LMFDB - Embedded newform 546.6.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,6,Mod(1,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, 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("546.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	546.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:	$87.5695656179$
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	$\chi$	$=$	546.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +93.0000 q^{5} -36.0000 q^{6} -49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +93.0000 q^{5} -36.0000 q^{6} -49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +372.000 q^{10} -302.000 q^{11} -144.000 q^{12} -169.000 q^{13} -196.000 q^{14} -837.000 q^{15} +256.000 q^{16} -488.000 q^{17} +324.000 q^{18} +2053.00 q^{19} +1488.00 q^{20} +441.000 q^{21} -1208.00 q^{22} +59.0000 q^{23} -576.000 q^{24} +5524.00 q^{25} -676.000 q^{26} -729.000 q^{27} -784.000 q^{28} +5871.00 q^{29} -3348.00 q^{30} +3861.00 q^{31} +1024.00 q^{32} +2718.00 q^{33} -1952.00 q^{34} -4557.00 q^{35} +1296.00 q^{36} +12388.0 q^{37} +8212.00 q^{38} +1521.00 q^{39} +5952.00 q^{40} +2602.00 q^{41} +1764.00 q^{42} -14221.0 q^{43} -4832.00 q^{44} +7533.00 q^{45} +236.000 q^{46} -21645.0 q^{47} -2304.00 q^{48} +2401.00 q^{49} +22096.0 q^{50} +4392.00 q^{51} -2704.00 q^{52} -7781.00 q^{53} -2916.00 q^{54} -28086.0 q^{55} -3136.00 q^{56} -18477.0 q^{57} +23484.0 q^{58} +19072.0 q^{59} -13392.0 q^{60} +13954.0 q^{61} +15444.0 q^{62} -3969.00 q^{63} +4096.00 q^{64} -15717.0 q^{65} +10872.0 q^{66} +2694.00 q^{67} -7808.00 q^{68} -531.000 q^{69} -18228.0 q^{70} +82032.0 q^{71} +5184.00 q^{72} +6503.00 q^{73} +49552.0 q^{74} -49716.0 q^{75} +32848.0 q^{76} +14798.0 q^{77} +6084.00 q^{78} +28535.0 q^{79} +23808.0 q^{80} +6561.00 q^{81} +10408.0 q^{82} +15019.0 q^{83} +7056.00 q^{84} -45384.0 q^{85} -56884.0 q^{86} -52839.0 q^{87} -19328.0 q^{88} +41979.0 q^{89} +30132.0 q^{90} +8281.00 q^{91} +944.000 q^{92} -34749.0 q^{93} -86580.0 q^{94} +190929. q^{95} -9216.00 q^{96} -57405.0 q^{97} +9604.00 q^{98} -24462.0 q^{99} +O(q^{100})\)

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
$5$	93.0000	1.66363	0.831817	−	0.555050i	$-0.187301\pi$
$5$	93.0000	1.66363	0.831817	+	0.555050i	$0.187301\pi$
$6$	−36.0000	−0.408248
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	372.000	1.17637
$11$	−302.000	−0.752532	−0.376266	−	0.926512i	$-0.622792\pi$
$11$	−302.000	−0.752532	−0.376266	+	0.926512i	$0.622792\pi$
$12$	−144.000	−0.288675
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	−196.000	−0.267261
$15$	−837.000	−0.960500
$16$	256.000	0.250000
$17$	−488.000	−0.409541	−0.204771	−	0.978810i	$-0.565645\pi$
$17$	−488.000	−0.409541	−0.204771	+	0.978810i	$0.565645\pi$
$18$	324.000	0.235702
$19$	2053.00	1.30468	0.652341	−	0.757925i	$-0.273787\pi$
$19$	2053.00	1.30468	0.652341	+	0.757925i	$0.273787\pi$
$20$	1488.00	0.831817
$21$	441.000	0.218218
$22$	−1208.00	−0.532121
$23$	59.0000	0.0232559	0.0116279	−	0.999932i	$-0.496299\pi$
$23$	59.0000	0.0232559	0.0116279	+	0.999932i	$0.496299\pi$
$24$	−576.000	−0.204124
$25$	5524.00	1.76768
$26$	−676.000	−0.196116
$27$	−729.000	−0.192450
$28$	−784.000	−0.188982
$29$	5871.00	1.29633	0.648167	−	0.761498i	$-0.275536\pi$
$29$	5871.00	1.29633	0.648167	+	0.761498i	$0.275536\pi$
$30$	−3348.00	−0.679176
$31$	3861.00	0.721598	0.360799	−	0.932644i	$-0.382504\pi$
$31$	3861.00	0.721598	0.360799	+	0.932644i	$0.382504\pi$
$32$	1024.00	0.176777
$33$	2718.00	0.434475
$34$	−1952.00	−0.289589
$35$	−4557.00	−0.628795
$36$	1296.00	0.166667
$37$	12388.0	1.48764	0.743818	−	0.668382i	$-0.233013\pi$
$37$	12388.0	1.48764	0.743818	+	0.668382i	$0.233013\pi$
$38$	8212.00	0.922550
$39$	1521.00	0.160128
$40$	5952.00	0.588184
$41$	2602.00	0.241740	0.120870	−	0.992668i	$-0.461432\pi$
$41$	2602.00	0.241740	0.120870	+	0.992668i	$0.461432\pi$
$42$	1764.00	0.154303
$43$	−14221.0	−1.17289	−0.586447	−	0.809987i	$-0.699474\pi$
$43$	−14221.0	−1.17289	−0.586447	+	0.809987i	$0.699474\pi$
$44$	−4832.00	−0.376266
$45$	7533.00	0.554545
$46$	236.000	0.0164444
$47$	−21645.0	−1.42927	−0.714633	−	0.699500i	$-0.753406\pi$
$47$	−21645.0	−1.42927	−0.714633	+	0.699500i	$0.753406\pi$
$48$	−2304.00	−0.144338
$49$	2401.00	0.142857
$50$	22096.0	1.24994
$51$	4392.00	0.236449
$52$	−2704.00	−0.138675
$53$	−7781.00	−0.380492	−0.190246	−	0.981736i	$-0.560929\pi$
$53$	−7781.00	−0.380492	−0.190246	+	0.981736i	$0.560929\pi$
$54$	−2916.00	−0.136083
$55$	−28086.0	−1.25194
$56$	−3136.00	−0.133631
$57$	−18477.0	−0.753259
$58$	23484.0	0.916647
$59$	19072.0	0.713290	0.356645	−	0.934240i	$-0.383921\pi$
$59$	19072.0	0.713290	0.356645	+	0.934240i	$0.383921\pi$
$60$	−13392.0	−0.480250
$61$	13954.0	0.480147	0.240073	−	0.970755i	$-0.422828\pi$
$61$	13954.0	0.480147	0.240073	+	0.970755i	$0.422828\pi$
$62$	15444.0	0.510247
$63$	−3969.00	−0.125988
$64$	4096.00	0.125000
$65$	−15717.0	−0.461409
$66$	10872.0	0.307220
$67$	2694.00	0.0733180	0.0366590	−	0.999328i	$-0.488328\pi$
$67$	2694.00	0.0733180	0.0366590	+	0.999328i	$0.488328\pi$
$68$	−7808.00	−0.204771
$69$	−531.000	−0.0134268
$70$	−18228.0	−0.444625
$71$	82032.0	1.93125	0.965623	−	0.259948i	$-0.0837054\pi$
$71$	82032.0	1.93125	0.965623	+	0.259948i	$0.0837054\pi$
$72$	5184.00	0.117851
$73$	6503.00	0.142826	0.0714129	−	0.997447i	$-0.477249\pi$
$73$	6503.00	0.142826	0.0714129	+	0.997447i	$0.477249\pi$
$74$	49552.0	1.05192
$75$	−49716.0	−1.02057
$76$	32848.0	0.652341
$77$	14798.0	0.284431
