LMFDB - Embedded newform 6.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 6 = 2 \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	6.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:	$0.962302918878$
Analytic rank:	$0$
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	$\chi$	$=$	6.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -66.0000 q^{5} -36.0000 q^{6} +176.000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -66.0000 q^{5} -36.0000 q^{6} +176.000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -264.000 q^{10} -60.0000 q^{11} -144.000 q^{12} -658.000 q^{13} +704.000 q^{14} +594.000 q^{15} +256.000 q^{16} -414.000 q^{17} +324.000 q^{18} +956.000 q^{19} -1056.00 q^{20} -1584.00 q^{21} -240.000 q^{22} +600.000 q^{23} -576.000 q^{24} +1231.00 q^{25} -2632.00 q^{26} -729.000 q^{27} +2816.00 q^{28} +5574.00 q^{29} +2376.00 q^{30} -3592.00 q^{31} +1024.00 q^{32} +540.000 q^{33} -1656.00 q^{34} -11616.0 q^{35} +1296.00 q^{36} -8458.00 q^{37} +3824.00 q^{38} +5922.00 q^{39} -4224.00 q^{40} +19194.0 q^{41} -6336.00 q^{42} +13316.0 q^{43} -960.000 q^{44} -5346.00 q^{45} +2400.00 q^{46} -19680.0 q^{47} -2304.00 q^{48} +14169.0 q^{49} +4924.00 q^{50} +3726.00 q^{51} -10528.0 q^{52} -31266.0 q^{53} -2916.00 q^{54} +3960.00 q^{55} +11264.0 q^{56} -8604.00 q^{57} +22296.0 q^{58} +26340.0 q^{59} +9504.00 q^{60} -31090.0 q^{61} -14368.0 q^{62} +14256.0 q^{63} +4096.00 q^{64} +43428.0 q^{65} +2160.00 q^{66} -16804.0 q^{67} -6624.00 q^{68} -5400.00 q^{69} -46464.0 q^{70} +6120.00 q^{71} +5184.00 q^{72} -25558.0 q^{73} -33832.0 q^{74} -11079.0 q^{75} +15296.0 q^{76} -10560.0 q^{77} +23688.0 q^{78} +74408.0 q^{79} -16896.0 q^{80} +6561.00 q^{81} +76776.0 q^{82} -6468.00 q^{83} -25344.0 q^{84} +27324.0 q^{85} +53264.0 q^{86} -50166.0 q^{87} -3840.00 q^{88} -32742.0 q^{89} -21384.0 q^{90} -115808. q^{91} +9600.00 q^{92} +32328.0 q^{93} -78720.0 q^{94} -63096.0 q^{95} -9216.00 q^{96} +166082. q^{97} +56676.0 q^{98} -4860.00 q^{99} +O(q^{100})\)

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	16.0000	0.500000
$5$	−66.0000	−1.18064	−0.590322	−	0.807168i	$-0.700999\pi$
$5$	−66.0000	−1.18064	−0.590322	+	0.807168i	$0.700999\pi$
$6$	−36.0000	−0.408248
$7$	176.000	1.35759	0.678793	−	0.734329i	$-0.262503\pi$
$7$	176.000	1.35759	0.678793	+	0.734329i	$0.262503\pi$
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	−264.000	−0.834841
$11$	−60.0000	−0.149510	−0.0747549	−	0.997202i	$-0.523817\pi$
$11$	−60.0000	−0.149510	−0.0747549	+	0.997202i	$0.523817\pi$
$12$	−144.000	−0.288675
$13$	−658.000	−1.07986	−0.539930	−	0.841710i	$-0.681549\pi$
$13$	−658.000	−1.07986	−0.539930	+	0.841710i	$0.681549\pi$
$14$	704.000	0.959959
$15$	594.000	0.681645
$16$	256.000	0.250000
$17$	−414.000	−0.347439	−0.173719	−	0.984795i	$-0.555579\pi$
$17$	−414.000	−0.347439	−0.173719	+	0.984795i	$0.555579\pi$
$18$	324.000	0.235702
$19$	956.000	0.607539	0.303769	−	0.952746i	$-0.401755\pi$
$19$	956.000	0.607539	0.303769	+	0.952746i	$0.401755\pi$
$20$	−1056.00	−0.590322
$21$	−1584.00	−0.783803
$22$	−240.000	−0.105719
$23$	600.000	0.236500	0.118250	−	0.992984i	$-0.462272\pi$
$23$	600.000	0.236500	0.118250	+	0.992984i	$0.462272\pi$
$24$	−576.000	−0.204124
$25$	1231.00	0.393920
$26$	−2632.00	−0.763576
$27$	−729.000	−0.192450
$28$	2816.00	0.678793
$29$	5574.00	1.23076	0.615378	−	0.788232i	$-0.289003\pi$
$29$	5574.00	1.23076	0.615378	+	0.788232i	$0.289003\pi$
$30$	2376.00	0.481996
$31$	−3592.00	−0.671324	−0.335662	−	0.941983i	$-0.608960\pi$
$31$	−3592.00	−0.671324	−0.335662	+	0.941983i	$0.608960\pi$
$32$	1024.00	0.176777
$33$	540.000	0.0863195
$34$	−1656.00	−0.245676
$35$	−11616.0	−1.60283
$36$	1296.00	0.166667
$37$	−8458.00	−1.01570	−0.507848	−	0.861447i	$-0.669559\pi$
$37$	−8458.00	−1.01570	−0.507848	+	0.861447i	$0.669559\pi$
$38$	3824.00	0.429595
$39$	5922.00	0.623458
$40$	−4224.00	−0.417421
$41$	19194.0	1.78322	0.891612	−	0.452800i	$-0.149575\pi$
$41$	19194.0	1.78322	0.891612	+	0.452800i	$0.149575\pi$
$42$	−6336.00	−0.554232
$43$	13316.0	1.09825	0.549127	−	0.835739i	$-0.314960\pi$
$43$	13316.0	1.09825	0.549127	+	0.835739i	$0.314960\pi$
$44$	−960.000	−0.0747549
$45$	−5346.00	−0.393548
$46$	2400.00	0.167231
$47$	−19680.0	−1.29951	−0.649756	−	0.760143i	$-0.725129\pi$
$47$	−19680.0	−1.29951	−0.649756	+	0.760143i	$0.725129\pi$
$48$	−2304.00	−0.144338
$49$	14169.0	0.843042
$50$	4924.00	0.278544
$51$	3726.00	0.200594
$52$	−10528.0	−0.539930
$53$	−31266.0	−1.52891	−0.764456	−	0.644676i	$-0.776992\pi$
$53$	−31266.0	−1.52891	−0.764456	+	0.644676i	$0.776992\pi$
$54$	−2916.00	−0.136083
$55$	3960.00	0.176518
$56$	11264.0	0.479979
$57$	−8604.00	−0.350763
$58$	22296.0	0.870276
$59$	26340.0	0.985112	0.492556	−	0.870281i	$-0.336063\pi$
$59$	26340.0	0.985112	0.492556	+	0.870281i	$0.336063\pi$
$60$	9504.00	0.340823
$61$	−31090.0	−1.06978	−0.534892	−	0.844920i	$-0.679648\pi$
$61$	−31090.0	−1.06978	−0.534892	+	0.844920i	$0.679648\pi$
$62$	−14368.0	−0.474698
$63$	14256.0	0.452529
$64$	4096.00	0.125000
$65$	43428.0	1.27493
$66$	2160.00	0.0610371
$67$	−16804.0	−0.457326	−0.228663	−	0.973506i	$-0.573435\pi$
$67$	−16804.0	−0.457326	−0.228663	+	0.973506i	$0.573435\pi$
$68$	−6624.00	−0.173719
$69$	−5400.00	−0.136544
$70$	−46464.0	−1.13337
$71$	6120.00	0.144081	0.0720403	−	0.997402i	$-0.477049\pi$
$71$	6120.00	0.144081	0.0720403	+	0.997402i	$0.477049\pi$
$72$	5184.00	0.117851
$73$	−25558.0	−0.561332	−0.280666	−	0.959806i	$-0.590555\pi$
$73$	−25558.0	−0.561332	−0.280666	+	0.959806i	$0.590555\pi$
$74$	−33832.0	−0.718205
$75$	−11079.0	−0.227430
$76$	15296.0	0.303769
$77$	−10560.0	−0.202972
$78$	23688.0	0.440851
$79$	74408.0	1.34138	0.670690	−	0.741738i	$-0.265998\pi$
