	435120.	0.331676
$281$	153018.	0.115605	0.0578025	−	0.998328i	$-0.481591\pi$
$281$	153018.	0.115605	0.0578025	+	0.998328i	$0.481591\pi$
$282$	0	0
$283$	715324.	0.530929	0.265465	−	0.964121i	$-0.414475\pi$
$283$	715324.	0.530929	0.265465	+	0.964121i	$0.414475\pi$
$284$	−103824.	−0.0763839
$285$	0	0
$286$	−88088.0	−0.0636798
$287$	26068.0	0.0186811
$288$	0	0
$289$	−1.39795e6	−0.984573
$290$	−439560.	−0.306919
$291$	0	0
$292$	680288.	0.466913
$293$	−347424.	−0.236424	−0.118212	−	0.992988i	$-0.537716\pi$
$293$	−347424.	−0.236424	−0.118212	+	0.992988i	$0.537716\pi$
$294$	0	0
$295$	2.04758e6	1.36989
$296$	−16560.0	−0.0109858
$297$	0	0
$298$	925460.	0.603694
$299$	158704.	0.102662
$300$	0	0
$301$	1.01312e6	0.644535
$302$	−15296.0	−0.00965074
$303$	0	0
$304$	−865920.	−0.537395
$305$	1.44389e6	0.888759
$306$	0	0
$307$	2.64043e6	1.59893	0.799463	−	0.600715i	$-0.205117\pi$
$307$	2.64043e6	1.59893	0.799463	+	0.600715i	$0.205117\pi$
$308$	−166012.	−0.0997155
$309$	0	0
$310$	1.30862e6	0.773407
$311$	947778.	0.555656	0.277828	−	0.960631i	$-0.410386\pi$
$311$	947778.	0.555656	0.277828	+	0.960631i	$0.410386\pi$
$312$	0	0
$313$	−248686.	−0.143480	−0.0717399	−	0.997423i	$-0.522855\pi$
$313$	−248686.	−0.143480	−0.0717399	+	0.997423i	$0.522855\pi$
$314$	−322964.	−0.184854
$315$	0	0
$316$	61600.0	0.0347027
$317$	−2.60904e6	−1.45825	−0.729125	−	0.684380i	$-0.760073\pi$
$317$	−2.60904e6	−1.45825	−0.729125	+	0.684380i	$0.760073\pi$
$318$	0	0
$319$	359370.	0.197727
$320$	−790912.	−0.431771
$321$	0	0
$322$	−42728.0	−0.0229653
$323$	195360.	0.104191
$324$	0	0
$325$	855764.	0.449413
$326$	358928.	0.187052
$327$	0	0
$328$	63840.0	0.0327649
$329$	−574378.	−0.292555
$330$	0	0
$331$	152332.	0.0764225	0.0382112	−	0.999270i	$-0.487834\pi$
$331$	152332.	0.0764225	0.0382112	+	0.999270i	$0.487834\pi$
$332$	2.08387e6	1.03759
$333$	0	0
$334$	−633696.	−0.310824
$335$	4.74251e6	2.30885
$336$	0	0
$337$	206558.	0.0990757	0.0495379	−	0.998772i	$-0.484225\pi$
$337$	206558.	0.0990757	0.0495379	+	0.998772i	$0.484225\pi$
$338$	−477594.	−0.227388
$339$	0	0
$340$	306656.	0.143865
$341$	−1.06988e6	−0.498253
$342$	0	0
$343$	−117649.	−0.0539949
$344$	2.48112e6	1.13045
$345$	0	0
$346$	350232.	0.157277
$347$	−2.01807e6	−0.899730	−0.449865	−	0.893097i	$-0.648528\pi$
$347$	−2.01807e6	−0.899730	−0.449865	+	0.893097i	$0.648528\pi$
$348$	0	0
$349$	−580440.	−0.255090	−0.127545	−	0.991833i	$-0.540710\pi$
$349$	−580440.	−0.255090	−0.127545	+	0.991833i	$0.540710\pi$
$350$	−230398.	−0.100533
$351$	0	0
$352$	−623392.	−0.268167
$353$	−572034.	−0.244335	−0.122167	−	0.992510i	$-0.538984\pi$
$353$	−572034.	−0.244335	−0.122167	+	0.992510i	$0.538984\pi$
$354$	0	0
$355$	274392.	0.115558
$356$	956760.	0.400109
$357$	0	0
$358$	139560.	0.0575511
$359$	−4.56544e6	−1.86959	−0.934795	−	0.355187i	$-0.884417\pi$
$359$	−4.56544e6	−1.86959	−0.934795	+	0.355187i	$0.884417\pi$
$360$	0	0
$361$	−733699.	−0.296312
$362$	−157276.	−0.0630799
$363$	0	0
$364$	499408.	0.197561
$365$	−1.79790e6	−0.706373
$366$	0	0
$367$	924358.	0.358241	0.179120	−	0.983827i	$-0.442675\pi$
$367$	924358.	0.358241	0.179120	+	0.983827i	$0.442675\pi$
$368$	286016.	0.110096
$369$	0	0
$370$	20424.0	0.00775598
$371$	258426.	0.0974768
$372$	0	0