$78$	6084.00	0.113228
$79$	28535.0	0.514411	0.257205	−	0.966357i	$-0.417198\pi$
$79$	28535.0	0.514411	0.257205	+	0.966357i	$0.417198\pi$
$80$	23808.0	0.415909
$81$	6561.00	0.111111
$82$	10408.0	0.170936
$83$	15019.0	0.239302	0.119651	−	0.992816i	$-0.461822\pi$
$83$	15019.0	0.239302	0.119651	+	0.992816i	$0.461822\pi$
$84$	7056.00	0.109109
$85$	−45384.0	−0.681327
$86$	−56884.0	−0.829362
$87$	−52839.0	−0.748439
$88$	−19328.0	−0.266060
$89$	41979.0	0.561768	0.280884	−	0.959742i	$-0.409372\pi$
$89$	41979.0	0.561768	0.280884	+	0.959742i	$0.409372\pi$
$90$	30132.0	0.392122
$91$	8281.00	0.104828
$92$	944.000	0.0116279
$93$	−34749.0	−0.416615
$94$	−86580.0	−1.01064
$95$	190929.	2.17052
$96$	−9216.00	−0.102062
$97$	−57405.0	−0.619470	−0.309735	−	0.950823i	$-0.600240\pi$
$97$	−57405.0	−0.619470	−0.309735	+	0.950823i	$0.600240\pi$
$98$	9604.00	0.101015
$99$	−24462.0	−0.250844
$100$	88384.0	0.883840
$101$	49902.0	0.486760	0.243380	−	0.969931i	$-0.421744\pi$
$101$	49902.0	0.486760	0.243380	+	0.969931i	$0.421744\pi$
$102$	17568.0	0.167194
$103$	−44608.0	−0.414305	−0.207152	−	0.978309i	$-0.566420\pi$
$103$	−44608.0	−0.414305	−0.207152	+	0.978309i	$0.566420\pi$
$104$	−10816.0	−0.0980581
$105$	41013.0	0.363035
$106$	−31124.0	−0.269049
$107$	−92556.0	−0.781529	−0.390765	−	0.920491i	$-0.627789\pi$
$107$	−92556.0	−0.781529	−0.390765	+	0.920491i	$0.627789\pi$
$108$	−11664.0	−0.0962250
$109$	71606.0	0.577276	0.288638	−	0.957438i	$-0.406798\pi$
$109$	71606.0	0.577276	0.288638	+	0.957438i	$0.406798\pi$
$110$	−112344.	−0.885255
$111$	−111492.	−0.858887
$112$	−12544.0	−0.0944911
$113$	99577.0	0.733606	0.366803	−	0.930299i	$-0.380452\pi$
$113$	99577.0	0.733606	0.366803	+	0.930299i	$0.380452\pi$
$114$	−73908.0	−0.532635
$115$	5487.00	0.0386893
$116$	93936.0	0.648167
$117$	−13689.0	−0.0924500
$118$	76288.0	0.504372
$119$	23912.0	0.154792
$120$	−53568.0	−0.339588
$121$	−69847.0	−0.433695
$122$	55816.0	0.339515
$123$	−23418.0	−0.139568
$124$	61776.0	0.360799
$125$	223107.	1.27714
$126$	−15876.0	−0.0890871
$127$	81172.0	0.446578	0.223289	−	0.974752i	$-0.428321\pi$
$127$	81172.0	0.446578	0.223289	+	0.974752i	$0.428321\pi$
$128$	16384.0	0.0883883
$129$	127989.	0.677171
$130$	−62868.0	−0.326266
$131$	145320.	0.739856	0.369928	−	0.929060i	$-0.379382\pi$
$131$	145320.	0.739856	0.369928	+	0.929060i	$0.379382\pi$
$132$	43488.0	0.217237
$133$	−100597.	−0.493124
$134$	10776.0	0.0518437
$135$	−67797.0	−0.320167
$136$	−31232.0	−0.144795
$137$	405024.	1.84365	0.921827	−	0.387602i	$-0.126696\pi$
$137$	405024.	1.84365	0.921827	+	0.387602i	$0.126696\pi$
$138$	−2124.00	−0.00949417
$139$	−22634.0	−0.0993629	−0.0496815	−	0.998765i	$-0.515821\pi$
$139$	−22634.0	−0.0993629	−0.0496815	+	0.998765i	$0.515821\pi$
$140$	−72912.0	−0.314397
$141$	194805.	0.825187
$142$	328128.	1.36560
$143$	51038.0	0.208715
$144$	20736.0	0.0833333
$145$	546003.	2.15663
$146$	26012.0	0.100993
$147$	−21609.0	−0.0824786
$148$	198208.	0.743818
$149$	121558.	0.448557	0.224279	−	0.974525i	$-0.427997\pi$
$149$	121558.	0.448557	0.224279	+	0.974525i	$0.427997\pi$
$150$	−198864.	−0.721652
$151$	386992.	1.38121	0.690605	−	0.723232i	$-0.257344\pi$
$151$	386992.	1.38121	0.690605	+	0.723232i	$0.257344\pi$
$152$	131392.	0.461275
$153$	−39528.0	−0.136514
$154$	59192.0	0.201123
$155$	359073.	1.20048
$156$	24336.0	0.0800641
$157$	−11008.0	−0.0356418	−0.0178209	−	0.999841i	$-0.505673\pi$
$157$	−11008.0	−0.0356418	−0.0178209	+	0.999841i	$0.505673\pi$
$158$	114140.	0.363743
$159$	70029.0	0.219677
$160$	95232.0	0.294092
$161$	−2891.00	−0.00878989
$162$	26244.0	0.0785674
$163$	−395468.	−1.16585	−0.582925	−	0.812526i	$-0.698092\pi$
$163$	−395468.	−1.16585	−0.582925	+	0.812526i	$0.698092\pi$
$164$	41632.0	0.120870
$165$	252774.	0.722807
$166$	60076.0	0.169212
$167$	−177287.	−0.491910	−0.245955	−	0.969281i	$-0.579102\pi$
$167$	−177287.	−0.491910	−0.245955	+	0.969281i	$0.579102\pi$
$168$	28224.0	0.0771517
$169$	28561.0	0.0769231
$170$	−181536.	−0.481771
$171$	166293.	0.434894
$172$	−227536.	−0.586447
$173$	466064.	1.18394	0.591971	−	0.805959i	$-0.298350\pi$
$173$	466064.	1.18394	0.591971	+	0.805959i	$0.298350\pi$
$174$	−211356.	−0.529226
$175$	−270676.	−0.668120
$176$	−77312.0	−0.188133
$177$	−171648.	−0.411818
$178$	167916.	0.397230
$179$	−708985.	−1.65388	−0.826941	−	0.562289i	$-0.809921\pi$
$179$	−708985.	−1.65388	−0.826941	+	0.562289i	$0.809921\pi$
$180$	120528.	0.277272
$181$	321862.	0.730253	0.365126	−	0.930958i	$-0.381026\pi$
$181$	321862.	0.730253	0.365126	+	0.930958i	$0.381026\pi$
$182$	33124.0	0.0741249
$183$	−125586.	−0.277213
$184$	3776.00	0.00822219
$185$	1.15208e6	2.47488
$186$	−138996.	−0.294591
$187$	147376.	0.308193
$188$	−346320.	−0.714633
$189$	35721.0	0.0727393
$190$	763716.	1.53479
$191$	−386880.	−0.767349	−0.383674	−	0.923468i	$-0.625342\pi$
$191$	−386880.	−0.767349	−0.383674	+	0.923468i	$0.625342\pi$
$192$	−36864.0	−0.0721688
$193$	−264854.	−0.511815	−0.255908	−	0.966701i	$-0.582374\pi$
$193$	−264854.	−0.511815	−0.255908	+	0.966701i	$0.582374\pi$
$194$	−229620.	−0.438032
$195$	141453.	0.266395
$196$	38416.0	0.0714286
$197$	−282062.	−0.517820	−0.258910	−	0.965901i	$-0.583363\pi$
$197$	−282062.	−0.517820	−0.258910	+	0.965901i	$0.583363\pi$
$198$	−97848.0	−0.177374
$199$	872688.	1.56216	0.781081	−	0.624429i	$-0.214668\pi$
$199$	872688.	1.56216	0.781081	+	0.624429i	$0.214668\pi$
$200$	353536.	0.624969
$201$	−24246.0	−0.0423302
$202$	199608.	0.344191
$203$	−287679.	−0.489968