$79$	74408.0	1.34138	0.670690	+	0.741738i	$0.265998\pi$
$80$	−16896.0	−0.295161
$81$	6561.00	0.111111
$82$	76776.0	1.26093
$83$	−6468.00	−0.103056	−0.0515282	−	0.998672i	$-0.516409\pi$
$83$	−6468.00	−0.103056	−0.0515282	+	0.998672i	$0.516409\pi$
$84$	−25344.0	−0.391902
$85$	27324.0	0.410201
$86$	53264.0	0.776583
$87$	−50166.0	−0.710577
$88$	−3840.00	−0.0528597
$89$	−32742.0	−0.438157	−0.219079	−	0.975707i	$-0.570305\pi$
$89$	−32742.0	−0.438157	−0.219079	+	0.975707i	$0.570305\pi$
$90$	−21384.0	−0.278280
$91$	−115808.	−1.46600
$92$	9600.00	0.118250
$93$	32328.0	0.387589
$94$	−78720.0	−0.918894
$95$	−63096.0	−0.717287
$96$	−9216.00	−0.102062
$97$	166082.	1.79223	0.896114	−	0.443824i	$-0.146378\pi$
$97$	166082.	1.79223	0.896114	+	0.443824i	$0.146378\pi$
$98$	56676.0	0.596120
$99$	−4860.00	−0.0498366
$100$	19696.0	0.196960
$101$	−22002.0	−0.214614	−0.107307	−	0.994226i	$-0.534223\pi$
$101$	−22002.0	−0.214614	−0.107307	+	0.994226i	$0.534223\pi$
$102$	14904.0	0.141841
$103$	−79264.0	−0.736178	−0.368089	−	0.929791i	$-0.619988\pi$
$103$	−79264.0	−0.736178	−0.368089	+	0.929791i	$0.619988\pi$
$104$	−42112.0	−0.381788
$105$	104544.	0.925392
$106$	−125064.	−1.08110
$107$	227988.	1.92510	0.962548	−	0.271110i	$-0.0873908\pi$
$107$	227988.	1.92510	0.962548	+	0.271110i	$0.0873908\pi$
$108$	−11664.0	−0.0962250
$109$	−8530.00	−0.0687674	−0.0343837	−	0.999409i	$-0.510947\pi$
$109$	−8530.00	−0.0687674	−0.0343837	+	0.999409i	$0.510947\pi$
$110$	15840.0	0.124817
$111$	76122.0	0.586412
$112$	45056.0	0.339397
$113$	−195438.	−1.43984	−0.719918	−	0.694059i	$-0.755821\pi$
$113$	−195438.	−1.43984	−0.719918	+	0.694059i	$0.755821\pi$
$114$	−34416.0	−0.248027
$115$	−39600.0	−0.279223
$116$	89184.0	0.615378
$117$	−53298.0	−0.359953
$118$	105360.	0.696580
$119$	−72864.0	−0.471678
$120$	38016.0	0.240998
$121$	−157451.	−0.977647
$122$	−124360.	−0.756452
$123$	−172746.	−1.02954
$124$	−57472.0	−0.335662
$125$	125004.	0.715565
$126$	57024.0	0.319986
$127$	173000.	0.951780	0.475890	−	0.879505i	$-0.342126\pi$
$127$	173000.	0.951780	0.475890	+	0.879505i	$0.342126\pi$
$128$	16384.0	0.0883883
$129$	−119844.	−0.634077
$130$	173712.	0.901512
$131$	151260.	0.770098	0.385049	−	0.922896i	$-0.374185\pi$
$131$	151260.	0.770098	0.385049	+	0.922896i	$0.374185\pi$
$132$	8640.00	0.0431597
$133$	168256.	0.824786
$134$	−67216.0	−0.323378
$135$	48114.0	0.227215
$136$	−26496.0	−0.122838
$137$	−128454.	−0.584718	−0.292359	−	0.956309i	$-0.594440\pi$
$137$	−128454.	−0.584718	−0.292359	+	0.956309i	$0.594440\pi$
$138$	−21600.0	−0.0965508
$139$	154196.	0.676918	0.338459	−	0.940981i	$-0.390094\pi$
$139$	154196.	0.676918	0.338459	+	0.940981i	$0.390094\pi$
$140$	−185856.	−0.801413
$141$	177120.	0.750274
$142$	24480.0	0.101880
$143$	39480.0	0.161450
$144$	20736.0	0.0833333
$145$	−367884.	−1.45308
$146$	−102232.	−0.396922
$147$	−127521.	−0.486730
$148$	−135328.	−0.507848
$149$	29454.0	0.108687	0.0543436	−	0.998522i	$-0.482693\pi$
$149$	29454.0	0.108687	0.0543436	+	0.998522i	$0.482693\pi$
$150$	−44316.0	−0.160817
$151$	−203872.	−0.727638	−0.363819	−	0.931470i	$-0.618527\pi$
$151$	−203872.	−0.727638	−0.363819	+	0.931470i	$0.618527\pi$
$152$	61184.0	0.214797
$153$	−33534.0	−0.115813
$154$	−42240.0	−0.143523
$155$	237072.	0.792594
$156$	94752.0	0.311729
$157$	136142.	0.440801	0.220401	−	0.975409i	$-0.429263\pi$
$157$	136142.	0.440801	0.220401	+	0.975409i	$0.429263\pi$
$158$	297632.	0.948499
$159$	281394.	0.882718
$160$	−67584.0	−0.208710
$161$	105600.	0.321070
$162$	26244.0	0.0785674
$163$	−171124.	−0.504478	−0.252239	−	0.967665i	$-0.581167\pi$
$163$	−171124.	−0.504478	−0.252239	+	0.967665i	$0.581167\pi$
$164$	307104.	0.891612
$165$	−35640.0	−0.101913
$166$	−25872.0	−0.0728718
$167$	−676200.	−1.87622	−0.938110	−	0.346336i	$-0.887426\pi$
$167$	−676200.	−1.87622	−0.938110	+	0.346336i	$0.887426\pi$
$168$	−101376.	−0.277116
$169$	61671.0	0.166098
$170$	109296.	0.290056
$171$	77436.0	0.202513
$172$	213056.	0.549127
$173$	133158.	0.338261	0.169131	−	0.985594i	$-0.445904\pi$
$173$	133158.	0.338261	0.169131	+	0.985594i	$0.445904\pi$
$174$	−200664.	−0.502454
$175$	216656.	0.534781
$176$	−15360.0	−0.0373774
$177$	−237060.	−0.568755
$178$	−130968.	−0.309824
$179$	−693396.	−1.61752	−0.808758	−	0.588141i	$-0.799860\pi$
$179$	−693396.	−1.61752	−0.808758	+	0.588141i	$0.799860\pi$
$180$	−85536.0	−0.196774
$181$	377174.	0.855747	0.427873	−	0.903839i	$-0.359263\pi$
$181$	377174.	0.855747	0.427873	+	0.903839i	$0.359263\pi$
$182$	−463232.	−1.03662
$183$	279810.	0.617640
$184$	38400.0	0.0836155
$185$	558228.	1.19917
$186$	129312.	0.274067
$187$	24840.0	0.0519455
$188$	−314880.	−0.649756
$189$	−128304.	−0.261268
$190$	−252384.	−0.507198
$191$	−265344.	−0.526291	−0.263145	−	0.964756i	$-0.584760\pi$
$191$	−265344.	−0.526291	−0.263145	+	0.964756i	$0.584760\pi$
$192$	−36864.0	−0.0721688
$193$	295298.	0.570647	0.285323	−	0.958431i	$-0.407899\pi$
$193$	295298.	0.570647	0.285323	+	0.958431i	$0.407899\pi$
$194$	664328.	1.26730
$195$	−390852.	−0.736081
$196$	226704.	0.421521
$197$	201294.	0.369543	0.184772	−	0.982781i	$-0.440845\pi$
$197$	201294.	0.369543	0.184772	+	0.982781i	$0.440845\pi$
$198$	−19440.0	−0.0352398
$199$	652448.	1.16792	0.583960	−	0.811782i	$-0.301502\pi$
$199$	652448.	1.16792	0.583960	+	0.811782i	$0.301502\pi$
$200$	78784.0	0.139272
$201$	151236.	0.264037
$202$	−88008.0	−0.151755
$203$	981024.	1.67086
$204$	59616.0	0.100297
$205$	−1.26680e6	−2.10535
$206$	−317056.	−0.520557
$207$	48600.0	0.0788334
$208$	−168448.	−0.269965
$209$	−57360.0	−0.0908330
$210$	418176.	0.654351
$211$	−1.14706e6	−1.77370	−0.886850	−	0.462058i	$-0.847111\pi$
$211$	−1.14706e6	−1.77370	−0.886850	+	0.462058i	$0.847111\pi$
$212$	−500256.	−0.764456
$213$	−55080.0	−0.0831850
