LMFDB - Embedded newform 546.6.a.c.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)

$f(q)$

$=$

$q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -61.0000 q^{5} -36.0000 q^{6} -49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -61.0000 q^{5} -36.0000 q^{6} -49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -244.000 q^{10} +181.000 q^{11} -144.000 q^{12} -169.000 q^{13} -196.000 q^{14} +549.000 q^{15} +256.000 q^{16} -1965.00 q^{17} +324.000 q^{18} +2193.00 q^{19} -976.000 q^{20} +441.000 q^{21} +724.000 q^{22} -4015.00 q^{23} -576.000 q^{24} +596.000 q^{25} -676.000 q^{26} -729.000 q^{27} -784.000 q^{28} -6841.00 q^{29} +2196.00 q^{30} +8992.00 q^{31} +1024.00 q^{32} -1629.00 q^{33} -7860.00 q^{34} +2989.00 q^{35} +1296.00 q^{36} -9753.00 q^{37} +8772.00 q^{38} +1521.00 q^{39} -3904.00 q^{40} -9900.00 q^{41} +1764.00 q^{42} +13975.0 q^{43} +2896.00 q^{44} -4941.00 q^{45} -16060.0 q^{46} +18808.0 q^{47} -2304.00 q^{48} +2401.00 q^{49} +2384.00 q^{50} +17685.0 q^{51} -2704.00 q^{52} +2082.00 q^{53} -2916.00 q^{54} -11041.0 q^{55} -3136.00 q^{56} -19737.0 q^{57} -27364.0 q^{58} +31700.0 q^{59} +8784.00 q^{60} +21577.0 q^{61} +35968.0 q^{62} -3969.00 q^{63} +4096.00 q^{64} +10309.0 q^{65} -6516.00 q^{66} -49694.0 q^{67} -31440.0 q^{68} +36135.0 q^{69} +11956.0 q^{70} +53500.0 q^{71} +5184.00 q^{72} +60137.0 q^{73} -39012.0 q^{74} -5364.00 q^{75} +35088.0 q^{76} -8869.00 q^{77} +6084.00 q^{78} +17678.0 q^{79} -15616.0 q^{80} +6561.00 q^{81} -39600.0 q^{82} +80658.0 q^{83} +7056.00 q^{84} +119865. q^{85} +55900.0 q^{86} +61569.0 q^{87} +11584.0 q^{88} +81690.0 q^{89} -19764.0 q^{90} +8281.00 q^{91} -64240.0 q^{92} -80928.0 q^{93} +75232.0 q^{94} -133773. q^{95} -9216.00 q^{96} +38054.0 q^{97} +9604.00 q^{98} +14661.0 q^{99} +O(q^{100})\)

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
$5$	−61.0000	−1.09120	−0.545601	−	0.838045i	$-0.683698\pi$
$5$	−61.0000	−1.09120	−0.545601	+	0.838045i	$0.683698\pi$
$6$	−36.0000	−0.408248
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	−244.000	−0.771596
$11$	181.000	0.451021	0.225511	−	0.974241i	$-0.427595\pi$
$11$	181.000	0.451021	0.225511	+	0.974241i	$0.427595\pi$
$12$	−144.000	−0.288675
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	−196.000	−0.267261
$15$	549.000	0.630005
$16$	256.000	0.250000
$17$	−1965.00	−1.64907	−0.824537	−	0.565808i	$-0.808564\pi$
$17$	−1965.00	−1.64907	−0.824537	+	0.565808i	$0.808564\pi$
$18$	324.000	0.235702
$19$	2193.00	1.39365	0.696826	−	0.717240i	$-0.254595\pi$
$19$	2193.00	1.39365	0.696826	+	0.717240i	$0.254595\pi$
$20$	−976.000	−0.545601
$21$	441.000	0.218218
$22$	724.000	0.318920
$23$	−4015.00	−1.58258	−0.791291	−	0.611440i	$-0.790590\pi$
$23$	−4015.00	−1.58258	−0.791291	+	0.611440i	$0.790590\pi$
$24$	−576.000	−0.204124
$25$	596.000	0.190720
$26$	−676.000	−0.196116
$27$	−729.000	−0.192450
$28$	−784.000	−0.188982
$29$	−6841.00	−1.51051	−0.755257	−	0.655429i	$-0.772488\pi$
$29$	−6841.00	−1.51051	−0.755257	+	0.655429i	$0.772488\pi$
$30$	2196.00	0.445481
$31$	8992.00	1.68055	0.840276	−	0.542159i	$-0.182393\pi$
$31$	8992.00	1.68055	0.840276	+	0.542159i	$0.182393\pi$
$32$	1024.00	0.176777
$33$	−1629.00	−0.260397
$34$	−7860.00	−1.16607
$35$	2989.00	0.412435
$36$	1296.00	0.166667
$37$	−9753.00	−1.17121	−0.585604	−	0.810597i	$-0.699143\pi$
$37$	−9753.00	−1.17121	−0.585604	+	0.810597i	$0.699143\pi$
$38$	8772.00	0.985461
$39$	1521.00	0.160128
$40$	−3904.00	−0.385798
$41$	−9900.00	−0.919762	−0.459881	−	0.887981i	$-0.652108\pi$
$41$	−9900.00	−0.919762	−0.459881	+	0.887981i	$0.652108\pi$
$42$	1764.00	0.154303
$43$	13975.0	1.15261	0.576303	−	0.817236i	$-0.304495\pi$
$43$	13975.0	1.15261	0.576303	+	0.817236i	$0.304495\pi$
$44$	2896.00	0.225511
$45$	−4941.00	−0.363734
$46$	−16060.0	−1.11905
$47$	18808.0	1.24193	0.620966	−	0.783837i	$-0.286740\pi$
$47$	18808.0	1.24193	0.620966	+	0.783837i	$0.286740\pi$
$48$	−2304.00	−0.144338
$49$	2401.00	0.142857
$50$	2384.00	0.134859
$51$	17685.0	0.952094
$52$	−2704.00	−0.138675
$53$	2082.00	0.101810	0.0509051	−	0.998703i	$-0.483789\pi$
$53$	2082.00	0.101810	0.0509051	+	0.998703i	$0.483789\pi$
$54$	−2916.00	−0.136083
$55$	−11041.0	−0.492155
$56$	−3136.00	−0.133631
$57$	−19737.0	−0.804626
$58$	−27364.0	−1.06809
$59$	31700.0	1.18558	0.592788	−	0.805359i	$-0.298027\pi$
$59$	31700.0	1.18558	0.592788	+	0.805359i	$0.298027\pi$
$60$	8784.00	0.315003
$61$	21577.0	0.742449	0.371224	−	0.928543i	$-0.378938\pi$
$61$	21577.0	0.742449	0.371224	+	0.928543i	$0.378938\pi$
$62$	35968.0	1.18833
$63$	−3969.00	−0.125988
$64$	4096.00	0.125000
$65$	10309.0	0.302645
$66$	−6516.00	−0.184129
$67$	−49694.0	−1.35244	−0.676218	−	0.736701i	$-0.736382\pi$
$67$	−49694.0	−1.35244	−0.676218	+	0.736701i	$0.736382\pi$
$68$	−31440.0	−0.824537
$69$	36135.0	0.913704
$70$	11956.0	0.291636
$71$	53500.0	1.25953	0.629764	−	0.776786i	$-0.283152\pi$
$71$	53500.0	1.25953	0.629764	+	0.776786i	$0.283152\pi$