$204$	70272.0	0.118224
$205$	241986.	0.402166
$206$	−178432.	−0.292958
$207$	4779.00	0.00775195
$208$	−43264.0	−0.0693375
$209$	−620006.	−0.981816
$210$	164052.	0.256704
$211$	117895.	0.182301	0.0911505	−	0.995837i	$-0.470946\pi$
$211$	117895.	0.182301	0.0911505	+	0.995837i	$0.470946\pi$
$212$	−124496.	−0.190246
$213$	−738288.	−1.11500
$214$	−370224.	−0.552625
$215$	−1.32255e6	−1.95127
$216$	−46656.0	−0.0680414
$217$	−189189.	−0.272739
$218$	286424.	0.408195
$219$	−58527.0	−0.0824605
$220$	−449376.	−0.625969
$221$	82472.0	0.113586
$222$	−445968.	−0.607325
$223$	−752569.	−1.01341	−0.506704	−	0.862120i	$-0.669136\pi$
$223$	−752569.	−1.01341	−0.506704	+	0.862120i	$0.669136\pi$
$224$	−50176.0	−0.0668153
$225$	447444.	0.589227
$226$	398308.	0.518738
$227$	484812.	0.624466	0.312233	−	0.950006i	$-0.398923\pi$
$227$	484812.	0.624466	0.312233	+	0.950006i	$0.398923\pi$
$228$	−295632.	−0.376629
$229$	−397746.	−0.501207	−0.250603	−	0.968090i	$-0.580629\pi$
$229$	−397746.	−0.501207	−0.250603	+	0.968090i	$0.580629\pi$
$230$	21948.0	0.0273574
$231$	−133182.	−0.164216
$232$	375744.	0.458323
$233$	−272541.	−0.328883	−0.164442	−	0.986387i	$-0.552582\pi$
$233$	−272541.	−0.328883	−0.164442	+	0.986387i	$0.552582\pi$
$234$	−54756.0	−0.0653720
$235$	−2.01299e6	−2.37778
$236$	305152.	0.356645
$237$	−256815.	−0.296995
$238$	95648.0	0.109454
$239$	−818896.	−0.927329	−0.463665	−	0.886011i	$-0.653466\pi$
$239$	−818896.	−0.927329	−0.463665	+	0.886011i	$0.653466\pi$
$240$	−214272.	−0.240125
$241$	1.34780e6	1.49479	0.747397	−	0.664378i	$-0.231303\pi$
$241$	1.34780e6	1.49479	0.747397	+	0.664378i	$0.231303\pi$
$242$	−279388.	−0.306669
$243$	−59049.0	−0.0641500
$244$	223264.	0.240073
$245$	223293.	0.237662
$246$	−93672.0	−0.0986897
$247$	−346957.	−0.361854
$248$	247104.	0.255124
$249$	−135171.	−0.138161
$250$	892428.	0.903074
$251$	423426.	0.424222	0.212111	−	0.977246i	$-0.431966\pi$
$251$	423426.	0.424222	0.212111	+	0.977246i	$0.431966\pi$
$252$	−63504.0	−0.0629941
$253$	−17818.0	−0.0175008
$254$	324688.	0.315778
$255$	408456.	0.393364
$256$	65536.0	0.0625000
$257$	−2.11090e6	−1.99358	−0.996792	−	0.0800385i	$-0.974496\pi$
$257$	−2.11090e6	−1.99358	−0.996792	+	0.0800385i	$0.974496\pi$
$258$	511956.	0.478832
$259$	−607012.	−0.562274
$260$	−251472.	−0.230705
$261$	475551.	0.432112
$262$	581280.	0.523157
$263$	−675943.	−0.602588	−0.301294	−	0.953531i	$-0.597419\pi$
$263$	−675943.	−0.602588	−0.301294	+	0.953531i	$0.597419\pi$
$264$	173952.	0.153610
$265$	−723633.	−0.633000
$266$	−402388.	−0.348691
$267$	−377811.	−0.324337
$268$	43104.0	0.0366590
$269$	897716.	0.756412	0.378206	−	0.925722i	$-0.376541\pi$
$269$	897716.	0.756412	0.378206	+	0.925722i	$0.376541\pi$
$270$	−271188.	−0.226392
$271$	−2.16653e6	−1.79201	−0.896006	−	0.444041i	$-0.853544\pi$
$271$	−2.16653e6	−1.79201	−0.896006	+	0.444041i	$0.853544\pi$
$272$	−124928.	−0.102385
$273$	−74529.0	−0.0605228
$274$	1.62010e6	1.30366
$275$	−1.66825e6	−1.33024
$276$	−8496.00	−0.00671339
$277$	1.30656e6	1.02313	0.511564	−	0.859245i	$-0.329066\pi$
$277$	1.30656e6	1.02313	0.511564	+	0.859245i	$0.329066\pi$
$278$	−90536.0	−0.0702602
$279$	312741.	0.240533
$280$	−291648.	−0.222313
$281$	1.64837e6	1.24535	0.622673	−	0.782482i	$-0.286047\pi$
$281$	1.64837e6	1.24535	0.622673	+	0.782482i	$0.286047\pi$
$282$	779220.	0.583495
$283$	834116.	0.619099	0.309550	−	0.950883i	$-0.399822\pi$
$283$	834116.	0.619099	0.309550	+	0.950883i	$0.399822\pi$
$284$	1.31251e6	0.965623
$285$	−1.71836e6	−1.25315
$286$	204152.	0.147584
$287$	−127498.	−0.0913689
$288$	82944.0	0.0589256
$289$	−1.18171e6	−0.832276
$290$	2.18401e6	1.52497
$291$	516645.	0.357651
$292$	104048.	0.0714129
$293$	241437.	0.164299	0.0821495	−	0.996620i	$-0.473822\pi$
$293$	241437.	0.164299	0.0821495	+	0.996620i	$0.473822\pi$
$294$	−86436.0	−0.0583212
$295$	1.77370e6	1.18665
$296$	792832.	0.525959
$297$	220158.	0.144825
$298$	486232.	0.317178
$299$	−9971.00	−0.00645002
$300$	−795456.	−0.510285
$301$	696829.	0.443313
$302$	1.54797e6	0.976663
$303$	−449118.	−0.281031
$304$	525568.	0.326171
$305$	1.29772e6	0.798789
$306$	−158112.	−0.0965298
$307$	1.67540e6	1.01455	0.507274	−	0.861785i	$-0.330653\pi$
$307$	1.67540e6	1.01455	0.507274	+	0.861785i	$0.330653\pi$
$308$	236768.	0.142215
$309$	401472.	0.239199
$310$	1.43629e6	0.848865
$311$	1.63906e6	0.960934	0.480467	−	0.877013i	$-0.340467\pi$
$311$	1.63906e6	0.960934	0.480467	+	0.877013i	$0.340467\pi$
$312$	97344.0	0.0566139
$313$	−1.96954e6	−1.13633	−0.568165	−	0.822915i	$-0.692346\pi$
$313$	−1.96954e6	−1.13633	−0.568165	+	0.822915i	$0.692346\pi$
$314$	−44032.0	−0.0252025
$315$	−369117.	−0.209598
$316$	456560.	0.257205
$317$	−1.22674e6	−0.685653	−0.342826	−	0.939399i	$-0.611384\pi$
$317$	−1.22674e6	−0.685653	−0.342826	+	0.939399i	$0.611384\pi$
$318$	280116.	0.155335
$319$	−1.77304e6	−0.975534
$320$	380928.	0.207954
$321$	833004.	0.451216
$322$	−11564.0	−0.00621539
$323$	−1.00186e6	−0.534321
$324$	104976.	0.0555556
$325$	−933556.	−0.490266
$326$	−1.58187e6	−0.824380
$327$	−644454.	−0.333290
$328$	166528.	0.0854678
$329$	1.06060e6	0.540212
$330$	1.01110e6	0.511102
$331$	−2.41427e6	−1.21120	−0.605600	−	0.795769i	$-0.707067\pi$
$331$	−2.41427e6	−1.21120	−0.605600	+	0.795769i	$0.707067\pi$
$332$	240304.	0.119651
$333$	1.00343e6	0.495879
$334$	−709148.	−0.347833
$335$	250542.	0.121974
$336$	112896.	0.0545545
$337$	617817.	0.296336	0.148168	−	0.988962i	$-0.452662\pi$
$337$	617817.	0.296336	0.148168	+	0.988962i	$0.452662\pi$