$214$	911952.	1.36125
$215$	−878856.	−1.29665
$216$	−46656.0	−0.0680414
$217$	−632192.	−0.911380
$218$	−34120.0	−0.0486259
$219$	230022.	0.324085
$220$	63360.0	0.0882589
$221$	272412.	0.375185
$222$	304488.	0.414656
$223$	701960.	0.945258	0.472629	−	0.881262i	$-0.343305\pi$
$223$	701960.	0.945258	0.472629	+	0.881262i	$0.343305\pi$
$224$	180224.	0.239990
$225$	99711.0	0.131307
$226$	−781752.	−1.01812
$227$	1.23611e6	1.59218	0.796089	−	0.605179i	$-0.206899\pi$
$227$	1.23611e6	1.59218	0.796089	+	0.605179i	$0.206899\pi$
$228$	−137664.	−0.175381
$229$	105830.	0.133358	0.0666792	−	0.997774i	$-0.478760\pi$
$229$	105830.	0.133358	0.0666792	+	0.997774i	$0.478760\pi$
$230$	−158400.	−0.197440
$231$	95040.0	0.117186
$232$	356736.	0.435138
$233$	−438678.	−0.529366	−0.264683	−	0.964335i	$-0.585267\pi$
$233$	−438678.	−0.529366	−0.264683	+	0.964335i	$0.585267\pi$
$234$	−213192.	−0.254525
$235$	1.29888e6	1.53426
$236$	421440.	0.492556
$237$	−669672.	−0.774446
$238$	−291456.	−0.333527
$239$	28464.0	0.0322330	0.0161165	−	0.999870i	$-0.494870\pi$
$239$	28464.0	0.0322330	0.0161165	+	0.999870i	$0.494870\pi$
$240$	152064.	0.170411
$241$	892562.	0.989910	0.494955	−	0.868919i	$-0.335185\pi$
$241$	892562.	0.989910	0.494955	+	0.868919i	$0.335185\pi$
$242$	−629804.	−0.691301
$243$	−59049.0	−0.0641500
$244$	−497440.	−0.534892
$245$	−935154.	−0.995332
$246$	−690984.	−0.727998
$247$	−629048.	−0.656057
$248$	−229888.	−0.237349
$249$	58212.0	0.0594996
$250$	500016.	0.505981
$251$	−110124.	−0.110331	−0.0551655	−	0.998477i	$-0.517569\pi$
$251$	−110124.	−0.110331	−0.0551655	+	0.998477i	$0.517569\pi$
$252$	228096.	0.226264
$253$	−36000.0	−0.0353591
$254$	692000.	0.673010
$255$	−245916.	−0.236830
$256$	65536.0	0.0625000
$257$	140802.	0.132977	0.0664884	−	0.997787i	$-0.478820\pi$
$257$	140802.	0.132977	0.0664884	+	0.997787i	$0.478820\pi$
$258$	−479376.	−0.448360
$259$	−1.48861e6	−1.37889
$260$	694848.	0.637465
$261$	451494.	0.410252
$262$	605040.	0.544541
$263$	−938760.	−0.836884	−0.418442	−	0.908244i	$-0.637424\pi$
$263$	−938760.	−0.836884	−0.418442	+	0.908244i	$0.637424\pi$
$264$	34560.0	0.0305186
$265$	2.06356e6	1.80510
$266$	673024.	0.583212
$267$	294678.	0.252970
$268$	−268864.	−0.228663
$269$	−1.11451e6	−0.939078	−0.469539	−	0.882912i	$-0.655580\pi$
$269$	−1.11451e6	−0.939078	−0.469539	+	0.882912i	$0.655580\pi$
$270$	192456.	0.160665
$271$	567704.	0.469568	0.234784	−	0.972048i	$-0.424562\pi$
$271$	567704.	0.469568	0.234784	+	0.972048i	$0.424562\pi$
$272$	−105984.	−0.0868596
$273$	1.04227e6	0.846398
$274$	−513816.	−0.413458
$275$	−73860.0	−0.0588949
$276$	−86400.0	−0.0682718
$277$	−1.21326e6	−0.950066	−0.475033	−	0.879968i	$-0.657564\pi$
$277$	−1.21326e6	−0.950066	−0.475033	+	0.879968i	$0.657564\pi$
$278$	616784.	0.478653
$279$	−290952.	−0.223775
$280$	−743424.	−0.566685
$281$	687738.	0.519586	0.259793	−	0.965664i	$-0.416346\pi$
$281$	687738.	0.519586	0.259793	+	0.965664i	$0.416346\pi$
$282$	708480.	0.530524
$283$	−830908.	−0.616718	−0.308359	−	0.951270i	$-0.599780\pi$
$283$	−830908.	−0.616718	−0.308359	+	0.951270i	$0.599780\pi$
$284$	97920.0	0.0720403
$285$	567864.	0.414126
$286$	157920.	0.114162
$287$	3.37814e6	2.42088
$288$	82944.0	0.0589256
$289$	−1.24846e6	−0.879286
$290$	−1.47154e6	−1.02749
$291$	−1.49474e6	−1.03474
$292$	−408928.	−0.280666
$293$	−1.31263e6	−0.893248	−0.446624	−	0.894722i	$-0.647374\pi$
$293$	−1.31263e6	−0.893248	−0.446624	+	0.894722i	$0.647374\pi$
$294$	−510084.	−0.344170
$295$	−1.73844e6	−1.16307
$296$	−541312.	−0.359102
$297$	43740.0	0.0287732
$298$	117816.	0.0768535
$299$	−394800.	−0.255387
$300$	−177264.	−0.113715
$301$	2.34362e6	1.49097
$302$	−815488.	−0.514518
$303$	198018.	0.123908
$304$	244736.	0.151885
$305$	2.05194e6	1.26303
$306$	−134136.	−0.0818921
$307$	1.69022e6	1.02352	0.511761	−	0.859128i	$-0.328993\pi$
$307$	1.69022e6	1.02352	0.511761	+	0.859128i	$0.328993\pi$
$308$	−168960.	−0.101486
$309$	713376.	0.425033
$310$	948288.	0.560449
$311$	−1.50204e6	−0.880604	−0.440302	−	0.897850i	$-0.645129\pi$
$311$	−1.50204e6	−0.880604	−0.440302	+	0.897850i	$0.645129\pi$
$312$	379008.	0.220426
$313$	810842.	0.467816	0.233908	−	0.972259i	$-0.424848\pi$
$313$	810842.	0.467816	0.233908	+	0.972259i	$0.424848\pi$
$314$	544568.	0.311694
$315$	−940896.	−0.534275
$316$	1.19053e6	0.670690
$317$	903558.	0.505019	0.252510	−	0.967594i	$-0.418744\pi$
$317$	903558.	0.505019	0.252510	+	0.967594i	$0.418744\pi$
$318$	1.12558e6	0.624176
$319$	−334440.	−0.184010
$320$	−270336.	−0.147580
$321$	−2.05189e6	−1.11146
$322$	422400.	0.227031
$323$	−395784.	−0.211082
$324$	104976.	0.0555556
$325$	−809998.	−0.425379
$326$	−684496.	−0.356720
$327$	76770.0	0.0397029
$328$	1.22842e6	0.630465
$329$	−3.46368e6	−1.76420
$330$	−142560.	−0.0720631
$331$	1.12197e6	0.562875	0.281438	−	0.959580i	$-0.409189\pi$
$331$	1.12197e6	0.562875	0.281438	+	0.959580i	$0.409189\pi$
$332$	−103488.	−0.0515282
$333$	−685098.	−0.338565
$334$	−2.70480e6	−1.32669
$335$	1.10906e6	0.539939
$336$	−405504.	−0.195951
$337$	−2.75217e6	−1.32008	−0.660041	−	0.751229i	$-0.729461\pi$
$337$	−2.75217e6	−1.32008	−0.660041	+	0.751229i	$0.729461\pi$
$338$	246684.	0.117449
$339$	1.75894e6	0.831289
$340$	437184.	0.205101
$341$	215520.	0.100369
$342$	309744.	0.143198
$343$	−464288.	−0.213085
$344$	852224.	0.388291
$345$	356400.	0.161209
$346$	532632.	0.239187
$347$	1.91749e6	0.854889	0.427445	−	0.904042i	$-0.359414\pi$
$347$	1.91749e6	0.854889	0.427445	+	0.904042i	$0.359414\pi$
$348$	−802656.	−0.355289
$349$	1.83659e6	0.807140	0.403570	−	0.914949i	$-0.367769\pi$
$349$	1.83659e6	0.807140	0.403570	+	0.914949i	$0.367769\pi$
$350$	866624.	0.378147
$351$	479682.	0.207819
$352$	−61440.0	−0.0264298
$353$	−622014.	−0.265683	−0.132841	−	0.991137i	$-0.542410\pi$