$72$	5184.00	0.117851
$73$	60137.0	1.32079	0.660396	−	0.750917i	$-0.270388\pi$
$73$	60137.0	1.32079	0.660396	+	0.750917i	$0.270388\pi$
$74$	−39012.0	−0.828169
$75$	−5364.00	−0.110112
$76$	35088.0	0.696826
$77$	−8869.00	−0.170470
$78$	6084.00	0.113228
$79$	17678.0	0.318688	0.159344	−	0.987223i	$-0.449062\pi$
$79$	17678.0	0.318688	0.159344	+	0.987223i	$0.449062\pi$
$80$	−15616.0	−0.272800
$81$	6561.00	0.111111
$82$	−39600.0	−0.650370
$83$	80658.0	1.28515	0.642573	−	0.766225i	$-0.277867\pi$
$83$	80658.0	1.28515	0.642573	+	0.766225i	$0.277867\pi$
$84$	7056.00	0.109109
$85$	119865.	1.79947
$86$	55900.0	0.815015
$87$	61569.0	0.872095
$88$	11584.0	0.159460
$89$	81690.0	1.09319	0.546593	−	0.837399i	$-0.315925\pi$
$89$	81690.0	1.09319	0.546593	+	0.837399i	$0.315925\pi$
$90$	−19764.0	−0.257199
$91$	8281.00	0.104828
$92$	−64240.0	−0.791291
$93$	−80928.0	−0.970267
$94$	75232.0	0.878179
$95$	−133773.	−1.52076
$96$	−9216.00	−0.102062
$97$	38054.0	0.410649	0.205325	−	0.978694i	$-0.434175\pi$
$97$	38054.0	0.410649	0.205325	+	0.978694i	$0.434175\pi$
$98$	9604.00	0.101015
$99$	14661.0	0.150340
$100$	9536.00	0.0953600
$101$	−39404.0	−0.384359	−0.192179	−	0.981360i	$-0.561556\pi$
$101$	−39404.0	−0.384359	−0.192179	+	0.981360i	$0.561556\pi$
$102$	70740.0	0.673232
$103$	134557.	1.24972	0.624861	−	0.780736i	$-0.285156\pi$
$103$	134557.	1.24972	0.624861	+	0.780736i	$0.285156\pi$
$104$	−10816.0	−0.0980581
$105$	−26901.0	−0.238120
$106$	8328.00	0.0719906
$107$	−111946.	−0.945255	−0.472628	−	0.881262i	$-0.656694\pi$
$107$	−111946.	−0.945255	−0.472628	+	0.881262i	$0.656694\pi$
$108$	−11664.0	−0.0962250
$109$	−43019.0	−0.346812	−0.173406	−	0.984850i	$-0.555477\pi$
$109$	−43019.0	−0.346812	−0.173406	+	0.984850i	$0.555477\pi$
$110$	−44164.0	−0.348006
$111$	87777.0	0.676197
$112$	−12544.0	−0.0944911
$113$	−12262.0	−0.0903369	−0.0451685	−	0.998979i	$-0.514382\pi$
$113$	−12262.0	−0.0903369	−0.0451685	+	0.998979i	$0.514382\pi$
$114$	−78948.0	−0.568956
$115$	244915.	1.72691
$116$	−109456.	−0.755257
$117$	−13689.0	−0.0924500
$118$	126800.	0.838329
$119$	96285.0	0.623292
$120$	35136.0	0.222741
$121$	−128290.	−0.796580
$122$	86308.0	0.524991
$123$	89100.0	0.531025
$124$	143872.	0.840276
$125$	154269.	0.883087
$126$	−15876.0	−0.0890871
$127$	−25382.0	−0.139642	−0.0698211	−	0.997560i	$-0.522243\pi$
$127$	−25382.0	−0.139642	−0.0698211	+	0.997560i	$0.522243\pi$
$128$	16384.0	0.0883883
$129$	−125775.	−0.665457
$130$	41236.0	0.214002
$131$	−19033.0	−0.0969012	−0.0484506	−	0.998826i	$-0.515428\pi$
$131$	−19033.0	−0.0969012	−0.0484506	+	0.998826i	$0.515428\pi$
$132$	−26064.0	−0.130199
$133$	−107457.	−0.526751
$134$	−198776.	−0.956317
$135$	44469.0	0.210002
$136$	−125760.	−0.583036
$137$	68079.0	0.309893	0.154946	−	0.987923i	$-0.450479\pi$
$137$	68079.0	0.309893	0.154946	+	0.987923i	$0.450479\pi$
$138$	144540.	0.646086
$139$	122980.	0.539880	0.269940	−	0.962877i	$-0.412996\pi$
$139$	122980.	0.539880	0.269940	+	0.962877i	$0.412996\pi$
$140$	47824.0	0.206218
$141$	−169272.	−0.717030
$142$	214000.	0.890621
$143$	−30589.0	−0.125091
$144$	20736.0	0.0833333
$145$	417301.	1.64827
$146$	240548.	0.933941
$147$	−21609.0	−0.0824786
$148$	−156048.	−0.585604
$149$	77976.0	0.287737	0.143868	−	0.989597i	$-0.454046\pi$
$149$	77976.0	0.287737	0.143868	+	0.989597i	$0.454046\pi$
$150$	−21456.0	−0.0778611
$151$	25911.0	0.0924787	0.0462394	−	0.998930i	$-0.485276\pi$
$151$	25911.0	0.0924787	0.0462394	+	0.998930i	$0.485276\pi$
$152$	140352.	0.492731
$153$	−159165.	−0.549691
$154$	−35476.0	−0.120540
$155$	−548512.	−1.83382
$156$	24336.0	0.0800641
$157$	−318931.	−1.03264	−0.516318	−	0.856397i	$-0.672698\pi$
$157$	−318931.	−1.03264	−0.516318	+	0.856397i	$0.672698\pi$
$158$	70712.0	0.225346
$159$	−18738.0	−0.0587801
$160$	−62464.0	−0.192899
$161$	196735.	0.598159
$162$	26244.0	0.0785674
$163$	539690.	1.59102	0.795509	−	0.605941i	$-0.207203\pi$
$163$	539690.	1.59102	0.795509	+	0.605941i	$0.207203\pi$
$164$	−158400.	−0.459881
$165$	99369.0	0.284146
$166$	322632.	0.908735
$167$	−303525.	−0.842177	−0.421088	−	0.907020i	$-0.638352\pi$
$167$	−303525.	−0.842177	−0.421088	+	0.907020i	$0.638352\pi$
$168$	28224.0	0.0771517
$169$	28561.0	0.0769231
$170$	479460.	1.27242
$171$	177633.	0.464551
$172$	223600.	0.576303
$173$	−329374.	−0.836708	−0.418354	−	0.908284i	$-0.637393\pi$
$173$	−329374.	−0.836708	−0.418354	+	0.908284i	$0.637393\pi$
$174$	246276.	0.616665
$175$	−29204.0	−0.0720854
$176$	46336.0	0.112755
$177$	−285300.	−0.684492
$178$	326760.	0.772999
$179$	594576.	1.38699	0.693497	−	0.720459i	$-0.256069\pi$
$179$	594576.	1.38699	0.693497	+	0.720459i	$0.256069\pi$
$180$	−79056.0	−0.181867
$181$	−15930.0	−0.0361426	−0.0180713	−	0.999837i	$-0.505753\pi$
$181$	−15930.0	−0.0361426	−0.0180713	+	0.999837i	$0.505753\pi$
$182$	33124.0	0.0741249
$183$	−194193.	−0.428653
$184$	−256960.	−0.559527
$185$	594933.	1.27802
$186$	−323712.	−0.686083
$187$	−355665.	−0.743767
$188$	300928.	0.620966