$338$	114244.	0.0543928
$339$	−896193.	−0.423548
$340$	−726144.	−0.340663
$341$	−1.16602e6	−0.543026
$342$	665172.	0.307517
$343$	−117649.	−0.0539949
$344$	−910144.	−0.414681
$345$	−49383.0	−0.0223373
$346$	1.86426e6	0.837173
$347$	−1.87077e6	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$347$	−1.87077e6	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$348$	−845424.	−0.374220
$349$	−3.75833e6	−1.65170	−0.825850	−	0.563890i	$-0.809304\pi$
$349$	−3.75833e6	−1.65170	−0.825850	+	0.563890i	$0.809304\pi$
$350$	−1.08270e6	−0.472432
$351$	123201.	0.0533761
$352$	−309248.	−0.133030
$353$	140774.	0.0601292	0.0300646	−	0.999548i	$-0.490429\pi$
$353$	140774.	0.0601292	0.0300646	+	0.999548i	$0.490429\pi$
$354$	−686592.	−0.291199
$355$	7.62898e6	3.21289
$356$	671664.	0.280884
$357$	−215208.	−0.0893692
$358$	−2.83594e6	−1.16947
$359$	−1.78318e6	−0.730227	−0.365114	−	0.930963i	$-0.618970\pi$
$359$	−1.78318e6	−0.730227	−0.365114	+	0.930963i	$0.618970\pi$
$360$	482112.	0.196061
$361$	1.73871e6	0.702197
$362$	1.28745e6	0.516367
$363$	628623.	0.250394
$364$	132496.	0.0524142
$365$	604779.	0.237610
$366$	−502344.	−0.196019
$367$	−3.07850e6	−1.19309	−0.596546	−	0.802579i	$-0.703461\pi$
$367$	−3.07850e6	−1.19309	−0.596546	+	0.802579i	$0.703461\pi$
$368$	15104.0	0.00581397
$369$	210762.	0.0805798
$370$	4.60834e6	1.75001
$371$	381269.	0.143813
$372$	−555984.	−0.208308
$373$	−3.45143e6	−1.28448	−0.642240	−	0.766503i	$-0.721995\pi$
$373$	−3.45143e6	−1.28448	−0.642240	+	0.766503i	$0.721995\pi$
$374$	589504.	0.217925
$375$	−2.00796e6	−0.737357
$376$	−1.38528e6	−0.505322
$377$	−992199.	−0.359539
$378$	142884.	0.0514344
$379$	−2.49923e6	−0.893732	−0.446866	−	0.894601i	$-0.647460\pi$
$379$	−2.49923e6	−0.893732	−0.446866	+	0.894601i	$0.647460\pi$
$380$	3.05486e6	1.08526
$381$	−730548.	−0.257832
$382$	−1.54752e6	−0.542598
$383$	3.94648e6	1.37472	0.687359	−	0.726318i	$-0.258770\pi$
$383$	3.94648e6	1.37472	0.687359	+	0.726318i	$0.258770\pi$
$384$	−147456.	−0.0510310
$385$	1.37621e6	0.473188
$386$	−1.05942e6	−0.361908
$387$	−1.15190e6	−0.390965
$388$	−918480.	−0.309735
$389$	−4.05665e6	−1.35923	−0.679616	−	0.733568i	$-0.737854\pi$
$389$	−4.05665e6	−1.35923	−0.679616	+	0.733568i	$0.737854\pi$
$390$	565812.	0.188370
$391$	−28792.0	−0.00952423
$392$	153664.	0.0505076
$393$	−1.30788e6	−0.427156
$394$	−1.12825e6	−0.366154
$395$	2.65376e6	0.855792
$396$	−391392.	−0.125422
$397$	3.34947e6	1.06660	0.533298	−	0.845927i	$-0.320952\pi$
$397$	3.34947e6	1.06660	0.533298	+	0.845927i	$0.320952\pi$
$398$	3.49075e6	1.10462
$399$	905373.	0.284705
$400$	1.41414e6	0.441920
$401$	6.00207e6	1.86397	0.931987	−	0.362491i	$-0.118074\pi$
$401$	6.00207e6	1.86397	0.931987	+	0.362491i	$0.118074\pi$
$402$	−96984.0	−0.0299319
$403$	−652509.	−0.200135
$404$	798432.	0.243380
$405$	610173.	0.184848
$406$	−1.15072e6	−0.346460
$407$	−3.74118e6	−1.11949
$408$	281088.	0.0835972
$409$	2.16245e6	0.639201	0.319600	−	0.947552i	$-0.396451\pi$
$409$	2.16245e6	0.639201	0.319600	+	0.947552i	$0.396451\pi$
$410$	967944.	0.284374
$411$	−3.64522e6	−1.06443
$412$	−713728.	−0.207152
$413$	−934528.	−0.269598
$414$	19116.0	0.00548146
$415$	1.39677e6	0.398111
$416$	−173056.	−0.0490290
$417$	203706.	0.0573672
$418$	−2.48002e6	−0.694249
$419$	−4.99448e6	−1.38981	−0.694905	−	0.719102i	$-0.744554\pi$
$419$	−4.99448e6	−1.38981	−0.694905	+	0.719102i	$0.744554\pi$
$420$	656208.	0.181517
$421$	46040.0	0.0126599	0.00632995	−	0.999980i	$-0.497985\pi$
$421$	46040.0	0.0126599	0.00632995	+	0.999980i	$0.497985\pi$
$422$	471580.	0.128906
$423$	−1.75324e6	−0.476422
$424$	−497984.	−0.134524
$425$	−2.69571e6	−0.723938
$426$	−2.95315e6	−0.788428
$427$	−683746.	−0.181478
$428$	−1.48090e6	−0.390765
$429$	−459342.	−0.120502
$430$	−5.29021e6	−1.37975
$431$	−6.96428e6	−1.80586	−0.902928	−	0.429792i	$-0.858587\pi$
$431$	−6.96428e6	−1.80586	−0.902928	+	0.429792i	$0.858587\pi$
$432$	−186624.	−0.0481125
$433$	5.61426e6	1.43904	0.719520	−	0.694471i	$-0.244362\pi$
$433$	5.61426e6	1.43904	0.719520	+	0.694471i	$0.244362\pi$
$434$	−756756.	−0.192855
$435$	−4.91403e6	−1.24513
$436$	1.14570e6	0.288638
$437$	121127.	0.0303415
$438$	−234108.	−0.0583084
$439$	1.88335e6	0.466413	0.233207	−	0.972427i	$-0.425078\pi$
$439$	1.88335e6	0.466413	0.233207	+	0.972427i	$0.425078\pi$
$440$	−1.79750e6	−0.442627
$441$	194481.	0.0476190
$442$	329888.	0.0803176
$443$	−6.97660e6	−1.68902	−0.844509	−	0.535541i	$-0.820108\pi$
$443$	−6.97660e6	−1.68902	−0.844509	+	0.535541i	$0.820108\pi$
$444$	−1.78387e6	−0.429444
$445$	3.90405e6	0.934577
$446$	−3.01028e6	−0.716587
$447$	−1.09402e6	−0.258975
$448$	−200704.	−0.0472456
$449$	1.42088e6	0.332614	0.166307	−	0.986074i	$-0.446816\pi$
$449$	1.42088e6	0.332614	0.166307	+	0.986074i	$0.446816\pi$
$450$	1.78978e6	0.416646
$451$	−785804.	−0.181917
$452$	1.59323e6	0.366803
$453$	−3.48293e6	−0.797442
$454$	1.93925e6	0.441564
$455$	770133.	0.174396
$456$	−1.18253e6	−0.266317
$457$	180272.	0.0403773	0.0201887	−	0.999796i	$-0.493573\pi$
$457$	180272.	0.0403773	0.0201887	+	0.999796i	$0.493573\pi$
$458$	−1.59098e6	−0.354407
$459$	355752.	0.0788162
$460$	87792.0	0.0193446
$461$	−3.59082e6	−0.786940	−0.393470	−	0.919337i	$-0.628725\pi$
$461$	−3.59082e6	−0.786940	−0.393470	+	0.919337i	$0.628725\pi$
$462$	−532728.	−0.116118
$463$	5.31597e6	1.15247	0.576235	−	0.817284i	$-0.304521\pi$
$463$	5.31597e6	1.15247	0.576235	+	0.817284i	$0.304521\pi$
$464$	1.50298e6	0.324084
$465$	−3.23166e6	−0.693095