$353$	−622014.	−0.265683	−0.132841	+	0.991137i	$0.542410\pi$
$354$	−948240.	−0.402170
$355$	−403920.	−0.170108
$356$	−523872.	−0.219079
$357$	655776.	0.272323
$358$	−2.77358e6	−1.14376
$359$	3.74062e6	1.53182	0.765909	−	0.642949i	$-0.222289\pi$
$359$	3.74062e6	1.53182	0.765909	+	0.642949i	$0.222289\pi$
$360$	−342144.	−0.139140
$361$	−1.56216e6	−0.630897
$362$	1.50870e6	0.605104
$363$	1.41706e6	0.564445
$364$	−1.85293e6	−0.733002
$365$	1.68683e6	0.662733
$366$	1.11924e6	0.436738
$367$	16232.0	0.00629081	0.00314541	−	0.999995i	$-0.498999\pi$
$367$	16232.0	0.00629081	0.00314541	+	0.999995i	$0.498999\pi$
$368$	153600.	0.0591251
$369$	1.55471e6	0.594408
$370$	2.23291e6	0.847944
$371$	−5.50282e6	−2.07563
$372$	517248.	0.193795
$373$	293606.	0.109268	0.0546340	−	0.998506i	$-0.482601\pi$
$373$	293606.	0.109268	0.0546340	+	0.998506i	$0.482601\pi$
$374$	99360.0	0.0367310
$375$	−1.12504e6	−0.413131
$376$	−1.25952e6	−0.459447
$377$	−3.66769e6	−1.32904
$378$	−513216.	−0.184744
$379$	3.18012e6	1.13722	0.568611	−	0.822607i	$-0.307481\pi$
$379$	3.18012e6	1.13722	0.568611	+	0.822607i	$0.307481\pi$
$380$	−1.00954e6	−0.358643
$381$	−1.55700e6	−0.549511
$382$	−1.06138e6	−0.372144
$383$	−2.97984e6	−1.03800	−0.518998	−	0.854775i	$-0.673695\pi$
$383$	−2.97984e6	−1.03800	−0.518998	+	0.854775i	$0.673695\pi$
$384$	−147456.	−0.0510310
$385$	696960.	0.239638
$386$	1.18119e6	0.403508
$387$	1.07860e6	0.366085
$388$	2.65731e6	0.896114
$389$	3.45977e6	1.15924	0.579620	−	0.814887i	$-0.303201\pi$
$389$	3.45977e6	1.15924	0.579620	+	0.814887i	$0.303201\pi$
$390$	−1.56341e6	−0.520488
$391$	−248400.	−0.0821693
$392$	906816.	0.298060
$393$	−1.36134e6	−0.444616
$394$	805176.	0.261307
$395$	−4.91093e6	−1.58369
$396$	−77760.0	−0.0249183
$397$	−3.90416e6	−1.24323	−0.621615	−	0.783323i	$-0.713523\pi$
$397$	−3.90416e6	−1.24323	−0.621615	+	0.783323i	$0.713523\pi$
$398$	2.60979e6	0.825844
$399$	−1.51430e6	−0.476191
$400$	315136.	0.0984800
$401$	5.44115e6	1.68978	0.844890	−	0.534940i	$-0.179666\pi$
$401$	5.44115e6	1.68978	0.844890	+	0.534940i	$0.179666\pi$
$402$	604944.	0.186702
$403$	2.36354e6	0.724936
$404$	−352032.	−0.107307
$405$	−433026.	−0.131183
$406$	3.92410e6	1.18148
$407$	507480.	0.151856
$408$	238464.	0.0709206
$409$	1.96995e6	0.582299	0.291150	−	0.956678i	$-0.405962\pi$
$409$	1.96995e6	0.582299	0.291150	+	0.956678i	$0.405962\pi$
$410$	−5.06722e6	−1.48871
$411$	1.15609e6	0.337587
$412$	−1.26822e6	−0.368089
$413$	4.63584e6	1.33738
$414$	194400.	0.0557437
$415$	426888.	0.121673
$416$	−673792.	−0.190894
$417$	−1.38776e6	−0.390819
$418$	−229440.	−0.0642286
$419$	139020.	0.0386850	0.0193425	−	0.999813i	$-0.493843\pi$
$419$	139020.	0.0386850	0.0193425	+	0.999813i	$0.493843\pi$
$420$	1.67270e6	0.462696
$421$	4.32743e6	1.18994	0.594970	−	0.803748i	$-0.297164\pi$
$421$	4.32743e6	1.18994	0.594970	+	0.803748i	$0.297164\pi$
$422$	−4.58824e6	−1.25419
$423$	−1.59408e6	−0.433171
$424$	−2.00102e6	−0.540552
$425$	−509634.	−0.136863
$426$	−220320.	−0.0588207
$427$	−5.47184e6	−1.45232
$428$	3.64781e6	0.962548
$429$	−355320.	−0.0932130
$430$	−3.51542e6	−0.916867
$431$	−2.79936e6	−0.725881	−0.362941	−	0.931812i	$-0.618227\pi$
$431$	−2.79936e6	−0.725881	−0.362941	+	0.931812i	$0.618227\pi$
$432$	−186624.	−0.0481125
$433$	−5.90241e6	−1.51290	−0.756449	−	0.654052i	$-0.773068\pi$
$433$	−5.90241e6	−1.51290	−0.756449	+	0.654052i	$0.773068\pi$
$434$	−2.52877e6	−0.644443
$435$	3.31096e6	0.838939
$436$	−136480.	−0.0343837
$437$	573600.	0.143683
$438$	920088.	0.229163
$439$	−446512.	−0.110579	−0.0552894	−	0.998470i	$-0.517608\pi$
$439$	−446512.	−0.110579	−0.0552894	+	0.998470i	$0.517608\pi$
$440$	253440.	0.0624085
$441$	1.14769e6	0.281014
$442$	1.08965e6	0.265296
$443$	3.49525e6	0.846193	0.423096	−	0.906085i	$-0.360943\pi$
$443$	3.49525e6	0.846193	0.423096	+	0.906085i	$0.360943\pi$
$444$	1.21795e6	0.293206
$445$	2.16097e6	0.517308
$446$	2.80784e6	0.668398
$447$	−265086.	−0.0627506
$448$	720896.	0.169698
$449$	−1.20613e6	−0.282343	−0.141171	−	0.989985i	$-0.545087\pi$
$449$	−1.20613e6	−0.282343	−0.141171	+	0.989985i	$0.545087\pi$
$450$	398844.	0.0928478
$451$	−1.15164e6	−0.266609
$452$	−3.12701e6	−0.719918
$453$	1.83485e6	0.420102
$454$	4.94443e6	1.12584
$455$	7.64333e6	1.73083
$456$	−550656.	−0.124013
$457$	233546.	0.0523097	0.0261548	−	0.999658i	$-0.491674\pi$
$457$	233546.	0.0523097	0.0261548	+	0.999658i	$0.491674\pi$
$458$	423320.	0.0942986
$459$	301806.	0.0668646
$460$	−633600.	−0.139611
$461$	−1.74489e6	−0.382398	−0.191199	−	0.981551i	$-0.561238\pi$
$461$	−1.74489e6	−0.382398	−0.191199	+	0.981551i	$0.561238\pi$
$462$	380160.	0.0828632
$463$	−2.91786e6	−0.632576	−0.316288	−	0.948663i	$-0.602437\pi$
$463$	−2.91786e6	−0.632576	−0.316288	+	0.948663i	$0.602437\pi$
$464$	1.42694e6	0.307689
$465$	−2.13365e6	−0.457605
$466$	−1.75471e6	−0.374318
$467$	−5.31076e6	−1.12684	−0.563422	−	0.826169i	$-0.690516\pi$
$467$	−5.31076e6	−1.12684	−0.563422	+	0.826169i	$0.690516\pi$
$468$	−852768.	−0.179977
$469$	−2.95750e6	−0.620859
$470$	5.19552e6	1.08489
$471$	−1.22528e6	−0.254497
$472$	1.68576e6	0.348290
$473$	−798960.	−0.164200
$474$	−2.67869e6	−0.547616
$475$	1.17684e6	0.239322
$476$	−1.16582e6	−0.235839
$477$	−2.53255e6	−0.509638
$478$	113856.	0.0227922
$479$	2.34466e6	0.466918	0.233459	−	0.972367i	$-0.424996\pi$
$479$	2.34466e6	0.466918	0.233459	+	0.972367i	$0.424996\pi$
$480$	608256.	0.120499
$481$	5.56536e6	1.09681
$482$	3.57025e6	0.699972
$483$	−950400.	−0.185370
$484$	−2.51922e6	−0.488823
$485$	−1.09614e7	−2.11598
$486$	−236196.	−0.0453609
$487$	9.81531e6	1.87535	0.937674	−	0.347517i	$-0.112975\pi$
$487$	9.81531e6	1.87535	0.937674	+	0.347517i	$0.112975\pi$
$488$	−1.98976e6	−0.378226