$373$	4.92021e6	1.83110	0.915550	−	0.402205i	$-0.131756\pi$
$373$	4.92021e6	1.83110	0.915550	+	0.402205i	$0.131756\pi$
$374$	35816.0	0.0132403
$375$	0	0
$376$	−1.40664e6	−0.513113
$377$	−1.08108e6	−0.391746
$378$	0	0
$379$	3.97540e6	1.42162	0.710809	−	0.703385i	$-0.248329\pi$
$379$	3.97540e6	1.42162	0.710809	+	0.703385i	$0.248329\pi$
$380$	2.73504e6	0.971638
$381$	0	0
$382$	1.85442e6	0.650203
$383$	982846.	0.342364	0.171182	−	0.985239i	$-0.445241\pi$
$383$	982846.	0.342364	0.171182	+	0.985239i	$0.445241\pi$
$384$	0	0
$385$	438746.	0.150856
$386$	1.75495e6	0.599509
$387$	0	0
$388$	−4.24810e6	−1.43257
$389$	744090.	0.249317	0.124658	−	0.992200i	$-0.460217\pi$
$389$	744090.	0.249317	0.124658	+	0.992200i	$0.460217\pi$
$390$	0	0
$391$	−64528.0	−0.0213455
$392$	−288120.	−0.0947018
$393$	0	0
$394$	1.48920e6	0.483296
$395$	−162800.	−0.0525003
$396$	0	0
$397$	5.73024e6	1.82472	0.912360	−	0.409388i	$-0.134258\pi$
$397$	5.73024e6	1.82472	0.912360	+	0.409388i	$0.134258\pi$
$398$	2.15862e6	0.683075
$399$	0	0
$400$	1.54226e6	0.481955
$401$	−4.26756e6	−1.32531	−0.662657	−	0.748923i	$-0.730571\pi$
$401$	−4.26756e6	−1.32531	−0.662657	+	0.748923i	$0.730571\pi$
$402$	0	0
$403$	3.21849e6	0.987164
$404$	3.27186e6	0.997334
$405$	0	0
$406$	291060.	0.0876330
$407$	−16698.0	−0.00499664
$408$	0	0
$409$	−5.81772e6	−1.71967	−0.859834	−	0.510574i	$-0.829433\pi$
$409$	−5.81772e6	−1.71967	−0.859834	+	0.510574i	$0.829433\pi$
$410$	−78736.0	−0.0231320
$411$	0	0
$412$	−2.90343e6	−0.842692
$413$	−1.35583e6	−0.391138
$414$	0	0
$415$	−5.50738e6	−1.56973
$416$	1.87533e6	0.531305
$417$	0	0
$418$	319440.	0.0894229
$419$	4.38823e6	1.22111	0.610554	−	0.791974i	$-0.290947\pi$
$419$	4.38823e6	1.22111	0.610554	+	0.791974i	$0.290947\pi$
$420$	0	0
$421$	−1.48456e6	−0.408218	−0.204109	−	0.978948i	$-0.565430\pi$
$421$	−1.48456e6	−0.408218	−0.204109	+	0.978948i	$0.565430\pi$
$422$	1.45754e6	0.398419
$423$	0	0
$424$	632880.	0.170965
$425$	−347948.	−0.0934420
$426$	0	0
$427$	−956088.	−0.253763
$428$	−2.71858e6	−0.717352
$429$	0	0
$430$	−3.06005e6	−0.798100
$431$	206448.	0.0535325	0.0267662	−	0.999642i	$-0.491479\pi$
$431$	206448.	0.0535325	0.0267662	+	0.999642i	$0.491479\pi$
$432$	0	0
$433$	−5.67867e6	−1.45555	−0.727774	−	0.685817i	$-0.759445\pi$
$433$	−5.67867e6	−1.45555	−0.727774	+	0.685817i	$0.759445\pi$
$434$	−866516.	−0.220827
$435$	0	0
$436$	−1.48204e6	−0.373374
$437$	−575520.	−0.144164
$438$	0	0
$439$	−4.43666e6	−1.09874	−0.549370	−	0.835579i	$-0.685132\pi$
$439$	−4.43666e6	−1.09874	−0.549370	+	0.835579i	$0.685132\pi$
$440$	1.07448e6	0.264586
$441$	0	0
$442$	−107744.	−0.0262324
$443$	−6.17328e6	−1.49454	−0.747269	−	0.664522i	$-0.768635\pi$
$443$	−6.17328e6	−1.49454	−0.747269	+	0.664522i	$0.768635\pi$
$444$	0	0
$445$	−2.52858e6	−0.605308
$446$	76748.0	0.0182696
$447$	0	0
$448$	523712.	0.123281
$449$	4.93105e6	1.15431	0.577156	−	0.816634i	$-0.304162\pi$
$449$	4.93105e6	1.15431	0.577156	+	0.816634i	$0.304162\pi$
$450$	0	0
$451$	64372.0	0.0149024
$452$	−2.25473e6	−0.519096
$453$	0	0
$454$	−646536.	−0.147215
$455$	−1.31986e6	−0.298883
$456$	0	0
$457$	−4.15030e6	−0.929585	−0.464793	−	0.885420i	$-0.653871\pi$
$457$	−4.15030e6	−0.929585	−0.464793	+	0.885420i	$0.653871\pi$
$458$	1.62738e6	0.362514
$459$	0	0
$460$	−903392.	−0.199059