$466$	−1.09016e6	−0.232556
$467$	−2.77316e6	−0.588413	−0.294207	−	0.955742i	$-0.595055\pi$
$467$	−2.77316e6	−0.588413	−0.294207	+	0.955742i	$0.595055\pi$
$468$	−219024.	−0.0462250
$469$	−132006.	−0.0277116
$470$	−8.05194e6	−1.68134
$471$	99072.0	0.0205778
$472$	1.22061e6	0.252186
$473$	4.29474e6	0.882641
$474$	−1.02726e6	−0.210007
$475$	1.13408e7	2.30626
$476$	382592.	0.0773960
$477$	−630261.	−0.126831
$478$	−3.27558e6	−0.655721
$479$	3.72434e6	0.741671	0.370835	−	0.928699i	$-0.379071\pi$
$479$	3.72434e6	0.741671	0.370835	+	0.928699i	$0.379071\pi$
$480$	−857088.	−0.169794
$481$	−2.09357e6	−0.412596
$482$	5.39118e6	1.05698
$483$	26019.0	0.00507484
$484$	−1.11755e6	−0.216847
$485$	−5.33866e6	−1.03057
$486$	−236196.	−0.0453609
$487$	9.57663e6	1.82974	0.914872	−	0.403743i	$-0.132291\pi$
$487$	9.57663e6	1.82974	0.914872	+	0.403743i	$0.132291\pi$
$488$	893056.	0.169758
$489$	3.55921e6	0.673103
$490$	893172.	0.168052
$491$	−8.71582e6	−1.63157	−0.815783	−	0.578358i	$-0.803694\pi$
$491$	−8.71582e6	−1.63157	−0.815783	+	0.578358i	$0.803694\pi$
$492$	−374688.	−0.0697842
$493$	−2.86505e6	−0.530902
$494$	−1.38783e6	−0.255869
$495$	−2.27497e6	−0.417313
$496$	988416.	0.180400
$497$	−4.01957e6	−0.729942
$498$	−540684.	−0.0976945
$499$	−808088.	−0.145280	−0.0726402	−	0.997358i	$-0.523142\pi$
$499$	−808088.	−0.145280	−0.0726402	+	0.997358i	$0.523142\pi$
$500$	3.56971e6	0.638569
$501$	1.59558e6	0.284004
$502$	1.69370e6	0.299970
$503$	−9.61039e6	−1.69364	−0.846820	−	0.531880i	$-0.821486\pi$
$503$	−9.61039e6	−1.69364	−0.846820	+	0.531880i	$0.821486\pi$
$504$	−254016.	−0.0445435
$505$	4.64089e6	0.809790
$506$	−71272.0	−0.0123749
$507$	−257049.	−0.0444116
$508$	1.29875e6	0.223289
$509$	4.29919e6	0.735516	0.367758	−	0.929922i	$-0.380125\pi$
$509$	4.29919e6	0.735516	0.367758	+	0.929922i	$0.380125\pi$
$510$	1.63382e6	0.278150
$511$	−318647.	−0.0539831
$512$	262144.	0.0441942
$513$	−1.49664e6	−0.251086
$514$	−8.44359e6	−1.40968
$515$	−4.14854e6	−0.689251
$516$	2.04782e6	0.338586
$517$	6.53679e6	1.07557
$518$	−2.42805e6	−0.397588
$519$	−4.19458e6	−0.683549
$520$	−1.00589e6	−0.163133
$521$	−1.03603e6	−0.167216	−0.0836079	−	0.996499i	$-0.526644\pi$
$521$	−1.03603e6	−0.167216	−0.0836079	+	0.996499i	$0.526644\pi$
$522$	1.90220e6	0.305549
$523$	−732526.	−0.117103	−0.0585516	−	0.998284i	$-0.518648\pi$
$523$	−732526.	−0.117103	−0.0585516	+	0.998284i	$0.518648\pi$
$524$	2.32512e6	0.369928
$525$	2.43608e6	0.385739
$526$	−2.70377e6	−0.426094
$527$	−1.88417e6	−0.295524
$528$	695808.	0.108619
$529$	−6.43286e6	−0.999459
$530$	−2.89453e6	−0.447599
$531$	1.54483e6	0.237763
$532$	−1.60955e6	−0.246562
$533$	−439738.	−0.0670465
$534$	−1.51124e6	−0.229341
$535$	−8.60771e6	−1.30018
$536$	172416.	0.0259218
$537$	6.38086e6	0.954869
$538$	3.59086e6	0.534864
$539$	−725102.	−0.107505
$540$	−1.08475e6	−0.160083
$541$	−669764.	−0.0983849	−0.0491925	−	0.998789i	$-0.515665\pi$
$541$	−669764.	−0.0983849	−0.0491925	+	0.998789i	$0.515665\pi$
$542$	−8.66611e6	−1.26714
$543$	−2.89676e6	−0.421612
$544$	−499712.	−0.0723973
$545$	6.65936e6	0.960376
$546$	−298116.	−0.0427960
$547$	482197.	0.0689059	0.0344529	−	0.999406i	$-0.489031\pi$
$547$	482197.	0.0689059	0.0344529	+	0.999406i	$0.489031\pi$
$548$	6.48038e6	0.921827
$549$	1.13027e6	0.160049
$550$	−6.67299e6	−0.940619
$551$	1.20532e7	1.69131
$552$	−33984.0	−0.00474708
$553$	−1.39821e6	−0.194429
$554$	5.22624e6	0.723461
$555$	−1.03688e7	−1.42887
$556$	−362144.	−0.0496815
$557$	−328440.	−0.0448557	−0.0224279	−	0.999748i	$-0.507140\pi$
$557$	−328440.	−0.0448557	−0.0224279	+	0.999748i	$0.507140\pi$
$558$	1.25096e6	0.170082
$559$	2.40335e6	0.325302
$560$	−1.16659e6	−0.157199
$561$	−1.32638e6	−0.177935
$562$	6.59350e6	0.880592
$563$	9.51348e6	1.26494	0.632468	−	0.774587i	$-0.282042\pi$
$563$	9.51348e6	1.26494	0.632468	+	0.774587i	$0.282042\pi$
$564$	3.11688e6	0.412593
$565$	9.26066e6	1.22045
$566$	3.33646e6	0.437769
$567$	−321489.	−0.0419961
$568$	5.25005e6	0.682798
$569$	−1.43629e7	−1.85978	−0.929890	−	0.367837i	$-0.880099\pi$
$569$	−1.43629e7	−1.85978	−0.929890	+	0.367837i	$0.880099\pi$
$570$	−6.87344e6	−0.886109
$571$	−1.86893e6	−0.239885	−0.119943	−	0.992781i	$-0.538271\pi$
$571$	−1.86893e6	−0.239885	−0.119943	+	0.992781i	$0.538271\pi$
$572$	816608.	0.104357
$573$	3.48192e6	0.443029
$574$	−509992.	−0.0646076
$575$	325916.	0.0411089
$576$	331776.	0.0416667
$577$	−7.96326e6	−0.995753	−0.497876	−	0.867248i	$-0.665887\pi$
$577$	−7.96326e6	−0.995753	−0.497876	+	0.867248i	$0.665887\pi$
$578$	−4.72685e6	−0.588508
$579$	2.38369e6	0.295497
$580$	8.73605e6	1.07831
$581$	−735931.	−0.0904475
$582$	2.06658e6	0.252898
$583$	2.34986e6	0.286333
$584$	416192.	0.0504965
$585$	−1.27308e6	−0.153803
$586$	965748.	0.116177
$587$	−1.78965e6	−0.214375	−0.107187	−	0.994239i	$-0.534184\pi$
$587$	−1.78965e6	−0.214375	−0.107187	+	0.994239i	$0.534184\pi$
$588$	−345744.	−0.0412393
$589$	7.92663e6	0.941457
$590$	7.09478e6	0.839091
$591$	2.53856e6	0.298964
$592$	3.17133e6	0.371909
$593$	−4.82506e6	−0.563464	−0.281732	−	0.959493i	$-0.590909\pi$
$593$	−4.82506e6	−0.563464	−0.281732	+	0.959493i	$0.590909\pi$
$594$	880632.	0.102407
$595$	2.22382e6	0.257517
$596$	1.94493e6	0.224279
$597$	−7.85419e6	−0.901915
$598$	−39884.0	−0.00456085
$599$	−1.18825e7	−1.35313	−0.676567	−	0.736381i	$-0.736533\pi$
$599$	−1.18825e7	−1.35313	−0.676567	+	0.736381i	$0.736533\pi$
$600$	−3.18182e6	−0.360826
$601$	−1.04865e7	−1.18425	−0.592126	−	0.805845i	$-0.701711\pi$