$489$	1.54012e6	0.291260
$490$	−3.74062e6	−0.703806
$491$	−5.94520e6	−1.11292	−0.556458	−	0.830876i	$-0.687840\pi$
$491$	−5.94520e6	−1.11292	−0.556458	+	0.830876i	$0.687840\pi$
$492$	−2.76394e6	−0.514772
$493$	−2.30764e6	−0.427612
$494$	−2.51619e6	−0.463902
$495$	320760.	0.0588393
$496$	−919552.	−0.167831
$497$	1.07712e6	0.195602
$498$	232848.	0.0420726
$499$	6.47832e6	1.16469	0.582346	−	0.812941i	$-0.302135\pi$
$499$	6.47832e6	1.16469	0.582346	+	0.812941i	$0.302135\pi$
$500$	2.00006e6	0.357782
$501$	6.08580e6	1.08324
$502$	−440496.	−0.0780158
$503$	4.71794e6	0.831444	0.415722	−	0.909492i	$-0.363529\pi$
$503$	4.71794e6	0.831444	0.415722	+	0.909492i	$0.363529\pi$
$504$	912384.	0.159993
$505$	1.45213e6	0.253383
$506$	−144000.	−0.0250027
$507$	−555039.	−0.0958967
$508$	2.76800e6	0.475890
$509$	−1.90771e6	−0.326375	−0.163188	−	0.986595i	$-0.552178\pi$
$509$	−1.90771e6	−0.326375	−0.163188	+	0.986595i	$0.552178\pi$
$510$	−983664.	−0.167464
$511$	−4.49821e6	−0.762057
$512$	262144.	0.0441942
$513$	−696924.	−0.116921
$514$	563208.	0.0940288
$515$	5.23142e6	0.869164
$516$	−1.91750e6	−0.317039
$517$	1.18080e6	0.194290
$518$	−5.95443e6	−0.975025
$519$	−1.19842e6	−0.195295
$520$	2.77939e6	0.450756
$521$	8.01974e6	1.29439	0.647196	−	0.762324i	$-0.275941\pi$
$521$	8.01974e6	1.29439	0.647196	+	0.762324i	$0.275941\pi$
$522$	1.80598e6	0.290092
$523$	1.91162e6	0.305596	0.152798	−	0.988257i	$-0.451172\pi$
$523$	1.91162e6	0.305596	0.152798	+	0.988257i	$0.451172\pi$
$524$	2.42016e6	0.385049
$525$	−1.94990e6	−0.308756
$526$	−3.75504e6	−0.591766
$527$	1.48709e6	0.233244
$528$	138240.	0.0215799
$529$	−6.07634e6	−0.944068
$530$	8.25422e6	1.27640
$531$	2.13354e6	0.328371
$532$	2.69210e6	0.412393
$533$	−1.26297e7	−1.92563
$534$	1.17871e6	0.178877
$535$	−1.50472e7	−2.27285
$536$	−1.07546e6	−0.161689
$537$	6.24056e6	0.933874
$538$	−4.45802e6	−0.664028
$539$	−850140.	−0.126043
$540$	769824.	0.113608
$541$	−1.19900e7	−1.76128	−0.880639	−	0.473788i	$-0.842886\pi$
$541$	−1.19900e7	−1.76128	−0.880639	+	0.473788i	$0.842886\pi$
$542$	2.27082e6	0.332035
$543$	−3.39457e6	−0.494066
$544$	−423936.	−0.0614190
$545$	562980.	0.0811898
$546$	4.16909e6	0.598494
$547$	4.45809e6	0.637061	0.318530	−	0.947913i	$-0.396811\pi$
$547$	4.45809e6	0.637061	0.318530	+	0.947913i	$0.396811\pi$
$548$	−2.05526e6	−0.292359
$549$	−2.51829e6	−0.356595
$550$	−295440.	−0.0416450
$551$	5.32874e6	0.747732
$552$	−345600.	−0.0482754
$553$	1.30958e7	1.82104
$554$	−4.85303e6	−0.671798
$555$	−5.02405e6	−0.692344
$556$	2.46714e6	0.338459
$557$	9.02612e6	1.23272	0.616358	−	0.787466i	$-0.288607\pi$
$557$	9.02612e6	1.23272	0.616358	+	0.787466i	$0.288607\pi$
$558$	−1.16381e6	−0.158233
$559$	−8.76193e6	−1.18596
$560$	−2.97370e6	−0.400707
$561$	−223560.	−0.0299907
$562$	2.75095e6	0.367403
$563$	6.84899e6	0.910658	0.455329	−	0.890323i	$-0.349522\pi$
$563$	6.84899e6	0.910658	0.455329	+	0.890323i	$0.349522\pi$
$564$	2.83392e6	0.375137
$565$	1.28989e7	1.69993
$566$	−3.32363e6	−0.436086
$567$	1.15474e6	0.150843
$568$	391680.	0.0509402
$569$	−5.46322e6	−0.707405	−0.353703	−	0.935358i	$-0.615077\pi$
$569$	−5.46322e6	−0.707405	−0.353703	+	0.935358i	$0.615077\pi$
$570$	2.27146e6	0.292831
$571$	−1.02324e7	−1.31337	−0.656684	−	0.754166i	$-0.728041\pi$
$571$	−1.02324e7	−1.31337	−0.656684	+	0.754166i	$0.728041\pi$
$572$	631680.	0.0807248
$573$	2.38810e6	0.303854
$574$	1.35126e7	1.71182
$575$	738600.	0.0931622
$576$	331776.	0.0416667
$577$	1.59437e7	1.99365	0.996825	−	0.0796186i	$-0.0253702\pi$
$577$	1.59437e7	1.99365	0.996825	+	0.0796186i	$0.0253702\pi$
$578$	−4.99384e6	−0.621749
$579$	−2.65768e6	−0.329463
$580$	−5.88614e6	−0.726542
$581$	−1.13837e6	−0.139908
$582$	−5.97895e6	−0.731674
$583$	1.87596e6	0.228587
$584$	−1.63571e6	−0.198461
$585$	3.51767e6	0.424977
$586$	−5.25050e6	−0.631622
$587$	−9.47713e6	−1.13522	−0.567612	−	0.823296i	$-0.692133\pi$
$587$	−9.47713e6	−1.13522	−0.567612	+	0.823296i	$0.692133\pi$
$588$	−2.04034e6	−0.243365
$589$	−3.43395e6	−0.407855
$590$	−6.95376e6	−0.822412
$591$	−1.81165e6	−0.213356
$592$	−2.16525e6	−0.253924
$593$	2.45349e6	0.286515	0.143258	−	0.989685i	$-0.454242\pi$
$593$	2.45349e6	0.286515	0.143258	+	0.989685i	$0.454242\pi$
$594$	174960.	0.0203457
$595$	4.80902e6	0.556884
$596$	471264.	0.0543436
$597$	−5.87203e6	−0.674299
$598$	−1.57920e6	−0.180586
$599$	−9.29978e6	−1.05902	−0.529512	−	0.848302i	$-0.677625\pi$
$599$	−9.29978e6	−1.05902	−0.529512	+	0.848302i	$0.677625\pi$
$600$	−709056.	−0.0804086
$601$	−1.14617e7	−1.29438	−0.647192	−	0.762327i	$-0.724057\pi$
$601$	−1.14617e7	−1.29438	−0.647192	+	0.762327i	$0.724057\pi$
$602$	9.37446e6	1.05428
$603$	−1.36112e6	−0.152442
$604$	−3.26195e6	−0.363819
$605$	1.03918e7	1.15425
$606$	792072.	0.0876159
$607$	1.12784e7	1.24244	0.621219	−	0.783637i	$-0.286638\pi$
$607$	1.12784e7	1.24244	0.621219	+	0.783637i	$0.286638\pi$
$608$	978944.	0.107399
$609$	−8.82922e6	−0.964670
$610$	8.20776e6	0.893100
$611$	1.29494e7	1.40329
$612$	−536544.	−0.0579064
$613$	93782.0	0.0100802	0.00504009	−	0.999987i	$-0.498396\pi$
$613$	93782.0	0.0100802	0.00504009	+	0.999987i	$0.498396\pi$
$614$	6.76088e6	0.723740
$615$	1.14012e7	1.21553
$616$	−675840.	−0.0717616
$617$	−1.49642e7	−1.58248	−0.791242	−	0.611504i	$-0.790565\pi$
$617$	−1.49642e7	−1.58248	−0.791242	+	0.611504i	$0.790565\pi$
$618$	2.85350e6	0.300543
$619$	−5.06888e6	−0.531723	−0.265861	−	0.964011i	$-0.585656\pi$
$619$	−5.06888e6	−0.531723	−0.265861	+	0.964011i	$0.585656\pi$
$620$	3.79315e6	0.396297
$621$	−437400.	−0.0455145
$622$	−6.00816e6	−0.622681
$623$	−5.76259e6	−0.594837
$624$	1.51603e6	0.155864
$625$	−1.20971e7	−1.23875
$626$	3.24337e6	0.330796
$627$	516240.	0.0524424
$628$	2.17827e6	0.220401
$629$	3.50161e6	0.352892