$461$	4.40345e6	0.965029	0.482515	−	0.875888i	$-0.339723\pi$
$461$	4.40345e6	0.965029	0.482515	+	0.875888i	$0.339723\pi$
$462$	0	0
$463$	6.82728e6	1.48012	0.740058	−	0.672544i	$-0.234798\pi$
$463$	6.82728e6	1.48012	0.740058	+	0.672544i	$0.234798\pi$
$464$	−1.94832e6	−0.420112
$465$	0	0
$466$	−2.21603e6	−0.472727
$467$	−6.88244e6	−1.46033	−0.730163	−	0.683272i	$-0.760556\pi$
$467$	−6.88244e6	−1.46033	−0.730163	+	0.683272i	$0.760556\pi$
$468$	0	0
$469$	−3.14031e6	−0.659236
$470$	1.73486e6	0.362259
$471$	0	0
$472$	−3.32040e6	−0.686018
$473$	2.50180e6	0.514161
$474$	0	0
$475$	−3.10332e6	−0.631092
$476$	−203056.	−0.0410770
$477$	0	0
$478$	2.62704e6	0.525892
$479$	−5.02430e6	−1.00055	−0.500273	−	0.865868i	$-0.666767\pi$
$479$	−5.02430e6	−1.00055	−0.500273	+	0.865868i	$0.666767\pi$
$480$	0	0
$481$	50232.0	0.00989960
$482$	−2.11214e6	−0.414099
$483$	0	0
$484$	−409948.	−0.0795455
$485$	1.12271e7	2.16728
$486$	0	0
$487$	−4.51601e6	−0.862845	−0.431422	−	0.902150i	$-0.641988\pi$
$487$	−4.51601e6	−0.862845	−0.431422	+	0.902150i	$0.641988\pi$
$488$	−2.34144e6	−0.445075
$489$	0	0
$490$	355348.	0.0668596
$491$	−5.55737e6	−1.04032	−0.520159	−	0.854070i	$-0.674127\pi$
$491$	−5.55737e6	−1.04032	−0.520159	+	0.854070i	$0.674127\pi$
$492$	0	0
$493$	439560.	0.0814518
$494$	−960960.	−0.177169
$495$	0	0
$496$	5.80035e6	1.05864
$497$	−181692.	−0.0329947
$498$	0	0
$499$	−3.49744e6	−0.628780	−0.314390	−	0.949294i	$-0.601800\pi$
$499$	−3.49744e6	−0.628780	−0.314390	+	0.949294i	$0.601800\pi$
$500$	1.60373e6	0.286884
$501$	0	0
$502$	−3.37828e6	−0.598325
$503$	−4.61280e6	−0.812915	−0.406457	−	0.913670i	$-0.633236\pi$
$503$	−4.61280e6	−0.812915	−0.406457	+	0.913670i	$0.633236\pi$
$504$	0	0
$505$	−8.64705e6	−1.50883
$506$	−105512.	−0.0183200
$507$	0	0
$508$	−6.18934e6	−1.06411
$509$	−7.41609e6	−1.26876	−0.634382	−	0.773020i	$-0.718745\pi$
$509$	−7.41609e6	−1.26876	−0.634382	+	0.773020i	$0.718745\pi$
$510$	0	0
$511$	1.19050e6	0.201687
$512$	5.89875e6	0.994455
$513$	0	0
$514$	−1.28388e6	−0.214346
$515$	7.67336e6	1.27487
$516$	0	0
$517$	−1.41836e6	−0.233378
$518$	−13524.0	−0.00221453
$519$	0	0
$520$	−3.23232e6	−0.524211
$521$	−9.75970e6	−1.57522	−0.787612	−	0.616172i	$-0.788683\pi$
$521$	−9.75970e6	−1.57522	−0.787612	+	0.616172i	$0.788683\pi$
$522$	0	0
$523$	1.66084e6	0.265506	0.132753	−	0.991149i	$-0.457618\pi$
$523$	1.66084e6	0.265506	0.132753	+	0.991149i	$0.457618\pi$
$524$	1.05202e6	0.167376
$525$	0	0
$526$	2.20299e6	0.347175
$527$	−1.30862e6	−0.205251
$528$	0	0
$529$	−6.24625e6	−0.970465
$530$	−780552.	−0.120701
$531$	0	0
$532$	−1.81104e6	−0.277427
$533$	−193648.	−0.0295253
$534$	0	0
$535$	7.18481e6	1.08525
$536$	−7.69056e6	−1.15623
$537$	0	0
$538$	4.30294e6	0.640929
$539$	−290521.	−0.0430730
$540$	0	0
$541$	6.97250e6	1.02423	0.512113	−	0.858918i	$-0.328863\pi$
$541$	6.97250e6	1.02423	0.512113	+	0.858918i	$0.328863\pi$
$542$	−2.16654e6	−0.316787
$543$	0	0
$544$	−762496.	−0.110469
$545$	3.91682e6	0.564862
$546$	0	0
$547$	3.08503e6	0.440850	0.220425	−	0.975404i	$-0.429256\pi$
$547$	3.08503e6	0.440850	0.220425	+	0.975404i	$0.429256\pi$
$548$	−8.13686e6	−1.15746
$549$	0	0
$550$	−568942.	−0.0801976
$551$	3.92040e6	0.550112
$552$	0	0
$553$	107800.	0.0149901
$554$	−4.44744e6	−0.615653
$555$	0	0