$601$	−1.04865e7	−1.18425	−0.592126	+	0.805845i	$0.701711\pi$
$602$	2.78732e6	0.313469
$603$	218214.	0.0244393
$604$	6.19187e6	0.690605
$605$	−6.49577e6	−0.721510
$606$	−1.79647e6	−0.198719
$607$	2.11548e6	0.233043	0.116522	−	0.993188i	$-0.462826\pi$
$607$	2.11548e6	0.233043	0.116522	+	0.993188i	$0.462826\pi$
$608$	2.10227e6	0.230638
$609$	2.58911e6	0.282883
$610$	5.19089e6	0.564829
$611$	3.65800e6	0.396407
$612$	−632448.	−0.0682569
$613$	−1.09830e6	−0.118051	−0.0590257	−	0.998256i	$-0.518799\pi$
$613$	−1.09830e6	−0.118051	−0.0590257	+	0.998256i	$0.518799\pi$
$614$	6.70160e6	0.717393
$615$	−2.17787e6	−0.232191
$616$	947072.	0.100561
$617$	−1.45577e7	−1.53950	−0.769750	−	0.638345i	$-0.779619\pi$
$617$	−1.45577e7	−1.53950	−0.769750	+	0.638345i	$0.779619\pi$
$618$	1.60589e6	0.169139
$619$	−108428.	−0.0113740	−0.00568702	−	0.999984i	$-0.501810\pi$
$619$	−108428.	−0.0113740	−0.00568702	+	0.999984i	$0.501810\pi$
$620$	5.74517e6	0.600238
$621$	−43011.0	−0.00447559
$622$	6.55623e6	0.679483
$623$	−2.05697e6	−0.212328
$624$	389376.	0.0400320
$625$	3.48645e6	0.357013
$626$	−7.87817e6	−0.803507
$627$	5.58005e6	0.566852
$628$	−176128.	−0.0178209
$629$	−6.04534e6	−0.609248
$630$	−1.47647e6	−0.148208
$631$	8.40559e6	0.840417	0.420209	−	0.907428i	$-0.361957\pi$
$631$	8.40559e6	0.840417	0.420209	+	0.907428i	$0.361957\pi$
$632$	1.82624e6	0.181872
$633$	−1.06106e6	−0.105252
$634$	−4.90696e6	−0.484830
$635$	7.54900e6	0.742942
$636$	1.12046e6	0.109839
$637$	−405769.	−0.0396214
$638$	−7.09217e6	−0.689807
$639$	6.64459e6	0.643748
$640$	1.52371e6	0.147046
$641$	5.45817e6	0.524689	0.262345	−	0.964974i	$-0.415504\pi$
$641$	5.45817e6	0.524689	0.262345	+	0.964974i	$0.415504\pi$
$642$	3.33202e6	0.319058
$643$	1.36889e7	1.30569	0.652844	−	0.757492i	$-0.273576\pi$
$643$	1.36889e7	1.30569	0.652844	+	0.757492i	$0.273576\pi$
$644$	−46256.0	−0.00439494
$645$	1.19030e7	1.12657
$646$	−4.00746e6	−0.377822
$647$	6.58287e6	0.618236	0.309118	−	0.951024i	$-0.399966\pi$
$647$	6.58287e6	0.618236	0.309118	+	0.951024i	$0.399966\pi$
$648$	419904.	0.0392837
$649$	−5.75974e6	−0.536774
$650$	−3.73422e6	−0.346671
$651$	1.70270e6	0.157466
$652$	−6.32749e6	−0.582925
$653$	−1.66239e7	−1.52563	−0.762816	−	0.646616i	$-0.776184\pi$
$653$	−1.66239e7	−1.52563	−0.762816	+	0.646616i	$0.776184\pi$
$654$	−2.57782e6	−0.235672
$655$	1.35148e7	1.23085
$656$	666112.	0.0604349
$657$	526743.	0.0476086
$658$	4.24242e6	0.381987
$659$	−2.17810e7	−1.95373	−0.976864	−	0.213860i	$-0.931396\pi$
$659$	−2.17810e7	−1.95373	−0.976864	+	0.213860i	$0.931396\pi$
$660$	4.04438e6	0.361404
$661$	4.49538e6	0.400187	0.200093	−	0.979777i	$-0.435875\pi$
$661$	4.49538e6	0.400187	0.200093	+	0.979777i	$0.435875\pi$
$662$	−9.65708e6	−0.856448
$663$	−742248.	−0.0655791
$664$	961216.	0.0846059
$665$	−9.35552e6	−0.820378
$666$	4.01371e6	0.350639
$667$	346389.	0.0301474
$668$	−2.83659e6	−0.245955
$669$	6.77312e6	0.585091
$670$	1.00217e6	0.0862489
$671$	−4.21411e6	−0.361326
$672$	451584.	0.0385758
$673$	−1.50204e7	−1.27833	−0.639164	−	0.769070i	$-0.720720\pi$
$673$	−1.50204e7	−1.27833	−0.639164	+	0.769070i	$0.720720\pi$
$674$	2.47127e6	0.209542
$675$	−4.02700e6	−0.340190
$676$	456976.	0.0384615
$677$	−3.20280e6	−0.268571	−0.134285	−	0.990943i	$-0.542874\pi$
$677$	−3.20280e6	−0.268571	−0.134285	+	0.990943i	$0.542874\pi$
$678$	−3.58477e6	−0.299493
$679$	2.81284e6	0.234138
$680$	−2.90458e6	−0.240885
$681$	−4.36331e6	−0.360536
$682$	−4.66409e6	−0.383978
$683$	−1.43886e7	−1.18023	−0.590115	−	0.807319i	$-0.700918\pi$
$683$	−1.43886e7	−1.18023	−0.590115	+	0.807319i	$0.700918\pi$
$684$	2.66069e6	0.217447
$685$	3.76672e7	3.06717
$686$	−470596.	−0.0381802
$687$	3.57971e6	0.289372
$688$	−3.64058e6	−0.293224
$689$	1.31499e6	0.105530
$690$	−197532.	−0.0157948
$691$	2.87339e6	0.228928	0.114464	−	0.993427i	$-0.463485\pi$
$691$	2.87339e6	0.228928	0.114464	+	0.993427i	$0.463485\pi$
$692$	7.45702e6	0.591971
$693$	1.19864e6	0.0948102
$694$	−7.48307e6	−0.589768
$695$	−2.10496e6	−0.165304
$696$	−3.38170e6	−0.264613
$697$	−1.26978e6	−0.0990023
$698$	−1.50333e7	−1.16793
$699$	2.45287e6	0.189881
$700$	−4.33082e6	−0.334060
$701$	4.93926e6	0.379635	0.189818	−	0.981819i	$-0.439210\pi$
$701$	4.93926e6	0.379635	0.189818	+	0.981819i	$0.439210\pi$
$702$	492804.	0.0377426
$703$	2.54326e7	1.94089
$704$	−1.23699e6	−0.0940666
$705$	1.81169e7	1.37281
$706$	563096.	0.0425178
$707$	−2.44520e6	−0.183978
$708$	−2.74637e6	−0.205909
$709$	6.77459e6	0.506136	0.253068	−	0.967448i	$-0.418560\pi$
$709$	6.77459e6	0.506136	0.253068	+	0.967448i	$0.418560\pi$
$710$	3.05159e7	2.27185
$711$	2.31134e6	0.171470
$712$	2.68666e6	0.198615
$713$	227799.	0.0167814
$714$	−860832.	−0.0631936
$715$	4.74653e6	0.347225
$716$	−1.13438e7	−0.826941
$717$	7.37006e6	0.535394
$718$	−7.13270e6	−0.516349
$719$	5.84618e6	0.421745	0.210872	−	0.977514i	$-0.432370\pi$
$719$	5.84618e6	0.421745	0.210872	+	0.977514i	$0.432370\pi$
$720$	1.92845e6	0.138636
$721$	2.18579e6	0.156592
$722$	6.95484e6	0.496528
$723$	−1.21302e7	−0.863019
$724$	5.14979e6	0.365126
$725$	3.24314e7	2.29150
$726$	2.51449e6	0.177055
$727$	2.15115e7	1.50951	0.754754	−	0.656008i	$-0.227756\pi$
$727$	2.15115e7	1.50951	0.754754	+	0.656008i	$0.227756\pi$
$728$	529984.	0.0370625
$729$	531441.	0.0370370
$730$	2.41912e6	0.168016
$731$	6.93985e6	0.480349
$732$	−2.00938e6	−0.138606
$733$	1.45662e7	1.00135	0.500677	−	0.865634i	$-0.333084\pi$
$733$	1.45662e7	1.00135	0.500677	+	0.865634i	$0.333084\pi$
$734$	−1.23140e7	−0.843644