$189$	35721.0	0.0727393
$190$	−535092.	−1.07534
$191$	377905.	0.749548	0.374774	−	0.927116i	$-0.377720\pi$
$191$	377905.	0.749548	0.374774	+	0.927116i	$0.377720\pi$
$192$	−36864.0	−0.0721688
$193$	364208.	0.703811	0.351906	−	0.936035i	$-0.385534\pi$
$193$	364208.	0.703811	0.351906	+	0.936035i	$0.385534\pi$
$194$	152216.	0.290373
$195$	−92781.0	−0.174732
$196$	38416.0	0.0714286
$197$	577692.	1.06055	0.530275	−	0.847826i	$-0.322089\pi$
$197$	577692.	1.06055	0.530275	+	0.847826i	$0.322089\pi$
$198$	58644.0	0.106307
$199$	366119.	0.655375	0.327687	−	0.944786i	$-0.393731\pi$
$199$	366119.	0.655375	0.327687	+	0.944786i	$0.393731\pi$
$200$	38144.0	0.0674297
$201$	447246.	0.780830
$202$	−157616.	−0.271783
$203$	335209.	0.570920
$204$	282960.	0.476047
$205$	603900.	1.00365
$206$	538228.	0.883687
$207$	−325215.	−0.527527
$208$	−43264.0	−0.0693375
$209$	396933.	0.628567
$210$	−107604.	−0.168376
$211$	−365707.	−0.565493	−0.282746	−	0.959195i	$-0.591245\pi$
$211$	−365707.	−0.565493	−0.282746	+	0.959195i	$0.591245\pi$
$212$	33312.0	0.0509051
$213$	−481500.	−0.727189
$214$	−447784.	−0.668396
$215$	−852475.	−1.25772
$216$	−46656.0	−0.0680414
$217$	−440608.	−0.635189
$218$	−172076.	−0.245233
$219$	−541233.	−0.762560
$220$	−176656.	−0.246077
$221$	332085.	0.457371
$222$	351108.	0.478144
$223$	−349194.	−0.470224	−0.235112	−	0.971968i	$-0.575546\pi$
$223$	−349194.	−0.470224	−0.235112	+	0.971968i	$0.575546\pi$
$224$	−50176.0	−0.0668153
$225$	48276.0	0.0635733
$226$	−49048.0	−0.0638778
$227$	−770442.	−0.992374	−0.496187	−	0.868216i	$-0.665267\pi$
$227$	−770442.	−0.992374	−0.496187	+	0.868216i	$0.665267\pi$
$228$	−315792.	−0.402313
$229$	899634.	1.13365	0.566823	−	0.823840i	$-0.308172\pi$
$229$	899634.	1.13365	0.566823	+	0.823840i	$0.308172\pi$
$230$	979660.	1.22111
$231$	79821.0	0.0984209
$232$	−437824.	−0.534047
$233$	−1.16188e6	−1.40208	−0.701040	−	0.713122i	$-0.747281\pi$
$233$	−1.16188e6	−1.40208	−0.701040	+	0.713122i	$0.747281\pi$
$234$	−54756.0	−0.0653720
$235$	−1.14729e6	−1.35520
$236$	507200.	0.592788
$237$	−159102.	−0.183994
$238$	385140.	0.440734
$239$	1.44191e6	1.63284	0.816418	−	0.577461i	$-0.195956\pi$
$239$	1.44191e6	1.63284	0.816418	+	0.577461i	$0.195956\pi$
$240$	140544.	0.157501
$241$	850158.	0.942881	0.471441	−	0.881898i	$-0.343734\pi$
$241$	850158.	0.942881	0.471441	+	0.881898i	$0.343734\pi$
$242$	−513160.	−0.563267
$243$	−59049.0	−0.0641500
$244$	345232.	0.371224
$245$	−146461.	−0.155886
$246$	356400.	0.375491
$247$	−370617.	−0.386530
$248$	575488.	0.594165
$249$	−725922.	−0.741979
$250$	617076.	0.624437
$251$	690147.	0.691444	0.345722	−	0.938337i	$-0.387634\pi$
$251$	690147.	0.691444	0.345722	+	0.938337i	$0.387634\pi$
$252$	−63504.0	−0.0629941
$253$	−726715.	−0.713777
$254$	−101528.	−0.0987419
$255$	−1.07878e6	−1.03893
$256$	65536.0	0.0625000
$257$	−738870.	−0.697807	−0.348903	−	0.937159i	$-0.613446\pi$
$257$	−738870.	−0.697807	−0.348903	+	0.937159i	$0.613446\pi$
$258$	−503100.	−0.470549
$259$	477897.	0.442675
$260$	164944.	0.151322
$261$	−554121.	−0.503504
$262$	−76132.0	−0.0685195
$263$	−1.09871e6	−0.979474	−0.489737	−	0.871870i	$-0.662907\pi$
$263$	−1.09871e6	−0.979474	−0.489737	+	0.871870i	$0.662907\pi$
$264$	−104256.	−0.0920643
$265$	−127002.	−0.111095
$266$	−429828.	−0.372469
$267$	−735210.	−0.631151
$268$	−795104.	−0.676218
$269$	−1.88245e6	−1.58614	−0.793071	−	0.609130i	$-0.791519\pi$
$269$	−1.88245e6	−1.58614	−0.793071	+	0.609130i	$0.791519\pi$
$270$	177876.	0.148494
$271$	−1.19853e6	−0.991344	−0.495672	−	0.868510i	$-0.665078\pi$
$271$	−1.19853e6	−0.991344	−0.495672	+	0.868510i	$0.665078\pi$
$272$	−503040.	−0.412269
$273$	−74529.0	−0.0605228
$274$	272316.	0.219127
$275$	107876.	0.0860187
$276$	578160.	0.456852
$277$	−268278.	−0.210080	−0.105040	−	0.994468i	$-0.533497\pi$
$277$	−268278.	−0.210080	−0.105040	+	0.994468i	$0.533497\pi$
$278$	491920.	0.381753
$279$	728352.	0.560184
$280$	191296.	0.145818
$281$	1.97239e6	1.49014	0.745070	−	0.666986i	$-0.232416\pi$
$281$	1.97239e6	1.49014	0.745070	+	0.666986i	$0.232416\pi$
$282$	−677088.	−0.507017
$283$	891712.	0.661848	0.330924	−	0.943657i	$-0.392640\pi$
$283$	891712.	0.661848	0.330924	+	0.943657i	$0.392640\pi$
$284$	856000.	0.629764
$285$	1.20396e6	0.878009
$286$	−122356.	−0.0884525
$287$	485100.	0.347637
$288$	82944.0	0.0589256
$289$	2.44137e6	1.71945
$290$	1.66920e6	1.16551
$291$	−342486.	−0.237088
$292$	962192.	0.660396
$293$	−1.90690e6	−1.29765	−0.648826	−	0.760937i	$-0.724740\pi$
$293$	−1.90690e6	−1.29765	−0.648826	+	0.760937i	$0.724740\pi$
$294$	−86436.0	−0.0583212
$295$	−1.93370e6	−1.29370
$296$	−624192.	−0.414084
$297$	−131949.	−0.0867991
$298$	311904.	0.203461
$299$	678535.	0.438929
$300$	−85824.0	−0.0550561
$301$	−684775.	−0.435644
$302$	103644.	0.0653923
$303$	354636.	0.221910
$304$	561408.	0.348413
$305$	−1.31620e6	−0.810161
$306$	−636660.	−0.388691
$307$	1.88730e6	1.14286	0.571431	−	0.820650i	$-0.306388\pi$