$556$	−1.02861e7	−1.41112
$557$	3.29052e6	0.449394	0.224697	−	0.974429i	$-0.427861\pi$
$557$	3.29052e6	0.449394	0.224697	+	0.974429i	$0.427861\pi$
$558$	0	0
$559$	−7.52606e6	−1.01868
$560$	−2.37866e6	−0.320525
$561$	0	0
$562$	306036.	0.0408725
$563$	−5.45754e6	−0.725648	−0.362824	−	0.931858i	$-0.618187\pi$
$563$	−5.45754e6	−0.725648	−0.362824	+	0.931858i	$0.618187\pi$
$564$	0	0
$565$	5.95892e6	0.785320
$566$	1.43065e6	0.187712
$567$	0	0
$568$	−444960.	−0.0578696
$569$	−7.28571e6	−0.943390	−0.471695	−	0.881762i	$-0.656358\pi$
$569$	−7.28571e6	−0.943390	−0.471695	+	0.881762i	$0.656358\pi$
$570$	0	0
$571$	1.07129e7	1.37504	0.687519	−	0.726166i	$-0.258700\pi$
$571$	1.07129e7	1.37504	0.687519	+	0.726166i	$0.258700\pi$
$572$	1.23323e6	0.157599
$573$	0	0
$574$	52136.0	0.00660477
$575$	1.02504e6	0.129291
$576$	0	0
$577$	4.22024e6	0.527713	0.263856	−	0.964562i	$-0.415006\pi$
$577$	4.22024e6	0.527713	0.263856	+	0.964562i	$0.415006\pi$
$578$	−2.79591e6	−0.348099
$579$	0	0
$580$	6.15384e6	0.759585
$581$	3.64678e6	0.448197
$582$	0	0
$583$	638154.	0.0777596
$584$	2.91552e6	0.353740
$585$	0	0
$586$	−694848.	−0.0835884
$587$	6.24180e6	0.747678	0.373839	−	0.927494i	$-0.378041\pi$
$587$	6.24180e6	0.747678	0.373839	+	0.927494i	$0.378041\pi$
$588$	0	0
$589$	−1.16714e7	−1.38623
$590$	4.09516e6	0.484329
$591$	0	0
$592$	90528.0	0.0106164
$593$	−493664.	−0.0576494	−0.0288247	−	0.999584i	$-0.509176\pi$
$593$	−493664.	−0.0576494	−0.0288247	+	0.999584i	$0.509176\pi$
$594$	0	0
$595$	536648.	0.0621437
$596$	−1.29564e7	−1.49407
$597$	0	0
$598$	317408.	0.0362965
$599$	8.28890e6	0.943908	0.471954	−	0.881623i	$-0.343549\pi$
$599$	8.28890e6	0.943908	0.471954	+	0.881623i	$0.343549\pi$
$600$	0	0
$601$	80612.0	0.00910361	0.00455180	−	0.999990i	$-0.498551\pi$
$601$	80612.0	0.00910361	0.00455180	+	0.999990i	$0.498551\pi$
$602$	2.02625e6	0.227877
$603$	0	0
$604$	214144.	0.0238844
$605$	1.08343e6	0.120341
$606$	0	0
$607$	914168.	0.100706	0.0503529	−	0.998731i	$-0.483965\pi$
$607$	914168.	0.100706	0.0503529	+	0.998731i	$0.483965\pi$
$608$	−6.80064e6	−0.746089
$609$	0	0
$610$	2.88778e6	0.314224
$611$	4.26681e6	0.462381
$612$	0	0
$613$	9.97327e6	1.07198	0.535990	−	0.844224i	$-0.319939\pi$
$613$	9.97327e6	1.07198	0.535990	+	0.844224i	$0.319939\pi$
$614$	5.28086e6	0.565306
$615$	0	0
$616$	−711480.	−0.0755459
$617$	4.60636e6	0.487130	0.243565	−	0.969885i	$-0.421683\pi$
$617$	4.60636e6	0.487130	0.243565	+	0.969885i	$0.421683\pi$
$618$	0	0
$619$	2.51711e6	0.264044	0.132022	−	0.991247i	$-0.457853\pi$
$619$	2.51711e6	0.264044	0.132022	+	0.991247i	$0.457853\pi$
$620$	−1.83206e7	−1.91408
$621$	0	0
$622$	1.89556e6	0.196454
$623$	1.67433e6	0.172831
$624$	0	0
$625$	−1.15853e7	−1.18633
$626$	−497372.	−0.0507277
$627$	0	0
$628$	4.52150e6	0.457492
$629$	−20424.0	−0.00205833
$630$	0	0
$631$	−2.02153e6	−0.202119	−0.101059	−	0.994880i	$-0.532223\pi$
$631$	−2.02153e6	−0.202119	−0.101059	+	0.994880i	$0.532223\pi$
$632$	264000.	0.0262912
$633$	0	0
$634$	−5.21808e6	−0.515570
$635$	1.63576e7	1.60984
$636$	0	0
$637$	873964.	0.0853385
$638$	718740.	0.0699069
$639$	0	0
$640$	−1.37818e7	−1.33001
$641$	−1.55870e7	−1.49837	−0.749183	−	0.662363i	$-0.769554\pi$
$641$	−1.55870e7	−1.49837	−0.749183	+	0.662363i	$0.769554\pi$
$642$	0	0