$735$	−2.00964e6	−0.137214
$736$	60416.0	0.00411109
$737$	−813588.	−0.0551742
$738$	843048.	0.0569786
$739$	−5.48841e6	−0.369688	−0.184844	−	0.982768i	$-0.559178\pi$
$739$	−5.48841e6	−0.369688	−0.184844	+	0.982768i	$0.559178\pi$
$740$	1.84333e7	1.23744
$741$	3.12261e6	0.208916
$742$	1.52508e6	0.101691
$743$	7.56646e6	0.502829	0.251415	−	0.967879i	$-0.419104\pi$
$743$	7.56646e6	0.502829	0.251415	+	0.967879i	$0.419104\pi$
$744$	−2.22394e6	−0.147296
$745$	1.13049e7	0.746235
$746$	−1.38057e7	−0.908265
$747$	1.21654e6	0.0797672
$748$	2.35802e6	0.154096
$749$	4.53524e6	0.295390
$750$	−8.03185e6	−0.521390
$751$	1.63386e7	1.05710	0.528549	−	0.848903i	$-0.322736\pi$
$751$	1.63386e7	1.05710	0.528549	+	0.848903i	$0.322736\pi$
$752$	−5.54112e6	−0.357316
$753$	−3.81083e6	−0.244925
$754$	−3.96880e6	−0.254232
$755$	3.59903e7	2.29783
$756$	571536.	0.0363696
$757$	5.90963e6	0.374818	0.187409	−	0.982282i	$-0.439991\pi$
$757$	5.90963e6	0.374818	0.187409	+	0.982282i	$0.439991\pi$
$758$	−9.99690e6	−0.631964
$759$	160362.	0.0101041
$760$	1.22195e7	0.767393
$761$	2.58402e7	1.61746	0.808730	−	0.588180i	$-0.200155\pi$
$761$	2.58402e7	1.61746	0.808730	+	0.588180i	$0.200155\pi$
$762$	−2.92219e6	−0.182315
$763$	−3.50869e6	−0.218190
$764$	−6.19008e6	−0.383674
$765$	−3.67610e6	−0.227109
$766$	1.57859e7	0.972072
$767$	−3.22317e6	−0.197831
$768$	−589824.	−0.0360844
$769$	6.11750e6	0.373042	0.186521	−	0.982451i	$-0.440279\pi$
$769$	6.11750e6	0.373042	0.186521	+	0.982451i	$0.440279\pi$
$770$	5.50486e6	0.334595
$771$	1.89981e7	1.15100
$772$	−4.23766e6	−0.255908
$773$	−1.45256e6	−0.0874351	−0.0437176	−	0.999044i	$-0.513920\pi$
$773$	−1.45256e6	−0.0874351	−0.0437176	+	0.999044i	$0.513920\pi$
$774$	−4.60760e6	−0.276454
$775$	2.13282e7	1.27556
$776$	−3.67392e6	−0.219016
$777$	5.46311e6	0.324629
$778$	−1.62266e7	−0.961122
$779$	5.34191e6	0.315393
$780$	2.26325e6	0.133197
$781$	−2.47737e7	−1.45332
$782$	−115168.	−0.00673465
$783$	−4.27996e6	−0.249480
$784$	614656.	0.0357143
$785$	−1.02374e6	−0.0592949
$786$	−5.23152e6	−0.302045
$787$	−2.84829e7	−1.63926	−0.819630	−	0.572893i	$-0.805821\pi$
$787$	−2.84829e7	−1.63926	−0.819630	+	0.572893i	$0.805821\pi$
$788$	−4.51299e6	−0.258910
$789$	6.08349e6	0.347905
$790$	1.06150e7	0.605136
$791$	−4.87927e6	−0.277277
$792$	−1.56557e6	−0.0886868
$793$	−2.35823e6	−0.133169
$794$	1.33979e7	0.754197
$795$	6.51270e6	0.365463
$796$	1.39630e7	0.781081
$797$	3.29521e6	0.183754	0.0918770	−	0.995770i	$-0.470713\pi$
$797$	3.29521e6	0.183754	0.0918770	+	0.995770i	$0.470713\pi$
$798$	3.62149e6	0.201317
$799$	1.05628e7	0.585343
$800$	5.65658e6	0.312485
$801$	3.40030e6	0.187256
$802$	2.40083e7	1.31803
$803$	−1.96391e6	−0.107481
$804$	−387936.	−0.0211651
$805$	−268863.	−0.0146232
$806$	−2.61004e6	−0.141517
$807$	−8.07944e6	−0.436715
$808$	3.19373e6	0.172096
$809$	−1.02681e7	−0.551590	−0.275795	−	0.961216i	$-0.588941\pi$
$809$	−1.02681e7	−0.551590	−0.275795	+	0.961216i	$0.588941\pi$
$810$	2.44069e6	0.130707
$811$	−3.00578e7	−1.60474	−0.802371	−	0.596825i	$-0.796429\pi$
$811$	−3.00578e7	−1.60474	−0.802371	+	0.596825i	$0.796429\pi$
$812$	−4.60286e6	−0.244984
$813$	1.94988e7	1.03462
$814$	−1.49647e7	−0.791602
$815$	−3.67785e7	−1.93955
$816$	1.12435e6	0.0591122
$817$	−2.91957e7	−1.53026
$818$	8.64979e6	0.451983
$819$	670761.	0.0349428
$820$	3.87178e6	0.201083
$821$	3.43710e7	1.77965	0.889825	−	0.456302i	$-0.150826\pi$
$821$	3.43710e7	1.77965	0.889825	+	0.456302i	$0.150826\pi$
$822$	−1.45809e7	−0.752668
$823$	2.95480e7	1.52065	0.760324	−	0.649544i	$-0.225040\pi$
$823$	2.95480e7	1.52065	0.760324	+	0.649544i	$0.225040\pi$
$824$	−2.85491e6	−0.146479
$825$	1.50142e7	0.768012
$826$	−3.73811e6	−0.190635
$827$	1.04233e7	0.529960	0.264980	−	0.964254i	$-0.414635\pi$
$827$	1.04233e7	0.529960	0.264980	+	0.964254i	$0.414635\pi$
$828$	76464.0	0.00387598
$829$	7.34305e6	0.371099	0.185550	−	0.982635i	$-0.440593\pi$
$829$	7.34305e6	0.371099	0.185550	+	0.982635i	$0.440593\pi$
$830$	5.58707e6	0.281507
$831$	−1.17590e7	−0.590704
$832$	−692224.	−0.0346688
$833$	−1.17169e6	−0.0585059
$834$	814824.	0.0405647
$835$	−1.64877e7	−0.818359
$836$	−9.92010e6	−0.490908
$837$	−2.81467e6	−0.138872
$838$	−1.99779e7	−0.982744
$839$	1.74785e7	0.857233	0.428617	−	0.903486i	$-0.359001\pi$
$839$	1.74785e7	0.857233	0.428617	+	0.903486i	$0.359001\pi$
$840$	2.62483e6	0.128352
$841$	1.39575e7	0.680483
$842$	184160.	0.00895190
$843$	−1.48354e7	−0.719001
$844$	1.88632e6	0.0911505
$845$	2.65617e6	0.127972
$846$	−7.01298e6	−0.336881
$847$	3.42250e6	0.163921
$848$	−1.99194e6	−0.0951230
$849$	−7.50704e6	−0.357437
$850$	−1.07828e7	−0.511901
$851$	730892.	0.0345963
$852$	−1.18126e7	−0.557502
$853$	−3.15389e7	−1.48414	−0.742069	−	0.670324i	$-0.766155\pi$
$853$	−3.15389e7	−1.48414	−0.742069	+	0.670324i	$0.766155\pi$
$854$	−2.73498e6	−0.128325
$855$	1.54652e7	0.723505
$856$	−5.92358e6	−0.276312
$857$	1.09861e7	0.510965	0.255482	−	0.966814i	$-0.417766\pi$
$857$	1.09861e7	0.510965	0.255482	+	0.966814i	$0.417766\pi$
$858$	−1.83737e6	−0.0852075
$859$	3.42347e6	0.158301	0.0791504	−	0.996863i	$-0.474779\pi$
$859$	3.42347e6	0.158301	0.0791504	+	0.996863i	$0.474779\pi$
$860$	−2.11608e7	−0.975634
$861$	1.14748e6	0.0527519
$862$	−2.78571e7	−1.27693
$863$	−1.85496e7	−0.847826	−0.423913	−	0.905703i	$-0.639344\pi$
$863$	−1.85496e7	−0.847826	−0.423913	+	0.905703i	$0.639344\pi$
$864$	−746496.	−0.0340207
$865$	4.33440e7	1.96965
$866$	2.24571e7	1.01756
$867$	1.06354e7	0.480515
$868$	−3.02702e6	−0.136369