$630$	−3.76358e6	−0.377790
$631$	1.55919e7	1.55892	0.779462	−	0.626450i	$-0.215493\pi$
$631$	1.55919e7	1.55892	0.779462	+	0.626450i	$0.215493\pi$
$632$	4.76211e6	0.474250
$633$	1.03235e7	1.02405
$634$	3.61423e6	0.357102
$635$	−1.14180e7	−1.12371
$636$	4.50230e6	0.441359
$637$	−9.32320e6	−0.910367
$638$	−1.33776e6	−0.130115
$639$	495720.	0.0480269
$640$	−1.08134e6	−0.104355
$641$	1.09701e7	1.05455	0.527274	−	0.849695i	$-0.323214\pi$
$641$	1.09701e7	1.05455	0.527274	+	0.849695i	$0.323214\pi$
$642$	−8.20757e6	−0.785917
$643$	−2.83704e6	−0.270607	−0.135303	−	0.990804i	$-0.543201\pi$
$643$	−2.83704e6	−0.270607	−0.135303	+	0.990804i	$0.543201\pi$
$644$	1.68960e6	0.160535
$645$	7.90970e6	0.748619
$646$	−1.58314e6	−0.149258
$647$	−6.05686e6	−0.568835	−0.284418	−	0.958700i	$-0.591800\pi$
$647$	−6.05686e6	−0.568835	−0.284418	+	0.958700i	$0.591800\pi$
$648$	419904.	0.0392837
$649$	−1.58040e6	−0.147284
$650$	−3.23999e6	−0.300788
$651$	5.68973e6	0.526186
$652$	−2.73798e6	−0.252239
$653$	−1.08892e6	−0.0999341	−0.0499671	−	0.998751i	$-0.515912\pi$
$653$	−1.08892e6	−0.0999341	−0.0499671	+	0.998751i	$0.515912\pi$
$654$	307080.	0.0280742
$655$	−9.98316e6	−0.909211
$656$	4.91366e6	0.445806
$657$	−2.07020e6	−0.187111
$658$	−1.38547e7	−1.24748
$659$	7.41803e6	0.665388	0.332694	−	0.943035i	$-0.392042\pi$
$659$	7.41803e6	0.665388	0.332694	+	0.943035i	$0.392042\pi$
$660$	−570240.	−0.0509563
$661$	767654.	0.0683379	0.0341690	−	0.999416i	$-0.489122\pi$
$661$	767654.	0.0683379	0.0341690	+	0.999416i	$0.489122\pi$
$662$	4.48789e6	0.398013
$663$	−2.45171e6	−0.216613
$664$	−413952.	−0.0364359
$665$	−1.11049e7	−0.973779
$666$	−2.74039e6	−0.239402
$667$	3.34440e6	0.291074
$668$	−1.08192e7	−0.938110
$669$	−6.31764e6	−0.545745
$670$	4.43626e6	0.381794
$671$	1.86540e6	0.159943
$672$	−1.62202e6	−0.138558
$673$	1.42263e6	0.121075	0.0605373	−	0.998166i	$-0.480719\pi$
$673$	1.42263e6	0.121075	0.0605373	+	0.998166i	$0.480719\pi$
$674$	−1.10087e7	−0.933439
$675$	−897399.	−0.0758099
$676$	986736.	0.0830490
$677$	−6.16231e6	−0.516739	−0.258370	−	0.966046i	$-0.583185\pi$
$677$	−6.16231e6	−0.516739	−0.258370	+	0.966046i	$0.583185\pi$
$678$	7.03577e6	0.587810
$679$	2.92304e7	2.43310
$680$	1.74874e6	0.145028
$681$	−1.11250e7	−0.919245
$682$	862080.	0.0709719
$683$	1.50621e7	1.23548	0.617739	−	0.786383i	$-0.288049\pi$
$683$	1.50621e7	1.23548	0.617739	+	0.786383i	$0.288049\pi$
$684$	1.23898e6	0.101256
$685$	8.47796e6	0.690343
$686$	−1.85715e6	−0.150674
$687$	−952470.	−0.0769945
$688$	3.40890e6	0.274563
$689$	2.05730e7	1.65101
$690$	1.42560e6	0.113992
$691$	−5.87636e6	−0.468180	−0.234090	−	0.972215i	$-0.575211\pi$
$691$	−5.87636e6	−0.468180	−0.234090	+	0.972215i	$0.575211\pi$
$692$	2.13053e6	0.169131
$693$	−855360.	−0.0676575
$694$	7.66997e6	0.604498
$695$	−1.01769e7	−0.799199
$696$	−3.21062e6	−0.251227
$697$	−7.94632e6	−0.619561
$698$	7.34636e6	0.570734
$699$	3.94810e6	0.305630
$700$	3.46650e6	0.267390
$701$	3.60077e6	0.276758	0.138379	−	0.990379i	$-0.455811\pi$
$701$	3.60077e6	0.276758	0.138379	+	0.990379i	$0.455811\pi$
$702$	1.91873e6	0.146950
$703$	−8.08585e6	−0.617074
$704$	−245760.	−0.0186887
$705$	−1.16899e7	−0.885806
$706$	−2.48806e6	−0.187866
$707$	−3.87235e6	−0.291358
$708$	−3.79296e6	−0.284377
$709$	9.22516e6	0.689221	0.344610	−	0.938746i	$-0.388011\pi$
$709$	9.22516e6	0.689221	0.344610	+	0.938746i	$0.388011\pi$
$710$	−1.61568e6	−0.120284
$711$	6.02705e6	0.447127
$712$	−2.09549e6	−0.154912
$713$	−2.15520e6	−0.158768
$714$	2.62310e6	0.192562
$715$	−2.60568e6	−0.190615
$716$	−1.10943e7	−0.808758
$717$	−256176.	−0.0186098
$718$	1.49625e7	1.08316
$719$	−2.63923e7	−1.90395	−0.951975	−	0.306177i	$-0.900950\pi$
$719$	−2.63923e7	−1.90395	−0.951975	+	0.306177i	$0.900950\pi$
$720$	−1.36858e6	−0.0983870
$721$	−1.39505e7	−0.999426
$722$	−6.24865e6	−0.446111
$723$	−8.03306e6	−0.571525
$724$	6.03478e6	0.427873
$725$	6.86159e6	0.484819
$726$	5.66824e6	0.399123
$727$	−9.79485e6	−0.687324	−0.343662	−	0.939093i	$-0.611667\pi$
$727$	−9.79485e6	−0.687324	−0.343662	+	0.939093i	$0.611667\pi$
$728$	−7.41171e6	−0.518311
$729$	531441.	0.0370370
$730$	6.74731e6	0.468623
$731$	−5.51282e6	−0.381576
$732$	4.47696e6	0.308820
$733$	4.07584e6	0.280193	0.140096	−	0.990138i	$-0.455259\pi$
$733$	4.07584e6	0.280193	0.140096	+	0.990138i	$0.455259\pi$
$734$	64928.0	0.00444828
$735$	8.41639e6	0.574655
$736$	614400.	0.0418077
$737$	1.00824e6	0.0683747
$738$	6.21886e6	0.420310
$739$	−1.65709e7	−1.11618	−0.558089	−	0.829781i	$-0.688465\pi$
$739$	−1.65709e7	−1.11618	−0.558089	+	0.829781i	$0.688465\pi$
$740$	8.93165e6	0.599587
$741$	5.66143e6	0.378775
$742$	−2.20113e7	−1.46769
$743$	1.44141e7	0.957892	0.478946	−	0.877844i	$-0.341019\pi$
$743$	1.44141e7	0.957892	0.478946	+	0.877844i	$0.341019\pi$
$744$	2.06899e6	0.137033
$745$	−1.94396e6	−0.128321
$746$	1.17442e6	0.0772641
$747$	−523908.	−0.0343521
$748$	397440.	0.0259727
$749$	4.01259e7	2.61349
$750$	−4.50014e6	−0.292128
$751$	1.67944e7	1.08659	0.543295	−	0.839542i	$-0.317177\pi$
$751$	1.67944e7	1.08659	0.543295	+	0.839542i	$0.317177\pi$
$752$	−5.03808e6	−0.324878
$753$	991116.	0.0636997
$754$	−1.46708e7	−0.939776
$755$	1.34556e7	0.859081
$756$	−2.05286e6	−0.130634
$757$	1.32943e7	0.843188	0.421594	−	0.906785i	$-0.361471\pi$
$757$	1.32943e7	0.843188	0.421594	+	0.906785i	$0.361471\pi$
$758$	1.27205e7	0.804137
$759$	324000.	0.0204146
$760$	−4.03814e6	−0.253599
$761$	−2.14786e6	−0.134445	−0.0672225	−	0.997738i	$-0.521414\pi$
$761$	−2.14786e6	−0.134445	−0.0672225	+	0.997738i	$0.521414\pi$
$762$	−6.22800e6	−0.388563
$763$	−1.50128e6	−0.0933577
$764$	−4.24550e6	−0.263145
$765$	2.21324e6	0.136734
$766$	−1.19194e7	−0.733975
$767$	−1.73317e7	−1.06378
$768$	−589824.	−0.0360844