$643$	−5.88755e6	−0.561574	−0.280787	−	0.959770i	$-0.590595\pi$
$643$	−5.88755e6	−0.561574	−0.280787	+	0.959770i	$0.590595\pi$
$644$	598192.	0.0568363
$645$	0	0
$646$	390720.	0.0368370
$647$	−5.58958e6	−0.524950	−0.262475	−	0.964939i	$-0.584539\pi$
$647$	−5.58958e6	−0.524950	−0.262475	+	0.964939i	$0.584539\pi$
$648$	0	0
$649$	−3.34807e6	−0.312020
$650$	1.71153e6	0.158891
$651$	0	0
$652$	−5.02499e6	−0.462931
$653$	−1.76227e7	−1.61730	−0.808648	−	0.588293i	$-0.799800\pi$
$653$	−1.76227e7	−1.61730	−0.808648	+	0.588293i	$0.799800\pi$
$654$	0	0
$655$	−2.78033e6	−0.253217
$656$	−348992.	−0.0316633
$657$	0	0
$658$	−1.14876e6	−0.103434
$659$	9.75566e6	0.875071	0.437535	−	0.899201i	$-0.355851\pi$
$659$	9.75566e6	0.875071	0.437535	+	0.899201i	$0.355851\pi$
$660$	0	0
$661$	−9.79522e6	−0.871988	−0.435994	−	0.899950i	$-0.643603\pi$
$661$	−9.79522e6	−0.871988	−0.435994	+	0.899950i	$0.643603\pi$
$662$	304664.	0.0270194
$663$	0	0
$664$	8.93088e6	0.786093
$665$	4.78632e6	0.419708
$666$	0	0
$667$	−1.29492e6	−0.112701
$668$	8.87174e6	0.769251
$669$	0	0
$670$	9.48502e6	0.816303
$671$	−2.36095e6	−0.202433
$672$	0	0
$673$	−4.72727e6	−0.402321	−0.201160	−	0.979558i	$-0.564471\pi$
$673$	−4.72727e6	−0.402321	−0.201160	+	0.979558i	$0.564471\pi$
$674$	413116.	0.0350286
$675$	0	0
$676$	6.68632e6	0.562756
$677$	1.93989e7	1.62669	0.813344	−	0.581783i	$-0.197645\pi$
$677$	1.93989e7	1.62669	0.813344	+	0.581783i	$0.197645\pi$
$678$	0	0
$679$	−7.43418e6	−0.618812
$680$	1.31424e6	0.108994
$681$	0	0
$682$	−2.13976e6	−0.176159
$683$	6.22004e6	0.510201	0.255100	−	0.966915i	$-0.417891\pi$
$683$	6.22004e6	0.510201	0.255100	+	0.966915i	$0.417891\pi$
$684$	0	0
$685$	2.15045e7	1.75107
$686$	−235298.	−0.0190901
$687$	0	0
$688$	−1.35635e7	−1.09244
$689$	−1.91974e6	−0.154061
$690$	0	0
$691$	6.04140e6	0.481330	0.240665	−	0.970608i	$-0.422635\pi$
$691$	6.04140e6	0.481330	0.240665	+	0.970608i	$0.422635\pi$
$692$	−4.90325e6	−0.389241
$693$	0	0
$694$	−4.03614e6	−0.318103
$695$	2.71846e7	2.13482
$696$	0	0
$697$	78736.0	0.00613891
$698$	−1.16088e6	−0.0901880
$699$	0	0
$700$	3.22557e6	0.248806
$701$	−3.97068e6	−0.305190	−0.152595	−	0.988289i	$-0.548763\pi$
$701$	−3.97068e6	−0.305190	−0.152595	+	0.988289i	$0.548763\pi$
$702$	0	0
$703$	−182160.	−0.0139016
$704$	1.29325e6	0.0983445
$705$	0	0
$706$	−1.14407e6	−0.0863853
$707$	5.72575e6	0.430808
$708$	0	0
$709$	5.15939e6	0.385463	0.192732	−	0.981252i	$-0.438265\pi$
$709$	5.15939e6	0.385463	0.192732	+	0.981252i	$0.438265\pi$
$710$	548784.	0.0408560
$711$	0	0
$712$	4.10040e6	0.303128
$713$	3.85511e6	0.283997
$714$	0	0
$715$	−3.25926e6	−0.238426
$716$	−1.95384e6	−0.142432
$717$	0	0
$718$	−9.13088e6	−0.661000
$719$	1.26734e7	0.914259	0.457129	−	0.889400i	$-0.348878\pi$
$719$	1.26734e7	0.914259	0.457129	+	0.889400i	$0.348878\pi$
$720$	0	0
$721$	−5.08101e6	−0.364009
$722$	−1.46740e6	−0.104762
$723$	0	0
$724$	2.20186e6	0.156115
$725$	−6.98247e6	−0.493360
$726$	0	0
$727$	1.32783e7	0.931762	0.465881	−	0.884847i	$-0.345737\pi$
$727$	1.32783e7	0.931762	0.465881	+	0.884847i	$0.345737\pi$
$728$	2.14032e6	0.149675
$729$	0	0
$730$	−3.59581e6	−0.249741
$731$	3.06005e6	0.211804
$732$	0	0
$733$	3.84050e6	0.264015	0.132007	−	0.991249i	$-0.457858\pi$