$307$	1.88730e6	1.14286	0.571431	+	0.820650i	$0.306388\pi$
$308$	−141904.	−0.0852350
$309$	−1.21101e6	−0.721527
$310$	−2.19405e6	−1.29671
$311$	467370.	0.274006	0.137003	−	0.990571i	$-0.456253\pi$
$311$	467370.	0.274006	0.137003	+	0.990571i	$0.456253\pi$
$312$	97344.0	0.0566139
$313$	−1.16817e6	−0.673976	−0.336988	−	0.941509i	$-0.609408\pi$
$313$	−1.16817e6	−0.673976	−0.336988	+	0.941509i	$0.609408\pi$
$314$	−1.27572e6	−0.730184
$315$	242109.	0.137478
$316$	282848.	0.159344
$317$	1.55444e6	0.868814	0.434407	−	0.900717i	$-0.356958\pi$
$317$	1.55444e6	0.868814	0.434407	+	0.900717i	$0.356958\pi$
$318$	−74952.0	−0.0415638
$319$	−1.23822e6	−0.681273
$320$	−249856.	−0.136400
$321$	1.00751e6	0.545743
$322$	786940.	0.422963
$323$	−4.30924e6	−2.29824
$324$	104976.	0.0555556
$325$	−100724.	−0.0528962
$326$	2.15876e6	1.12502
$327$	387171.	0.200232
$328$	−633600.	−0.325185
$329$	−921592.	−0.469406
$330$	397476.	0.200921
$331$	34274.0	0.0171947	0.00859735	−	0.999963i	$-0.497263\pi$
$331$	34274.0	0.0171947	0.00859735	+	0.999963i	$0.497263\pi$
$332$	1.29053e6	0.642573
$333$	−789993.	−0.390403
$334$	−1.21410e6	−0.595509
$335$	3.03133e6	1.47578
$336$	112896.	0.0545545
$337$	−2.46690e6	−1.18325	−0.591626	−	0.806213i	$-0.701514\pi$
$337$	−2.46690e6	−1.18325	−0.591626	+	0.806213i	$0.701514\pi$
$338$	114244.	0.0543928
$339$	110358.	0.0521560
$340$	1.91784e6	0.899736
$341$	1.62755e6	0.757965
$342$	710532.	0.328487
$343$	−117649.	−0.0539949
$344$	894400.	0.407508
$345$	−2.20424e6	−0.997034
$346$	−1.31750e6	−0.591642
$347$	907056.	0.404399	0.202200	−	0.979344i	$-0.435191\pi$
$347$	907056.	0.404399	0.202200	+	0.979344i	$0.435191\pi$
$348$	985104.	0.436048
$349$	−1.73591e6	−0.762893	−0.381447	−	0.924391i	$-0.624574\pi$
$349$	−1.73591e6	−0.762893	−0.381447	+	0.924391i	$0.624574\pi$
$350$	−116816.	−0.0509721
$351$	123201.	0.0533761
$352$	185344.	0.0797300
$353$	28438.0	0.0121468	0.00607341	−	0.999982i	$-0.498067\pi$
$353$	28438.0	0.0121468	0.00607341	+	0.999982i	$0.498067\pi$
$354$	−1.14120e6	−0.484009
$355$	−3.26350e6	−1.37440
$356$	1.30704e6	0.546593
$357$	−866565.	−0.359858
$358$	2.37830e6	0.980753
$359$	2.57096e6	1.05283	0.526417	−	0.850227i	$-0.323535\pi$
$359$	2.57096e6	1.05283	0.526417	+	0.850227i	$0.323535\pi$
$360$	−316224.	−0.128599
$361$	2.33315e6	0.942268
$362$	−63720.0	−0.0255567
$363$	1.15461e6	0.459906
$364$	132496.	0.0524142
$365$	−3.66836e6	−1.44125
$366$	−776772.	−0.303103
$367$	4.47692e6	1.73506	0.867529	−	0.497386i	$-0.165707\pi$
$367$	4.47692e6	1.73506	0.867529	+	0.497386i	$0.165707\pi$
$368$	−1.02784e6	−0.395645
$369$	−801900.	−0.306587
$370$	2.37973e6	0.903699
$371$	−102018.	−0.0384806
$372$	−1.29485e6	−0.485134
$373$	1.83453e6	0.682737	0.341368	−	0.939930i	$-0.389110\pi$
$373$	1.83453e6	0.682737	0.341368	+	0.939930i	$0.389110\pi$
$374$	−1.42266e6	−0.525923
$375$	−1.38842e6	−0.509851
$376$	1.20371e6	0.439089
$377$	1.15613e6	0.418941
$378$	142884.	0.0514344
$379$	825676.	0.295265	0.147632	−	0.989042i	$-0.452835\pi$
$379$	825676.	0.295265	0.147632	+	0.989042i	$0.452835\pi$
$380$	−2.14037e6	−0.760378
$381$	228438.	0.0806224
$382$	1.51162e6	0.530010
$383$	224451.	0.0781852	0.0390926	−	0.999236i	$-0.487553\pi$
$383$	224451.	0.0781852	0.0390926	+	0.999236i	$0.487553\pi$
$384$	−147456.	−0.0510310
$385$	541009.	0.186017
$386$	1.45683e6	0.497670
$387$	1.13198e6	0.384202
$388$	608864.	0.205325
$389$	−2.88421e6	−0.966390	−0.483195	−	0.875513i	$-0.660524\pi$
$389$	−2.88421e6	−0.966390	−0.483195	+	0.875513i	$0.660524\pi$
$390$	−371124.	−0.123554
$391$	7.88947e6	2.60979
$392$	153664.	0.0505076
$393$	171297.	0.0559459
$394$	2.31077e6	0.749921
$395$	−1.07836e6	−0.347752
$396$	234576.	0.0751702
$397$	−389614.	−0.124068	−0.0620338	−	0.998074i	$-0.519759\pi$
$397$	−389614.	−0.124068	−0.0620338	+	0.998074i	$0.519759\pi$
$398$	1.46448e6	0.463420
$399$	967113.	0.304120
$400$	152576.	0.0476800
$401$	−3.56896e6	−1.10836	−0.554180	−	0.832397i	$-0.686968\pi$
$401$	−3.56896e6	−1.10836	−0.554180	+	0.832397i	$0.686968\pi$
$402$	1.78898e6	0.552130
$403$	−1.51965e6	−0.466101
$404$	−630464.	−0.192179
$405$	−400221.	−0.121245
$406$	1.34084e6	0.403702
$407$	−1.76529e6	−0.528239
$408$	1.13184e6	0.336616
$409$	−2.23968e6	−0.662032	−0.331016	−	0.943625i	$-0.607391\pi$
$409$	−2.23968e6	−0.662032	−0.331016	+	0.943625i	$0.607391\pi$
$410$	2.41560e6	0.709685
$411$	−612711.	−0.178917
$412$	2.15291e6	0.624861
$413$	−1.55330e6	−0.448105
$414$	−1.30086e6	−0.373018
$415$	−4.92014e6	−1.40235
$416$	−173056.	−0.0490290
$417$	−1.10682e6	−0.311700
$418$	1.58773e6	0.444464
$419$	−3.64224e6	−1.01352	−0.506762	−	0.862086i	$-0.669158\pi$
$419$	−3.64224e6	−1.01352	−0.506762	+	0.862086i	$0.669158\pi$
$420$	−430416.	−0.119060
$421$	3.56339e6	0.979846	0.489923	−	0.871766i	$-0.337025\pi$
$421$	3.56339e6	0.979846	0.489923	+	0.871766i	$0.337025\pi$
$422$	−1.46283e6	−0.399864
$423$	1.52345e6	0.413977
$424$	133248.	0.0359953
$425$	−1.17114e6	−0.314511
$426$	−1.92600e6	−0.514200