$769$	−1.31059e7	−0.799193	−0.399596	−	0.916691i	$-0.630850\pi$
$769$	−1.31059e7	−0.799193	−0.399596	+	0.916691i	$0.630850\pi$
$770$	2.78784e6	0.169450
$771$	−1.26722e6	−0.0767742
$772$	4.72477e6	0.285323
$773$	−2.37154e7	−1.42752	−0.713759	−	0.700392i	$-0.753009\pi$
$773$	−2.37154e7	−1.42752	−0.713759	+	0.700392i	$0.753009\pi$
$774$	4.31438e6	0.258861
$775$	−4.42175e6	−0.264448
$776$	1.06292e7	0.633648
$777$	1.33975e7	0.796105
$778$	1.38391e7	0.819707
$779$	1.83495e7	1.08338
$780$	−6.25363e6	−0.368041
$781$	−367200.	−0.0215415
$782$	−993600.	−0.0581025
$783$	−4.06345e6	−0.236859
$784$	3.62726e6	0.210760
$785$	−8.98537e6	−0.520430
$786$	−5.44536e6	−0.314391
$787$	−8.40048e6	−0.483468	−0.241734	−	0.970343i	$-0.577716\pi$
$787$	−8.40048e6	−0.483468	−0.241734	+	0.970343i	$0.577716\pi$
$788$	3.22070e6	0.184772
$789$	8.44884e6	0.483175
$790$	−1.96437e7	−1.11984
$791$	−3.43971e7	−1.95470
$792$	−311040.	−0.0176199
$793$	2.04572e7	1.15522
$794$	−1.56166e7	−0.879097
$795$	−1.85720e7	−1.04218
$796$	1.04392e7	0.583960
$797$	5.41023e6	0.301696	0.150848	−	0.988557i	$-0.451800\pi$
$797$	5.41023e6	0.301696	0.150848	+	0.988557i	$0.451800\pi$
$798$	−6.05722e6	−0.336718
$799$	8.14752e6	0.451501
$800$	1.26054e6	0.0696359
$801$	−2.65210e6	−0.146052
$802$	2.17646e7	1.19485
$803$	1.53348e6	0.0839246
$804$	2.41978e6	0.132019
$805$	−6.96960e6	−0.379069
$806$	9.45414e6	0.512607
$807$	1.00306e7	0.542177
$808$	−1.40813e6	−0.0758776
$809$	−2.60777e7	−1.40087	−0.700436	−	0.713715i	$-0.747011\pi$
$809$	−2.60777e7	−1.40087	−0.700436	+	0.713715i	$0.747011\pi$
$810$	−1.73210e6	−0.0927601
$811$	1.90021e7	1.01449	0.507247	−	0.861800i	$-0.330663\pi$
$811$	1.90021e7	1.01449	0.507247	+	0.861800i	$0.330663\pi$
$812$	1.56964e7	0.835429
$813$	−5.10934e6	−0.271105
$814$	2.02992e6	0.107379
$815$	1.12942e7	0.595608
$816$	953856.	0.0501484
$817$	1.27301e7	0.667231
$818$	7.87978e6	0.411748
$819$	−9.38045e6	−0.488668
$820$	−2.02689e7	−1.05268
$821$	−3.10173e7	−1.60600	−0.803001	−	0.595978i	$-0.796764\pi$
$821$	−3.10173e7	−1.60600	−0.803001	+	0.595978i	$0.796764\pi$
$822$	4.62434e6	0.238710
$823$	−1.56290e7	−0.804323	−0.402162	−	0.915569i	$-0.631741\pi$
$823$	−1.56290e7	−0.804323	−0.402162	+	0.915569i	$0.631741\pi$
$824$	−5.07290e6	−0.260278
$825$	664740.	0.0340030
$826$	1.85434e7	0.945667
$827$	1.58421e7	0.805467	0.402733	−	0.915317i	$-0.368060\pi$
$827$	1.58421e7	0.805467	0.402733	+	0.915317i	$0.368060\pi$
$828$	777600.	0.0394167
$829$	2.06176e6	0.104196	0.0520980	−	0.998642i	$-0.483409\pi$
$829$	2.06176e6	0.104196	0.0520980	+	0.998642i	$0.483409\pi$
$830$	1.70755e6	0.0860357
$831$	1.09193e7	0.548521
$832$	−2.69517e6	−0.134983
$833$	−5.86597e6	−0.292905
$834$	−5.55106e6	−0.276351
$835$	4.46292e7	2.21515
$836$	−917760.	−0.0454165
$837$	2.61857e6	0.129196
$838$	556080.	0.0273544
$839$	3.03900e7	1.49048	0.745240	−	0.666796i	$-0.232335\pi$
$839$	3.03900e7	1.49048	0.745240	+	0.666796i	$0.232335\pi$
$840$	6.69082e6	0.327176
$841$	1.05583e7	0.514760
$842$	1.73097e7	0.841414
$843$	−6.18964e6	−0.299983
$844$	−1.83530e7	−0.886850
$845$	−4.07029e6	−0.196103
$846$	−6.37632e6	−0.306298
$847$	−2.77114e7	−1.32724
$848$	−8.00410e6	−0.382228
$849$	7.47817e6	0.356062
$850$	−2.03854e6	−0.0967768
$851$	−5.07480e6	−0.240212
$852$	−881280.	−0.0415925
$853$	−2.97738e7	−1.40108	−0.700538	−	0.713615i	$-0.747056\pi$
$853$	−2.97738e7	−1.40108	−0.700538	+	0.713615i	$0.747056\pi$
$854$	−2.18874e7	−1.02695
$855$	−5.11078e6	−0.239096
$856$	1.45912e7	0.680624
$857$	8.64100e6	0.401894	0.200947	−	0.979602i	$-0.435598\pi$
$857$	8.64100e6	0.401894	0.200947	+	0.979602i	$0.435598\pi$
$858$	−1.42128e6	−0.0659115
$859$	−3.35663e7	−1.55210	−0.776051	−	0.630670i	$-0.782780\pi$
$859$	−3.35663e7	−1.55210	−0.776051	+	0.630670i	$0.782780\pi$
$860$	−1.40617e7	−0.648323
$861$	−3.04033e7	−1.39770
$862$	−1.11974e7	−0.513276
$863$	3.90191e7	1.78341	0.891703	−	0.452621i	$-0.149511\pi$
$863$	3.90191e7	1.78341	0.891703	+	0.452621i	$0.149511\pi$
$864$	−746496.	−0.0340207
$865$	−8.78843e6	−0.399366
$866$	−2.36097e7	−1.06978
$867$	1.12361e7	0.507656
$868$	−1.01151e7	−0.455690
$869$	−4.46448e6	−0.200549
$870$	1.32438e7	0.593219
$871$	1.10570e7	0.493848
$872$	−545920.	−0.0243130
$873$	1.34526e7	0.597409
$874$	2.29440e6	0.101599
$875$	2.20007e7	0.971441
$876$	3.68035e6	0.162043
$877$	−1.81382e7	−0.796333	−0.398166	−	0.917313i	$-0.630353\pi$
$877$	−1.81382e7	−0.796333	−0.398166	+	0.917313i	$0.630353\pi$
$878$	−1.78605e6	−0.0781910
$879$	1.18136e7	0.515717
$880$	1.01376e6	0.0441294
$881$	3.05312e7	1.32527	0.662634	−	0.748943i	$-0.269438\pi$
$881$	3.05312e7	1.32527	0.662634	+	0.748943i	$0.269438\pi$
$882$	4.59076e6	0.198707
$883$	−4.35533e7	−1.87983	−0.939916	−	0.341405i	$-0.889097\pi$
$883$	−4.35533e7	−1.87983	−0.939916	+	0.341405i	$0.889097\pi$
$884$	4.35859e6	0.187593
$885$	1.56460e7	0.671497
$886$	1.39810e7	0.598348
$887$	−1.34152e7	−0.572515	−0.286257	−	0.958153i	$-0.592411\pi$
$887$	−1.34152e7	−0.572515	−0.286257	+	0.958153i	$0.592411\pi$
$888$	4.87181e6	0.207328
$889$	3.04480e7	1.29212
$890$	8.64389e6	0.365792
$891$	−393660.	−0.0166122
$892$	1.12314e7	0.472629
$893$	−1.88141e7	−0.789504
$894$	−1.06034e6	−0.0443714
$895$	4.57641e7	1.90971
$896$	2.88358e6	0.119995
$897$	3.55320e6	0.147448
$898$	−4.82450e6	−0.199647
$899$	−2.00218e7	−0.826236
$900$	1.59538e6	0.0656533
$901$	1.29441e7	0.531203
$902$	−4.60656e6	−0.188521
$903$	−2.10925e7	−0.860815
$904$	−1.25080e7	−0.509059
$905$	−2.48935e7	−1.01033
$906$	7.33939e6	0.297057
$907$	3.10816e6	0.125454	0.0627272	−	0.998031i	$-0.480020\pi$
$907$	3.10816e6	0.125454	0.0627272	+	0.998031i	$0.480020\pi$
$908$	1.97777e7	0.796089
$909$	−1.78216e6	−0.0715381
$910$	3.05733e7	1.22388