$733$	3.84050e6	0.264015	0.132007	+	0.991249i	$0.457858\pi$
$734$	1.84872e6	0.126657
$735$	0	0
$736$	2.24627e6	0.152851
$737$	−7.75465e6	−0.525888
$738$	0	0
$739$	1.19394e7	0.804215	0.402107	−	0.915592i	$-0.368278\pi$
$739$	1.19394e7	0.804215	0.402107	+	0.915592i	$0.368278\pi$
$740$	−285936.	−0.0191951
$741$	0	0
$742$	516852.	0.0344633
$743$	−2.53631e7	−1.68551	−0.842754	−	0.538298i	$-0.819067\pi$
$743$	−2.53631e7	−1.68551	−0.842754	+	0.538298i	$0.819067\pi$
$744$	0	0
$745$	3.42420e7	2.26031
$746$	9.84043e6	0.647391
$747$	0	0
$748$	−501424.	−0.0327681
$749$	−4.75751e6	−0.309867
$750$	0	0
$751$	1.43903e7	0.931046	0.465523	−	0.885036i	$-0.345866\pi$
$751$	1.43903e7	0.931046	0.465523	+	0.885036i	$0.345866\pi$
$752$	7.68963e6	0.495862
$753$	0	0
$754$	−2.16216e6	−0.138503
$755$	−565952.	−0.0361337
$756$	0	0
$757$	−1.85431e7	−1.17609	−0.588047	−	0.808827i	$-0.700103\pi$
$757$	−1.85431e7	−1.17609	−0.588047	+	0.808827i	$0.700103\pi$
$758$	7.95080e6	0.502618
$759$	0	0
$760$	1.17216e7	0.736127
$761$	−6.43123e6	−0.402562	−0.201281	−	0.979534i	$-0.564510\pi$
$761$	−6.43123e6	−0.402562	−0.201281	+	0.979534i	$0.564510\pi$
$762$	0	0
$763$	−2.59357e6	−0.161282
$764$	−2.59618e7	−1.60917
$765$	0	0
$766$	1.96569e6	0.121044
$767$	1.00719e7	0.618190
$768$	0	0
$769$	1.48249e7	0.904014	0.452007	−	0.892014i	$-0.350708\pi$
$769$	1.48249e7	0.904014	0.452007	+	0.892014i	$0.350708\pi$
$770$	877492.	0.0533355
$771$	0	0
$772$	−2.45693e7	−1.48371
$773$	−1.96325e7	−1.18175	−0.590876	−	0.806762i	$-0.701218\pi$
$773$	−1.96325e7	−1.18175	−0.590876	+	0.806762i	$0.701218\pi$
$774$	0	0
$775$	2.07875e7	1.24322
$776$	−1.82062e7	−1.08534
$777$	0	0
$778$	1.48818e6	0.0881468
$779$	702240.	0.0414612
$780$	0	0
$781$	−448668.	−0.0263207
$782$	−129056.	−0.00754677
$783$	0	0
$784$	1.57506e6	0.0915179
$785$	−1.19497e7	−0.692120
$786$	0	0
$787$	−9.48639e6	−0.545964	−0.272982	−	0.962019i	$-0.588010\pi$
$787$	−9.48639e6	−0.545964	−0.272982	+	0.962019i	$0.588010\pi$
$788$	−2.08489e7	−1.19610
$789$	0	0
$790$	−325600.	−0.0185617
$791$	−3.94577e6	−0.224229
$792$	0	0
$793$	7.10237e6	0.401070
$794$	1.14605e7	0.645136
$795$	0	0
$796$	−3.02207e7	−1.69052
$797$	−3.32326e7	−1.85318	−0.926592	−	0.376068i	$-0.877276\pi$
$797$	−3.32326e7	−1.85318	−0.926592	+	0.376068i	$0.877276\pi$
$798$	0	0
$799$	−1.73486e6	−0.0961383
$800$	1.21124e7	0.669119
$801$	0	0
$802$	−8.53512e6	−0.468569
$803$	2.93982e6	0.160891
$804$	0	0
$805$	−1.58094e6	−0.0859854
$806$	6.43698e6	0.349015
$807$	0	0
$808$	1.40222e7	0.755595
$809$	2.75792e7	1.48153	0.740764	−	0.671766i	$-0.234464\pi$
$809$	2.75792e7	1.48153	0.740764	+	0.671766i	$0.234464\pi$
$810$	0	0
$811$	−7.76107e6	−0.414352	−0.207176	−	0.978304i	$-0.566427\pi$
$811$	−7.76107e6	−0.414352	−0.207176	+	0.978304i	$0.566427\pi$
$812$	−4.07484e6	−0.216880
$813$	0	0
$814$	−33396.0	−0.00176658
$815$	1.32803e7	0.700350
$816$	0	0
$817$	2.72923e7	1.43049
$818$	−1.16354e7	−0.607994
$819$	0	0
$820$	1.10230e6	0.0572488
$821$	1.60578e7	0.831434	0.415717	−	0.909494i	$-0.363531\pi$
$821$	1.60578e7	0.831434	0.415717	+	0.909494i	$0.363531\pi$
$822$	0	0
$823$	1.04666e6	0.0538651	0.0269326	−	0.999637i	$-0.491426\pi$
$823$	1.04666e6	0.0538651	0.0269326	+	0.999637i	$0.491426\pi$
$824$	−1.24433e7	−0.638435
$825$	0	0
$826$	−2.71166e6	−0.138288