$427$	−1.05727e6	−0.280619
$428$	−1.79114e6	−0.472628
$429$	275301.	0.0722212
$430$	−3.40990e6	−0.889345
$431$	−3.35116e6	−0.868965	−0.434482	−	0.900680i	$-0.643069\pi$
$431$	−3.35116e6	−0.868965	−0.434482	+	0.900680i	$0.643069\pi$
$432$	−186624.	−0.0481125
$433$	−2.22932e6	−0.571416	−0.285708	−	0.958317i	$-0.592229\pi$
$433$	−2.22932e6	−0.571416	−0.285708	+	0.958317i	$0.592229\pi$
$434$	−1.76243e6	−0.449147
$435$	−3.75571e6	−0.951631
$436$	−688304.	−0.173406
$437$	−8.80490e6	−2.20557
$438$	−2.16493e6	−0.539211
$439$	−1.71901e6	−0.425712	−0.212856	−	0.977084i	$-0.568277\pi$
$439$	−1.71901e6	−0.425712	−0.212856	+	0.977084i	$0.568277\pi$
$440$	−706624.	−0.174003
$441$	194481.	0.0476190
$442$	1.32834e6	0.323410
$443$	−810318.	−0.196176	−0.0980881	−	0.995178i	$-0.531273\pi$
$443$	−810318.	−0.196176	−0.0980881	+	0.995178i	$0.531273\pi$
$444$	1.40443e6	0.338099
$445$	−4.98309e6	−1.19289
$446$	−1.39678e6	−0.332498
$447$	−701784.	−0.166125
$448$	−200704.	−0.0472456
$449$	7.83837e6	1.83489	0.917445	−	0.397863i	$-0.130248\pi$
$449$	7.83837e6	1.83489	0.917445	+	0.397863i	$0.130248\pi$
$450$	193104.	0.0449531
$451$	−1.79190e6	−0.414832
$452$	−196192.	−0.0451685
$453$	−233199.	−0.0533926
$454$	−3.08177e6	−0.701714
$455$	−505141.	−0.114389
$456$	−1.26317e6	−0.284478
$457$	7.79980e6	1.74700	0.873500	−	0.486824i	$-0.161845\pi$
$457$	7.79980e6	1.74700	0.873500	+	0.486824i	$0.161845\pi$
$458$	3.59854e6	0.801608
$459$	1.43248e6	0.317365
$460$	3.91864e6	0.863457
$461$	−3.10922e6	−0.681394	−0.340697	−	0.940173i	$-0.610663\pi$
$461$	−3.10922e6	−0.681394	−0.340697	+	0.940173i	$0.610663\pi$
$462$	319284.	0.0695941
$463$	−179321.	−0.0388757	−0.0194379	−	0.999811i	$-0.506188\pi$
$463$	−179321.	−0.0388757	−0.0194379	+	0.999811i	$0.506188\pi$
$464$	−1.75130e6	−0.377628
$465$	4.93661e6	1.05876
$466$	−4.64754e6	−0.991421
$467$	2.05028e6	0.435032	0.217516	−	0.976057i	$-0.430205\pi$
$467$	2.05028e6	0.435032	0.217516	+	0.976057i	$0.430205\pi$
$468$	−219024.	−0.0462250
$469$	2.43501e6	0.511173
$470$	−4.58915e6	−0.958270
$471$	2.87038e6	0.596193
$472$	2.02880e6	0.419164
$473$	2.52948e6	0.519849
$474$	−636408.	−0.130104
$475$	1.30703e6	0.265797
$476$	1.54056e6	0.311646
$477$	168642.	0.0339367
$478$	5.76763e6	1.15459
$479$	−8.08398e6	−1.60985	−0.804927	−	0.593374i	$-0.797796\pi$
$479$	−8.08398e6	−1.60985	−0.804927	+	0.593374i	$0.797796\pi$
$480$	562176.	0.111370
$481$	1.64826e6	0.324835
$482$	3.40063e6	0.666718
$483$	−1.77062e6	−0.345347
$484$	−2.05264e6	−0.398290
$485$	−2.32129e6	−0.448101
$486$	−236196.	−0.0453609
$487$	−8.20240e6	−1.56718	−0.783589	−	0.621279i	$-0.786613\pi$
$487$	−8.20240e6	−1.56718	−0.783589	+	0.621279i	$0.786613\pi$
$488$	1.38093e6	0.262495
$489$	−4.85721e6	−0.918575
$490$	−585844.	−0.110228
$491$	4.39525e6	0.822773	0.411386	−	0.911461i	$-0.365045\pi$
$491$	4.39525e6	0.822773	0.411386	+	0.911461i	$0.365045\pi$
$492$	1.42560e6	0.265512
$493$	1.34426e7	2.49095
$494$	−1.48247e6	−0.273318
$495$	−894321.	−0.164052
$496$	2.30195e6	0.420138
$497$	−2.62150e6	−0.476057
$498$	−2.90369e6	−0.524658
$499$	−1.09892e7	−1.97568	−0.987838	−	0.155488i	$-0.950305\pi$
$499$	−1.09892e7	−1.97568	−0.987838	+	0.155488i	$0.950305\pi$
$500$	2.46830e6	0.441544
$501$	2.73173e6	0.486231
$502$	2.76059e6	0.488925
$503$	−889646.	−0.156782	−0.0783912	−	0.996923i	$-0.524978\pi$
$503$	−889646.	−0.156782	−0.0783912	+	0.996923i	$0.524978\pi$
$504$	−254016.	−0.0445435
$505$	2.40364e6	0.419413
$506$	−2.90686e6	−0.504717
$507$	−257049.	−0.0444116
$508$	−406112.	−0.0698211
$509$	−4.84570e6	−0.829014	−0.414507	−	0.910046i	$-0.636046\pi$
$509$	−4.84570e6	−0.829014	−0.414507	+	0.910046i	$0.636046\pi$
$510$	−4.31514e6	−0.734631
$511$	−2.94671e6	−0.499213
$512$	262144.	0.0441942
$513$	−1.59870e6	−0.268209
$514$	−2.95548e6	−0.493424
$515$	−8.20798e6	−1.36370
$516$	−2.01240e6	−0.332729
$517$	3.40425e6	0.560138
$518$	1.91159e6	0.313018
$519$	2.96437e6	0.483074
$520$	659776.	0.107001
$521$	−3.79031e6	−0.611759	−0.305880	−	0.952070i	$-0.598951\pi$
$521$	−3.79031e6	−0.611759	−0.305880	+	0.952070i	$0.598951\pi$
$522$	−2.21648e6	−0.356031
$523$	−1.20848e6	−0.193191	−0.0965955	−	0.995324i	$-0.530795\pi$
$523$	−1.20848e6	−0.193191	−0.0965955	+	0.995324i	$0.530795\pi$
$524$	−304528.	−0.0484506
$525$	262836.	0.0416185
$526$	−4.39483e6	−0.692593
$527$	−1.76693e7	−2.77136
$528$	−417024.	−0.0650993
$529$	9.68388e6	1.50456
$530$	−508008.	−0.0785563
$531$	2.56770e6	0.395192
$532$	−1.71931e6	−0.263376
$533$	1.67310e6	0.255096
$534$	−2.94084e6	−0.446291
$535$	6.82871e6	1.03146
$536$	−3.18042e6	−0.478159
$537$	−5.35118e6	−0.800782
$538$	−7.52978e6	−1.12157
$539$	434581.	0.0644316
$540$	711504.	0.105001
$541$	417077.	0.0612665	0.0306333	−	0.999531i	$-0.490248\pi$
$541$	417077.	0.0612665	0.0306333	+	0.999531i	$0.490248\pi$
$542$	−4.79410e6	−0.700986
$543$	143370.	0.0208669
$544$	−2.01216e6	−0.291518
$545$	2.62416e6	0.378442
$546$	−298116.	−0.0427960