$869$	−8.61757e6	−0.387111
$870$	−1.96561e7	−0.880439
$871$	−455286.	−0.0203348
$872$	4.58278e6	0.204098
$873$	−4.64980e6	−0.206490
$874$	484508.	0.0214547
$875$	−1.09322e7	−0.482713
$876$	−936432.	−0.0412303
$877$	1.97177e7	0.865680	0.432840	−	0.901471i	$-0.357512\pi$
$877$	1.97177e7	0.865680	0.432840	+	0.901471i	$0.357512\pi$
$878$	7.53342e6	0.329804
$879$	−2.17293e6	−0.0948580
$880$	−7.19002e6	−0.312985
$881$	3.30056e7	1.43268	0.716338	−	0.697754i	$-0.245817\pi$
$881$	3.30056e7	1.43268	0.716338	+	0.697754i	$0.245817\pi$
$882$	777924.	0.0336718
$883$	1.31820e7	0.568958	0.284479	−	0.958682i	$-0.408179\pi$
$883$	1.31820e7	0.568958	0.284479	+	0.958682i	$0.408179\pi$
$884$	1.31955e6	0.0567931
$885$	−1.59633e7	−0.685115
$886$	−2.79064e7	−1.19432
$887$	2.56309e7	1.09384	0.546921	−	0.837184i	$-0.315800\pi$
$887$	2.56309e7	1.09384	0.546921	+	0.837184i	$0.315800\pi$
$888$	−7.13549e6	−0.303663
$889$	−3.97743e6	−0.168790
$890$	1.56162e7	0.660846
$891$	−1.98142e6	−0.0836147
$892$	−1.20411e7	−0.506704
$893$	−4.44372e7	−1.86474
$894$	−4.37609e6	−0.183123
$895$	−6.59356e7	−2.75145
$896$	−802816.	−0.0334077
$897$	89739.0	0.00372392
$898$	5.68350e6	0.235194
$899$	2.26679e7	0.935433
$900$	7.15910e6	0.294613
$901$	3.79713e6	0.155827
$902$	−3.14322e6	−0.128635
$903$	−6.27146e6	−0.255947
$904$	6.37293e6	0.259369
$905$	2.99332e7	1.21487
$906$	−1.39317e7	−0.563876
$907$	3.17787e6	0.128268	0.0641339	−	0.997941i	$-0.479572\pi$
$907$	3.17787e6	0.128268	0.0641339	+	0.997941i	$0.479572\pi$
$908$	7.75699e6	0.312233
$909$	4.04206e6	0.162253
$910$	3.08053e6	0.123317
$911$	−1.99453e7	−0.796240	−0.398120	−	0.917333i	$-0.630337\pi$
$911$	−1.99453e7	−0.796240	−0.398120	+	0.917333i	$0.630337\pi$
$912$	−4.73011e6	−0.188315
$913$	−4.53574e6	−0.180082
$914$	721088.	0.0285511
$915$	−1.16795e7	−0.461181
$916$	−6.36394e6	−0.250603
$917$	−7.12068e6	−0.279639
$918$	1.42301e6	0.0557315
$919$	−4.09094e7	−1.59784	−0.798922	−	0.601434i	$-0.794596\pi$
$919$	−4.09094e7	−1.59784	−0.798922	+	0.601434i	$0.794596\pi$
$920$	351168.	0.0136787
$921$	−1.50786e7	−0.585749
$922$	−1.43633e7	−0.556451
$923$	−1.38634e7	−0.535631
$924$	−2.13091e6	−0.0821080
$925$	6.84313e7	2.62967
$926$	2.12639e7	0.814920
$927$	−3.61325e6	−0.138102
$928$	6.01190e6	0.229162
$929$	1.07776e7	0.409714	0.204857	−	0.978792i	$-0.434327\pi$
$929$	1.07776e7	0.409714	0.204857	+	0.978792i	$0.434327\pi$
$930$	−1.29266e7	−0.490092
$931$	4.92925e6	0.186383
$932$	−4.36066e6	−0.164442
$933$	−1.47515e7	−0.554795
$934$	−1.10926e7	−0.416071
$935$	1.37060e7	0.512721
$936$	−876096.	−0.0326860
$937$	−4.00147e7	−1.48892	−0.744459	−	0.667668i	$-0.767293\pi$
$937$	−4.00147e7	−1.48892	−0.744459	+	0.667668i	$0.767293\pi$
$938$	−528024.	−0.0195951
$939$	1.77259e7	0.656060
$940$	−3.22078e7	−1.18889
$941$	−2.44799e7	−0.901230	−0.450615	−	0.892718i	$-0.648795\pi$
$941$	−2.44799e7	−0.901230	−0.450615	+	0.892718i	$0.648795\pi$
$942$	396288.	0.0145507
$943$	153518.	0.00562186
$944$	4.88243e6	0.178323
$945$	3.32205e6	0.121012
$946$	1.71790e7	0.624122
$947$	−4.00398e7	−1.45083	−0.725416	−	0.688311i	$-0.758352\pi$
$947$	−4.00398e7	−1.45083	−0.725416	+	0.688311i	$0.758352\pi$
$948$	−4.10904e6	−0.148498
$949$	−1.09901e6	−0.0396127
$950$	4.53631e7	1.63077
$951$	1.10407e7	0.395862
$952$	1.53037e6	0.0547272
$953$	−2.17862e7	−0.777051	−0.388526	−	0.921438i	$-0.627015\pi$
$953$	−2.17862e7	−0.777051	−0.388526	+	0.921438i	$0.627015\pi$
$954$	−2.52104e6	−0.0896829
$955$	−3.59798e7	−1.27659
$956$	−1.31023e7	−0.463665
$957$	1.59574e7	0.563225
$958$	1.48974e7	0.524440
$959$	−1.98462e7	−0.696836
$960$	−3.42835e6	−0.120062
$961$	−1.37218e7	−0.479296
$962$	−8.37429e6	−0.291750
$963$	−7.49704e6	−0.260510
$964$	2.15647e7	0.747397
$965$	−2.46314e7	−0.851474
$966$	104076.	0.00358846
$967$	3.66362e7	1.25992	0.629962	−	0.776626i	$-0.283071\pi$
$967$	3.66362e7	1.25992	0.629962	+	0.776626i	$0.283071\pi$
$968$	−4.47021e6	−0.153334
$969$	9.01678e6	0.308491
$970$	−2.13547e7	−0.728724
$971$	−4.65132e7	−1.58317	−0.791587	−	0.611057i	$-0.790745\pi$
$971$	−4.65132e7	−1.58317	−0.791587	+	0.611057i	$0.790745\pi$
$972$	−944784.	−0.0320750
$973$	1.10907e6	0.0375557
$974$	3.83065e7	1.29382
$975$	8.40200e6	0.283055
$976$	3.57222e6	0.120037
$977$	5.12581e7	1.71801	0.859006	−	0.511966i	$-0.171082\pi$
$977$	5.12581e7	1.71801	0.859006	+	0.511966i	$0.171082\pi$
$978$	1.42368e7	0.475956
$979$	−1.26777e7	−0.422749
$980$	3.57269e6	0.118831
$981$	5.80009e6	0.192425
$982$	−3.48633e7	−1.15369
$983$	7.49074e6	0.247253	0.123626	−	0.992329i	$-0.460548\pi$
$983$	7.49074e6	0.247253	0.123626	+	0.992329i	$0.460548\pi$
$984$	−1.49875e6	−0.0493449
$985$	−2.62318e7	−0.861464
$986$	−1.14602e7	−0.375405
$987$	−9.54544e6	−0.311891
$988$	−5.55131e6	−0.180927
$989$	−839039.	−0.0272767
$990$	−9.09986e6	−0.295085
$991$	−1.00629e6	−0.0325492	−0.0162746	−	0.999868i	$-0.505181\pi$
$991$	−1.00629e6	−0.0325492	−0.0162746	+	0.999868i	$0.505181\pi$
$992$	3.95366e6	0.127562
$993$	2.17284e7	0.699287
$994$	−1.60783e7	−0.516147
$995$	8.11600e7	2.59887
$996$	−2.16274e6	−0.0690805
$997$	−5.15011e7	−1.64089	−0.820443	−	0.571728i	$-0.806273\pi$
$997$	−5.15011e7	−1.64089	−0.820443	+	0.571728i	$0.806273\pi$
$998$	−3.23235e6	−0.102729
$999$	−9.03085e6	−0.286296

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.6.a.e.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.6.a.e.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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