$827$	−8.91799e6	−0.453423	−0.226711	−	0.973962i	$-0.572797\pi$
$827$	−8.91799e6	−0.453423	−0.226711	+	0.973962i	$0.572797\pi$
$828$	0	0
$829$	−2.53821e7	−1.28275	−0.641374	−	0.767229i	$-0.721635\pi$
$829$	−2.53821e7	−1.28275	−0.641374	+	0.767229i	$0.721635\pi$
$830$	−1.10148e7	−0.554983
$831$	0	0
$832$	−3.89043e6	−0.194845
$833$	−355348.	−0.0177436
$834$	0	0
$835$	−2.34468e7	−1.16377
$836$	−4.47216e6	−0.221310
$837$	0	0
$838$	8.77646e6	0.431727
$839$	3.10636e7	1.52351	0.761757	−	0.647863i	$-0.224337\pi$
$839$	3.10636e7	1.52351	0.761757	+	0.647863i	$0.224337\pi$
$840$	0	0
$841$	−1.16902e7	−0.569946
$842$	−2.96912e6	−0.144327
$843$	0	0
$844$	−2.04056e7	−0.986038
$845$	−1.76710e7	−0.851371
$846$	0	0
$847$	−717409.	−0.0343604
$848$	−3.45974e6	−0.165217
$849$	0	0
$850$	−695896.	−0.0330367
$851$	60168.0	0.00284801
$852$	0	0
$853$	2.24337e7	1.05567	0.527835	−	0.849347i	$-0.323004\pi$
$853$	2.24337e7	1.05567	0.527835	+	0.849347i	$0.323004\pi$
$854$	−1.91218e6	−0.0897187
$855$	0	0
$856$	−1.16510e7	−0.543476
$857$	−4.76449e6	−0.221597	−0.110799	−	0.993843i	$-0.535341\pi$
$857$	−4.76449e6	−0.221597	−0.110799	+	0.993843i	$0.535341\pi$
$858$	0	0
$859$	468030.	0.0216417	0.0108208	−	0.999941i	$-0.496556\pi$
$859$	468030.	0.0216417	0.0108208	+	0.999941i	$0.496556\pi$
$860$	4.28407e7	1.97520
$861$	0	0
$862$	412896.	0.0189266
$863$	1.20487e7	0.550697	0.275349	−	0.961344i	$-0.411207\pi$
$863$	1.20487e7	0.550697	0.275349	+	0.961344i	$0.411207\pi$
$864$	0	0
$865$	1.29586e7	0.588867
$866$	−1.13573e7	−0.514614
$867$	0	0
$868$	1.21312e7	0.546519
$869$	266200.	0.0119580
$870$	0	0
$871$	2.33280e7	1.04192
$872$	−6.35160e6	−0.282873
$873$	0	0
$874$	−1.15104e6	−0.0509697
$875$	2.80652e6	0.123922
$876$	0	0
$877$	3.61718e6	0.158807	0.0794037	−	0.996843i	$-0.474698\pi$
$877$	3.61718e6	0.158807	0.0794037	+	0.996843i	$0.474698\pi$
$878$	−8.87332e6	−0.388463
$879$	0	0
$880$	−5.87382e6	−0.255690
$881$	2.27025e7	0.985448	0.492724	−	0.870186i	$-0.336001\pi$
$881$	2.27025e7	0.985448	0.492724	+	0.870186i	$0.336001\pi$
$882$	0	0
$883$	2.41926e7	1.04419	0.522097	−	0.852886i	$-0.325150\pi$
$883$	2.41926e7	1.04419	0.522097	+	0.852886i	$0.325150\pi$
$884$	1.50842e6	0.0649218
$885$	0	0
$886$	−1.23466e7	−0.528399
$887$	4.06125e7	1.73321	0.866604	−	0.498997i	$-0.166298\pi$
$887$	4.06125e7	1.73321	0.866604	+	0.498997i	$0.166298\pi$
$888$	0	0
$889$	−1.08314e7	−0.459651
$890$	−5.05716e6	−0.214009
$891$	0	0
$892$	−1.07447e6	−0.0452150
$893$	−1.54730e7	−0.649302
$894$	0	0
$895$	5.16372e6	0.215479
$896$	9.12576e6	0.379751
$897$	0	0
$898$	9.86210e6	0.408111
$899$	−2.62607e7	−1.08370
$900$	0	0
$901$	780552.	0.0320324
$902$	128744.	0.00526879
$903$	0	0
$904$	−9.66312e6	−0.393275
$905$	−5.81921e6	−0.236180
$906$	0	0
$907$	−1.98235e7	−0.800132	−0.400066	−	0.916486i	$-0.631013\pi$
$907$	−1.98235e7	−0.800132	−0.400066	+	0.916486i	$0.631013\pi$
$908$	9.05150e6	0.364339
$909$	0	0
$910$	−2.63973e6	−0.105671
$911$	2.21209e7	0.883094	0.441547	−	0.897238i	$-0.354430\pi$
$911$	2.21209e7	0.883094	0.441547	+	0.897238i	$0.354430\pi$
$912$	0	0
$913$	9.00530e6	0.357537
$914$	−8.30060e6	−0.328658
$915$	0	0
$916$	−2.27833e7	−0.897177
$917$	1.84103e6	0.0722998
$918$	0	0
$919$	4.42949e7	1.73008	0.865038	−	0.501706i	$-0.167294\pi$