$547$	2.71599e6	0.388114	0.194057	−	0.980990i	$-0.437835\pi$
$547$	2.71599e6	0.388114	0.194057	+	0.980990i	$0.437835\pi$
$548$	1.08926e6	0.154946
$549$	1.74774e6	0.247483
$550$	431504.	0.0608244
$551$	−1.50023e7	−2.10513
$552$	2.31264e6	0.323043
$553$	−866222.	−0.120453
$554$	−1.07311e6	−0.148549
$555$	−5.35440e6	−0.737867
$556$	1.96768e6	0.269940
$557$	−6.09143e6	−0.831919	−0.415960	−	0.909383i	$-0.636554\pi$
$557$	−6.09143e6	−0.831919	−0.415960	+	0.909383i	$0.636554\pi$
$558$	2.91341e6	0.396110
$559$	−2.36178e6	−0.319675
$560$	765184.	0.103109
$561$	3.20098e6	0.429414
$562$	7.88956e6	1.05369
$563$	−6.69101e6	−0.889653	−0.444826	−	0.895617i	$-0.646735\pi$
$563$	−6.69101e6	−0.889653	−0.444826	+	0.895617i	$0.646735\pi$
$564$	−2.70835e6	−0.358515
$565$	747982.	0.0985757
$566$	3.56685e6	0.467997
$567$	−321489.	−0.0419961
$568$	3.42400e6	0.445310
$569$	−1.35780e7	−1.75814	−0.879072	−	0.476689i	$-0.841837\pi$
$569$	−1.35780e7	−1.75814	−0.879072	+	0.476689i	$0.841837\pi$
$570$	4.81583e6	0.620846
$571$	6.89755e6	0.885329	0.442665	−	0.896687i	$-0.354033\pi$
$571$	6.89755e6	0.885329	0.442665	+	0.896687i	$0.354033\pi$
$572$	−489424.	−0.0625454
$573$	−3.40114e6	−0.432752
$574$	1.94040e6	0.245817
$575$	−2.39294e6	−0.301830
$576$	331776.	0.0416667
$577$	−1.30376e7	−1.63026	−0.815130	−	0.579278i	$-0.803335\pi$
$577$	−1.30376e7	−1.63026	−0.815130	+	0.579278i	$0.803335\pi$
$578$	9.76547e6	1.21583
$579$	−3.27787e6	−0.406346
$580$	6.67682e6	0.824137
$581$	−3.95224e6	−0.485739
$582$	−1.36994e6	−0.167647
$583$	376842.	0.0459185
$584$	3.84877e6	0.466971
$585$	835029.	0.100882
$586$	−7.62759e6	−0.917579
$587$	1.15172e7	1.37960	0.689800	−	0.724000i	$-0.257698\pi$
$587$	1.15172e7	1.37960	0.689800	+	0.724000i	$0.257698\pi$
$588$	−345744.	−0.0412393
$589$	1.97195e7	2.34211
$590$	−7.73480e6	−0.914785
$591$	−5.19923e6	−0.612308
$592$	−2.49677e6	−0.292802
$593$	7.40953e6	0.865275	0.432638	−	0.901568i	$-0.357583\pi$
$593$	7.40953e6	0.865275	0.432638	+	0.901568i	$0.357583\pi$
$594$	−527796.	−0.0613762
$595$	−5.87338e6	−0.680136
$596$	1.24762e6	0.143868
$597$	−3.29507e6	−0.378381
$598$	2.71414e6	0.310370
$599$	1.24429e7	1.41694	0.708472	−	0.705738i	$-0.249385\pi$
$599$	1.24429e7	1.41694	0.708472	+	0.705738i	$0.249385\pi$
$600$	−343296.	−0.0389306
$601$	1.61413e7	1.82285	0.911427	−	0.411461i	$-0.134981\pi$
$601$	1.61413e7	1.82285	0.911427	+	0.411461i	$0.134981\pi$
$602$	−2.73910e6	−0.308047
$603$	−4.02521e6	−0.450812
$604$	414576.	0.0462394
$605$	7.82569e6	0.869229
$606$	1.41854e6	0.156914
$607$	−1.73120e7	−1.90711	−0.953555	−	0.301219i	$-0.902606\pi$
$607$	−1.73120e7	−1.90711	−0.953555	+	0.301219i	$0.902606\pi$
$608$	2.24563e6	0.246365
$609$	−3.01688e6	−0.329621
$610$	−5.26479e6	−0.572870
$611$	−3.17855e6	−0.344450
$612$	−2.54664e6	−0.274846
$613$	710405.	0.0763580	0.0381790	−	0.999271i	$-0.487844\pi$
$613$	710405.	0.0763580	0.0381790	+	0.999271i	$0.487844\pi$
$614$	7.54918e6	0.808126
$615$	−5.43510e6	−0.579455
$616$	−567616.	−0.0602702
$617$	1.29679e7	1.37138	0.685690	−	0.727894i	$-0.259501\pi$
$617$	1.29679e7	1.37138	0.685690	+	0.727894i	$0.259501\pi$
$618$	−4.84405e6	−0.510197
$619$	1.06122e7	1.11321	0.556606	−	0.830777i	$-0.312103\pi$
$619$	1.06122e7	1.11321	0.556606	+	0.830777i	$0.312103\pi$
$620$	−8.77619e6	−0.916910
$621$	2.92694e6	0.304568
$622$	1.86948e6	0.193751
$623$	−4.00281e6	−0.413185
$624$	389376.	0.0400320
$625$	−1.12729e7	−1.15435
$626$	−4.67267e6	−0.476573
$627$	−3.57240e6	−0.362903
$628$	−5.10290e6	−0.516318
$629$	1.91646e7	1.93141
$630$	968436.	0.0972119
$631$	5.60267e6	0.560173	0.280086	−	0.959975i	$-0.409637\pi$
$631$	5.60267e6	0.560173	0.280086	+	0.959975i	$0.409637\pi$
$632$	1.13139e6	0.112673
$633$	3.29136e6	0.326487
$634$	6.21778e6	0.614344
$635$	1.54830e6	0.152378
$636$	−299808.	−0.0293901
$637$	−405769.	−0.0396214
$638$	−4.95288e6	−0.481733
$639$	4.33350e6	0.419843
$640$	−999424.	−0.0964495
$641$	−4.72434e6	−0.454147	−0.227073	−	0.973878i	$-0.572916\pi$
$641$	−4.72434e6	−0.454147	−0.227073	+	0.973878i	$0.572916\pi$
$642$	4.03006e6	0.385899
$643$	2.08328e7	1.98711	0.993553	−	0.113369i	$-0.0361642\pi$
$643$	2.08328e7	1.98711	0.993553	+	0.113369i	$0.0361642\pi$
$644$	3.14776e6	0.299080
$645$	7.67228e6	0.726148
$646$	−1.72370e7	−1.62510
$647$	2.11450e7	1.98585	0.992927	−	0.118728i	$-0.0378817\pi$
$647$	2.11450e7	1.98585	0.992927	+	0.118728i	$0.0378817\pi$
$648$	419904.	0.0392837
$649$	5.73770e6	0.534720
$650$	−402896.	−0.0374033
$651$	3.96547e6	0.366727
$652$	8.63504e6	0.795509
$653$	1.80532e7	1.65680	0.828401	−	0.560136i	$-0.189251\pi$
$653$	1.80532e7	1.65680	0.828401	+	0.560136i	$0.189251\pi$
$654$	1.54868e6	0.141585
$655$	1.16101e6	0.105739
$656$	−2.53440e6	−0.229941
$657$	4.87110e6	0.440264
$658$	−3.68637e6	−0.331920
$659$	1.49957e7	1.34510	0.672549	−	0.740052i	$-0.265199\pi$
$659$	1.49957e7	1.34510	0.672549	+	0.740052i	$0.265199\pi$
$660$	1.58990e6	0.142073
$661$	1.70046e7	1.51378	0.756892	−	0.653540i	$-0.226717\pi$