$911$	1.19035e6	0.0475203	0.0237602	−	0.999718i	$-0.492436\pi$
$911$	1.19035e6	0.0475203	0.0237602	+	0.999718i	$0.492436\pi$
$912$	−2.20262e6	−0.0876906
$913$	388080.	0.0154079
$914$	934184.	0.0369885
$915$	−1.84675e7	−0.729213
$916$	1.69328e6	0.0666792
$917$	2.66218e7	1.04547
$918$	1.20722e6	0.0472804
$919$	−4.71996e7	−1.84353	−0.921764	−	0.387752i	$-0.873252\pi$
$919$	−4.71996e7	−1.84353	−0.921764	+	0.387752i	$0.873252\pi$
$920$	−2.53440e6	−0.0987201
$921$	−1.52120e7	−0.590931
$922$	−6.97956e6	−0.270396
$923$	−4.02696e6	−0.155587
$924$	1.52064e6	0.0585931
$925$	−1.04118e7	−0.400103
$926$	−1.16715e7	−0.447299
$927$	−6.42038e6	−0.245393
$928$	5.70778e6	0.217569
$929$	1.33595e6	0.0507870	0.0253935	−	0.999678i	$-0.491916\pi$
$929$	1.33595e6	0.0507870	0.0253935	+	0.999678i	$0.491916\pi$
$930$	−8.53459e6	−0.323575
$931$	1.35456e7	0.512180
$932$	−7.01885e6	−0.264683
$933$	1.35184e7	0.508417
$934$	−2.12430e7	−0.796800
$935$	−1.63944e6	−0.0613291
$936$	−3.41107e6	−0.127263
$937$	1.47238e7	0.547861	0.273931	−	0.961749i	$-0.411676\pi$
$937$	1.47238e7	0.547861	0.273931	+	0.961749i	$0.411676\pi$
$938$	−1.18300e7	−0.439014
$939$	−7.29758e6	−0.270094
$940$	2.07821e7	0.767131
$941$	−2.69196e7	−0.991049	−0.495525	−	0.868594i	$-0.665024\pi$
$941$	−2.69196e7	−0.991049	−0.495525	+	0.868594i	$0.665024\pi$
$942$	−4.90111e6	−0.179956
$943$	1.15164e7	0.421733
$944$	6.74304e6	0.246278
$945$	8.46806e6	0.308464
$946$	−3.19584e6	−0.116107
$947$	−3.73160e6	−0.135214	−0.0676068	−	0.997712i	$-0.521536\pi$
$947$	−3.73160e6	−0.135214	−0.0676068	+	0.997712i	$0.521536\pi$
$948$	−1.07148e7	−0.387223
$949$	1.68172e7	0.606160
$950$	4.70734e6	0.169226
$951$	−8.13202e6	−0.291573
$952$	−4.66330e6	−0.166763
$953$	2.18735e7	0.780166	0.390083	−	0.920780i	$-0.372446\pi$
$953$	2.18735e7	0.780166	0.390083	+	0.920780i	$0.372446\pi$
$954$	−1.01302e7	−0.360368
$955$	1.75127e7	0.621362
$956$	455424.	0.0161165
$957$	3.00996e6	0.106238
$958$	9.37862e6	0.330161
$959$	−2.26079e7	−0.793805
$960$	2.43302e6	0.0852056
$961$	−1.57267e7	−0.549324
$962$	2.22615e7	0.775561
$963$	1.84670e7	0.641699
$964$	1.42810e7	0.494955
$965$	−1.94897e7	−0.673730
$966$	−3.80160e6	−0.131076
$967$	1.76025e7	0.605352	0.302676	−	0.953093i	$-0.402120\pi$
$967$	1.76025e7	0.605352	0.302676	+	0.953093i	$0.402120\pi$
$968$	−1.00769e7	−0.345650
$969$	3.56206e6	0.121868
$970$	−4.38456e7	−1.49623
$971$	1.67317e7	0.569497	0.284749	−	0.958602i	$-0.408090\pi$
$971$	1.67317e7	0.569497	0.284749	+	0.958602i	$0.408090\pi$
$972$	−944784.	−0.0320750
$973$	2.71385e7	0.918975
$974$	3.92612e7	1.32607
$975$	7.28998e6	0.245592
$976$	−7.95904e6	−0.267446
$977$	5.55382e7	1.86147	0.930733	−	0.365699i	$-0.119170\pi$
$977$	5.55382e7	1.86147	0.930733	+	0.365699i	$0.119170\pi$
$978$	6.16046e6	0.205952
$979$	1.96452e6	0.0655088
$980$	−1.49625e7	−0.497666
$981$	−690930.	−0.0229225
$982$	−2.37808e7	−0.786951
$983$	−3.86784e7	−1.27669	−0.638344	−	0.769751i	$-0.720380\pi$
$983$	−3.86784e7	−1.27669	−0.638344	+	0.769751i	$0.720380\pi$
$984$	−1.10557e7	−0.363999
$985$	−1.32854e7	−0.436299
$986$	−9.23054e6	−0.302367
$987$	3.11731e7	1.01856
$988$	−1.00648e7	−0.328028
$989$	7.98960e6	0.259737
$990$	1.28304e6	0.0416056
$991$	9.58498e6	0.310033	0.155016	−	0.987912i	$-0.450457\pi$
$991$	9.58498e6	0.310033	0.155016	+	0.987912i	$0.450457\pi$
$992$	−3.67821e6	−0.118674
$993$	−1.00977e7	−0.324976
$994$	4.30848e6	0.138311
$995$	−4.30616e7	−1.37890
$996$	931392.	0.0297498
$997$	−1.03650e7	−0.330242	−0.165121	−	0.986273i	$-0.552802\pi$
$997$	−1.03650e7	−0.330242	−0.165121	+	0.986273i	$0.552802\pi$
$998$	2.59133e7	0.823561
$999$	6.16588e6	0.195471

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6.6.a.a.1.1	✓	1
3.2	odd	2		18.6.a.b.1.1		1
4.3	odd	2		48.6.a.c.1.1		1
5.2	odd	4		150.6.c.b.49.2		2
5.3	odd	4		150.6.c.b.49.1		2
5.4	even	2		150.6.a.d.1.1		1
7.2	even	3		294.6.e.g.67.1		2
7.3	odd	6		294.6.e.a.79.1		2
7.4	even	3		294.6.e.g.79.1		2
7.5	odd	6		294.6.e.a.67.1		2
7.6	odd	2		294.6.a.m.1.1		1
8.3	odd	2		192.6.a.g.1.1		1
8.5	even	2		192.6.a.o.1.1		1
9.2	odd	6		162.6.c.h.109.1		2
9.4	even	3		162.6.c.e.55.1		2
9.5	odd	6		162.6.c.h.55.1		2
9.7	even	3		162.6.c.e.109.1		2
11.10	odd	2		726.6.a.a.1.1		1
12.11	even	2		144.6.a.j.1.1		1
13.12	even	2		1014.6.a.c.1.1		1
15.2	even	4		450.6.c.j.199.1		2
15.8	even	4		450.6.c.j.199.2		2
15.14	odd	2		450.6.a.m.1.1		1
16.3	odd	4		768.6.d.p.385.2		2
16.5	even	4		768.6.d.c.385.2		2
16.11	odd	4		768.6.d.p.385.1		2
16.13	even	4		768.6.d.c.385.1		2
21.20	even	2		882.6.a.a.1.1		1
24.5	odd	2		576.6.a.j.1.1		1
24.11	even	2		576.6.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.6.a.a.1.1	✓	1	1.1	even	1	trivial
18.6.a.b.1.1		1	3.2	odd	2
48.6.a.c.1.1		1	4.3	odd	2
144.6.a.j.1.1		1	12.11	even	2
150.6.a.d.1.1		1	5.4	even	2
150.6.c.b.49.1		2	5.3	odd	4
150.6.c.b.49.2		2	5.2	odd	4
162.6.c.e.55.1		2	9.4	even	3
162.6.c.e.109.1		2	9.7	even	3
162.6.c.h.55.1		2	9.5	odd	6
162.6.c.h.109.1		2	9.2	odd	6
192.6.a.g.1.1		1	8.3	odd	2
192.6.a.o.1.1		1	8.5	even	2
294.6.a.m.1.1		1	7.6	odd	2
294.6.e.a.67.1		2	7.5	odd	6
294.6.e.a.79.1		2	7.3	odd	6
294.6.e.g.67.1		2	7.2	even	3
294.6.e.g.79.1		2	7.4	even	3
450.6.a.m.1.1		1	15.14	odd	2
450.6.c.j.199.1		2	15.2	even	4
450.6.c.j.199.2		2	15.8	even	4
576.6.a.i.1.1		1	24.11	even	2
576.6.a.j.1.1		1	24.5	odd	2
726.6.a.a.1.1		1	11.10	odd	2
768.6.d.c.385.1		2	16.13	even	4
768.6.d.c.385.2		2	16.5	even	4
768.6.d.p.385.1		2	16.11	odd	4
768.6.d.p.385.2		2	16.3	odd	4
882.6.a.a.1.1		1	21.20	even	2
1014.6.a.c.1.1		1	13.12	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:	\( N \)	\(=\)	\( 6 = 2 \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	6.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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