$919$	4.42949e7	1.73008	0.865038	+	0.501706i	$0.167294\pi$
$920$	−3.87168e6	−0.150810
$921$	0	0
$922$	8.80690e6	0.341189
$923$	1.34971e6	0.0521479
$924$	0	0
$925$	324438.	0.0124674
$926$	1.36546e7	0.523300
$927$	0	0
$928$	−1.53014e7	−0.583260
$929$	−1.13166e7	−0.430207	−0.215104	−	0.976591i	$-0.569009\pi$
$929$	−1.13166e7	−0.430207	−0.215104	+	0.976591i	$0.569009\pi$
$930$	0	0
$931$	−3.16932e6	−0.119837
$932$	3.10244e7	1.16994
$933$	0	0
$934$	−1.37649e7	−0.516304
$935$	1.32519e6	0.0495735
$936$	0	0
$937$	3.37578e7	1.25610	0.628052	−	0.778172i	$-0.283853\pi$
$937$	3.37578e7	1.25610	0.628052	+	0.778172i	$0.283853\pi$
$938$	−6.28062e6	−0.233075
$939$	0	0
$940$	−2.42880e7	−0.896544
$941$	3.30036e7	1.21503	0.607516	−	0.794307i	$-0.292166\pi$
$941$	3.30036e7	1.21503	0.607516	+	0.794307i	$0.292166\pi$
$942$	0	0
$943$	−231952.	−0.00849413
$944$	1.81515e7	0.662953
$945$	0	0
$946$	5.00359e6	0.181783
$947$	−1.92599e7	−0.697876	−0.348938	−	0.937146i	$-0.613458\pi$
$947$	−1.92599e7	−0.697876	−0.348938	+	0.937146i	$0.613458\pi$
$948$	0	0
$949$	−8.84374e6	−0.318765
$950$	−6.20664e6	−0.223125
$951$	0	0
$952$	−870240.	−0.0311205
$953$	−1.18503e7	−0.422665	−0.211332	−	0.977414i	$-0.567780\pi$
$953$	−1.18503e7	−0.422665	−0.211332	+	0.977414i	$0.567780\pi$
$954$	0	0
$955$	6.86134e7	2.43445
$956$	−3.67786e7	−1.30152
$957$	0	0
$958$	−1.00486e7	−0.353746
$959$	−1.42395e7	−0.499975
$960$	0	0
$961$	4.95518e7	1.73082
$962$	100464.	0.00350004
$963$	0	0
$964$	2.95699e7	1.02484
$965$	6.49331e7	2.24465
$966$	0	0
$967$	−1.44196e7	−0.495892	−0.247946	−	0.968774i	$-0.579755\pi$
$967$	−1.44196e7	−0.495892	−0.247946	+	0.968774i	$0.579755\pi$
$968$	−1.75692e6	−0.0602648
$969$	0	0
$970$	2.24543e7	0.766248
$971$	1.37494e7	0.467990	0.233995	−	0.972238i	$-0.424820\pi$
$971$	1.37494e7	0.467990	0.233995	+	0.972238i	$0.424820\pi$
$972$	0	0
$973$	−1.80006e7	−0.609545
$974$	−9.03202e6	−0.305062
$975$	0	0
$976$	1.27999e7	0.430112
$977$	9.18802e6	0.307954	0.153977	−	0.988074i	$-0.450792\pi$
$977$	9.18802e6	0.307954	0.153977	+	0.988074i	$0.450792\pi$
$978$	0	0
$979$	4.13457e6	0.137871
$980$	−4.97487e6	−0.165469
$981$	0	0
$982$	−1.11147e7	−0.367808
$983$	3.43732e7	1.13458	0.567292	−	0.823517i	$-0.307991\pi$
$983$	3.43732e7	1.13458	0.567292	+	0.823517i	$0.307991\pi$
$984$	0	0
$985$	5.51005e7	1.80953
$986$	879120.	0.0287976
$987$	0	0
$988$	1.34534e7	0.438471
$989$	−9.01474e6	−0.293064
$990$	0	0
$991$	−4.32179e7	−1.39791	−0.698956	−	0.715164i	$-0.746352\pi$
$991$	−4.32179e7	−1.39791	−0.698956	+	0.715164i	$0.746352\pi$
$992$	4.55540e7	1.46976
$993$	0	0
$994$	−363384.	−0.0116654
$995$	7.98689e7	2.55753
$996$	0	0
$997$	2.54793e7	0.811801	0.405901	−	0.913917i	$-0.366958\pi$
$997$	2.54793e7	0.811801	0.405901	+	0.913917i	$0.366958\pi$
$998$	−6.99488e6	−0.222307
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.6.a.a.1.1		1
3.2	odd	2		77.6.a.a.1.1	✓	1
21.20	even	2		539.6.a.d.1.1		1
33.32	even	2		847.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.6.a.a.1.1	✓	1	3.2	odd	2
539.6.a.d.1.1		1	21.20	even	2
693.6.a.a.1.1		1	1.1	even	1	trivial
847.6.a.a.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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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