$661$	1.70046e7	1.51378	0.756892	+	0.653540i	$0.226717\pi$
$662$	137096.	0.0121585
$663$	−2.98876e6	−0.264063
$664$	5.16211e6	0.454367
$665$	6.55488e6	0.574792
$666$	−3.15997e6	−0.276056
$667$	2.74666e7	2.39051
$668$	−4.85640e6	−0.421088
$669$	3.14275e6	0.271484
$670$	1.21253e7	1.04353
$671$	3.90544e6	0.334860
$672$	451584.	0.0385758
$673$	−1.41794e6	−0.120676	−0.0603378	−	0.998178i	$-0.519218\pi$
$673$	−1.41794e6	−0.120676	−0.0603378	+	0.998178i	$0.519218\pi$
$674$	−9.86760e6	−0.836685
$675$	−434484.	−0.0367041
$676$	456976.	0.0384615
$677$	1.54183e6	0.129290	0.0646448	−	0.997908i	$-0.479409\pi$
$677$	1.54183e6	0.129290	0.0646448	+	0.997908i	$0.479409\pi$
$678$	441432.	0.0368799
$679$	−1.86465e6	−0.155211
$680$	7.67136e6	0.636209
$681$	6.93398e6	0.572947
$682$	6.51021e6	0.535962
$683$	1.04218e7	0.854849	0.427425	−	0.904051i	$-0.359421\pi$
$683$	1.04218e7	0.854849	0.427425	+	0.904051i	$0.359421\pi$
$684$	2.84213e6	0.232275
$685$	−4.15282e6	−0.338156
$686$	−470596.	−0.0381802
$687$	−8.09671e6	−0.654510
$688$	3.57760e6	0.288151
$689$	−351858.	−0.0282371
$690$	−8.81694e6	−0.705010
$691$	1.15759e7	0.922277	0.461139	−	0.887328i	$-0.347441\pi$
$691$	1.15759e7	0.922277	0.461139	+	0.887328i	$0.347441\pi$
$692$	−5.26998e6	−0.418354
$693$	−718389.	−0.0568233
$694$	3.62822e6	0.285953
$695$	−7.50178e6	−0.589118
$696$	3.94042e6	0.308332
$697$	1.94535e7	1.51676
$698$	−6.94364e6	−0.539447
$699$	1.04570e7	0.809492
$700$	−467264.	−0.0360427
$701$	−9.83750e6	−0.756118	−0.378059	−	0.925781i	$-0.623408\pi$
$701$	−9.83750e6	−0.756118	−0.378059	+	0.925781i	$0.623408\pi$
$702$	492804.	0.0377426
$703$	−2.13883e7	−1.63226
$704$	741376.	0.0563776
$705$	1.03256e7	0.782424
$706$	113752.	0.00858909
$707$	1.93080e6	0.145274
$708$	−4.56480e6	−0.342246
$709$	4.19856e6	0.313679	0.156839	−	0.987624i	$-0.449870\pi$
$709$	4.19856e6	0.313679	0.156839	+	0.987624i	$0.449870\pi$
$710$	−1.30540e7	−0.971847
$711$	1.43192e6	0.106229
$712$	5.22816e6	0.386499
$713$	−3.61029e7	−2.65961
$714$	−3.46626e6	−0.254458
$715$	1.86593e6	0.136499
$716$	9.51322e6	0.693497
$717$	−1.29772e7	−0.942719
$718$	1.02839e7	0.744466
$719$	1.78190e7	1.28547	0.642734	−	0.766089i	$-0.277800\pi$
$719$	1.78190e7	1.28547	0.642734	+	0.766089i	$0.277800\pi$
$720$	−1.26490e6	−0.0909334
$721$	−6.59329e6	−0.472350
$722$	9.33260e6	0.666284
$723$	−7.65142e6	−0.544373
$724$	−254880.	−0.0180713
$725$	−4.07724e6	−0.288085
$726$	4.61844e6	0.325202
$727$	1.62499e7	1.14029	0.570145	−	0.821544i	$-0.306887\pi$
$727$	1.62499e7	1.14029	0.570145	+	0.821544i	$0.306887\pi$
$728$	529984.	0.0370625
$729$	531441.	0.0370370
$730$	−1.46734e7	−1.01912
$731$	−2.74609e7	−1.90073
$732$	−3.10709e6	−0.214326
$733$	2.21669e7	1.52386	0.761931	−	0.647659i	$-0.224252\pi$
$733$	2.21669e7	1.52386	0.761931	+	0.647659i	$0.224252\pi$
$734$	1.79077e7	1.22687
$735$	1.31815e6	0.0900008
$736$	−4.11136e6	−0.279763
$737$	−8.99461e6	−0.609977
$738$	−3.20760e6	−0.216790
$739$	−1.08832e7	−0.733070	−0.366535	−	0.930404i	$-0.619456\pi$
$739$	−1.08832e7	−0.733070	−0.366535	+	0.930404i	$0.619456\pi$
$740$	9.51893e6	0.639012
$741$	3.33555e6	0.223163
$742$	−408072.	−0.0272099
$743$	6.74145e6	0.448004	0.224002	−	0.974589i	$-0.428088\pi$
$743$	6.74145e6	0.448004	0.224002	+	0.974589i	$0.428088\pi$
$744$	−5.17939e6	−0.343041
$745$	−4.75654e6	−0.313979
$746$	7.33813e6	0.482768
$747$	6.53330e6	0.428382
$748$	−5.69064e6	−0.371884
$749$	5.48535e6	0.357273
$750$	−5.55368e6	−0.360519
$751$	−1.30972e7	−0.847381	−0.423691	−	0.905807i	$-0.639266\pi$
$751$	−1.30972e7	−0.847381	−0.423691	+	0.905807i	$0.639266\pi$
$752$	4.81485e6	0.310483
$753$	−6.21132e6	−0.399206
$754$	4.62452e6	0.296236
$755$	−1.58057e6	−0.100913
$756$	571536.	0.0363696
$757$	1.46675e6	0.0930288	0.0465144	−	0.998918i	$-0.485189\pi$
$757$	1.46675e6	0.0930288	0.0465144	+	0.998918i	$0.485189\pi$
$758$	3.30270e6	0.208784
$759$	6.54044e6	0.412100
$760$	−8.56147e6	−0.537668
$761$	−2.36094e7	−1.47782	−0.738912	−	0.673802i	$-0.764660\pi$
$761$	−2.36094e7	−1.47782	−0.738912	+	0.673802i	$0.764660\pi$
$762$	913752.	0.0570087
$763$	2.10793e6	0.131083
$764$	6.04648e6	0.374774
$765$	9.70906e6	0.599824
$766$	897804.	0.0552853
$767$	−5.35730e6	−0.328819
$768$	−589824.	−0.0360844
$769$	−8.19998e6	−0.500031	−0.250016	−	0.968242i	$-0.580436\pi$
$769$	−8.19998e6	−0.500031	−0.250016	+	0.968242i	$0.580436\pi$
$770$	2.16404e6	0.131534
$771$	6.64983e6	0.402879
$772$	5.82733e6	0.351906
$773$	−398383.	−0.0239802	−0.0119901	−	0.999928i	$-0.503817\pi$
$773$	−398383.	−0.0239802	−0.0119901	+	0.999928i	$0.503817\pi$
$774$	4.52790e6	0.271672
$775$	5.35923e6	0.320515
$776$	2.43546e6	0.145186
$777$	−4.30107e6	−0.255578
$778$	−1.15368e7	−0.683341
$779$	−2.17107e7	−1.28183
$780$	−1.48450e6	−0.0873660
$781$	9.68350e6	0.568074
$782$	3.15579e7	1.84540
$783$	4.98709e6	0.290698
$784$	614656.	0.0357143
$785$	1.94548e7	1.12681
$786$	685188.	0.0395597
$787$	5.46374e6	0.314451	0.157225	−	0.987563i	$-0.449745\pi$
$787$	5.46374e6	0.314451	0.157225	+	0.987563i	$0.449745\pi$
$788$	9.24307e6	0.530275
$789$	9.88837e6	0.565500
$790$	−4.31343e6	−0.245898
$791$	600838.	0.0341441
$792$	938304.	0.0531533
$793$	−3.64651e6	−0.205918
$794$	−1.55846e6	−0.0877290
$795$	1.14302e6	0.0641409
$796$	5.85790e6	0.327687
$797$	1.19408e7	0.665866	0.332933	−	0.942950i	$-0.391962\pi$
$797$	1.19408e7	0.665866	0.332933	+	0.942950i	$0.391962\pi$
$798$	3.86845e6	0.215045
$799$	−3.69577e7	−2.04804
$800$	610304.	0.0337149
$801$	6.61689e6	0.364395
$802$	−1.42758e7	−0.783729
$803$	1.08848e7	0.595705
$804$	7.15594e6	0.390415
$805$	−1.20008e7	−0.652712
$806$	−6.07859e6	−0.329583
$807$	1.69420e7	0.915759
$808$	−2.52186e6	−0.135891
$809$	2.52350e7	1.35560	0.677802	−	0.735245i	$-0.262933\pi$
$809$	2.52350e7	1.35560	0.677802	+	0.735245i	$0.262933\pi$
$810$	−1.60088e6	−0.0857329
$811$	1.81758e7	0.970379	0.485190	−	0.874409i	$-0.338751\pi$
$811$	1.81758e7	0.970379	0.485190	+	0.874409i	$0.338751\pi$
$812$	5.36334e6	0.285460
$813$	1.07867e7	0.572353
$814$	−7.06117e6	−0.373522
$815$	−3.29211e7	−1.73612
$816$	4.52736e6	0.238023
$817$	3.06472e7	1.60633
$818$	−8.95874e6	−0.468127
$819$	670761.	0.0349428
$820$	9.66240e6	0.501823
$821$	−9.54813e6	−0.494379	−0.247190	−	0.968967i	$-0.579507\pi$
$821$	−9.54813e6	−0.494379	−0.247190	+	0.968967i	$0.579507\pi$
$822$	−2.45084e6	−0.126513
$823$	−3.63485e7	−1.87063	−0.935313	−	0.353821i	$-0.884882\pi$
$823$	−3.63485e7	−1.87063	−0.935313	+	0.353821i	$0.884882\pi$
$824$	8.61165e6	0.441843
$825$	−970884.	−0.0496629
$826$	−6.21320e6	−0.316858
$827$	2.47175e7	1.25673	0.628364	−	0.777919i	$-0.283725\pi$
$827$	2.47175e7	1.25673	0.628364	+	0.777919i	$0.283725\pi$
$828$	−5.20344e6	−0.263764
$829$	−4.96726e6	−0.251033	−0.125516	−	0.992092i	$-0.540059\pi$
$829$	−4.96726e6	−0.251033	−0.125516	+	0.992092i	$0.540059\pi$
$830$	−1.96806e7	−0.991613
$831$	2.41450e6	0.121290
$832$	−692224.	−0.0346688
$833$	−4.71797e6	−0.235582
$834$	−4.42728e6	−0.220405
$835$	1.85150e7	0.918984
$836$	6.35093e6	0.314283
$837$	−6.55517e6	−0.323422
$838$	−1.45690e7	−0.716669
$839$	−3.95361e7	−1.93905	−0.969525	−	0.244994i	$-0.921214\pi$
$839$	−3.95361e7	−1.93905	−0.969525	+	0.244994i	$0.921214\pi$
$840$	−1.72166e6	−0.0841880
$841$	2.62881e7	1.28165
$842$	1.42535e7	0.692855
$843$	−1.77515e7	−0.860333
$844$	−5.85131e6	−0.282746
$845$	−1.74222e6	−0.0839386
$846$	6.09379e6	0.292726
$847$	6.28621e6	0.301079
$848$	532992.	0.0254525
$849$	−8.02541e6	−0.382118
$850$	−4.68456e6	−0.222393
$851$	3.91583e7	1.85353
$852$	−7.70400e6	−0.363594
$853$	1.93130e7	0.908821	0.454410	−	0.890793i	$-0.349850\pi$
$853$	1.93130e7	0.908821	0.454410	+	0.890793i	$0.349850\pi$
$854$	−4.22909e6	−0.198428
$855$	−1.08356e7	−0.506919
$856$	−7.16454e6	−0.334198
$857$	−2.77975e6	−0.129287	−0.0646434	−	0.997908i	$-0.520591\pi$
$857$	−2.77975e6	−0.129287	−0.0646434	+	0.997908i	$0.520591\pi$
$858$	1.10120e6	0.0510681
$859$	2.30081e7	1.06389	0.531947	−	0.846778i	$-0.321460\pi$
$859$	2.30081e7	1.06389	0.531947	+	0.846778i	$0.321460\pi$
$860$	−1.36396e7	−0.628862
$861$	−4.36590e6	−0.200709
$862$	−1.34046e7	−0.614451
$863$	−3.83225e7	−1.75157	−0.875784	−	0.482704i	$-0.839655\pi$
$863$	−3.83225e7	−1.75157	−0.875784	+	0.482704i	$0.839655\pi$
$864$	−746496.	−0.0340207
$865$	2.00918e7	0.913017
$866$	−8.91728e6	−0.404052
$867$	−2.19723e7	−0.992723
$868$	−7.04973e6	−0.317595
$869$	3.19972e6	0.143735
$870$	−1.50228e7	−0.672905
$871$	8.39829e6	0.375098
$872$	−2.75322e6	−0.122617
$873$	3.08237e6	0.136883
$874$	−3.52196e7	−1.55957
$875$	−7.55918e6	−0.333776
$876$	−8.65973e6	−0.381280
$877$	4.29434e6	0.188537	0.0942686	−	0.995547i	$-0.469949\pi$
$877$	4.29434e6	0.188537	0.0942686	+	0.995547i	$0.469949\pi$
$878$	−6.87603e6	−0.301024
$879$	1.71621e7	0.749200
$880$	−2.82650e6	−0.123039
$881$	3.00824e7	1.30579	0.652894	−	0.757449i	$-0.273555\pi$
$881$	3.00824e7	1.30579	0.652894	+	0.757449i	$0.273555\pi$
$882$	777924.	0.0336718
$883$	6.52270e6	0.281531	0.140765	−	0.990043i	$-0.455044\pi$
$883$	6.52270e6	0.281531	0.140765	+	0.990043i	$0.455044\pi$
$884$	5.31336e6	0.228685
$885$	1.74033e7	0.746919
$886$	−3.24127e6	−0.138717
$887$	1.37522e7	0.586900	0.293450	−	0.955974i	$-0.405197\pi$
$887$	1.37522e7	0.586900	0.293450	+	0.955974i	$0.405197\pi$
$888$	5.61773e6	0.239072
$889$	1.24372e6	0.0527798
$890$	−1.99324e7	−0.843497
$891$	1.18754e6	0.0501135
$892$	−5.58710e6	−0.235112
$893$	4.12459e7	1.73082
$894$	−2.80714e6	−0.117468
$895$	−3.62691e7	−1.51349
$896$	−802816.	−0.0334077
$897$	−6.10681e6	−0.253416
$898$	3.13535e7	1.29746
$899$	−6.15143e7	−2.53850
$900$	772416.	0.0317867
$901$	−4.09113e6	−0.167893
$902$	−7.16760e6	−0.293331
$903$	6.16298e6	0.251519
$904$	−784768.	−0.0319389
$905$	971730.	0.0394388
$906$	−932796.	−0.0377543
$907$	−1.39117e7	−0.561514	−0.280757	−	0.959779i	$-0.590586\pi$
$907$	−1.39117e7	−0.561514	−0.280757	+	0.959779i	$0.590586\pi$
$908$	−1.23271e7	−0.496187
$909$	−3.19172e6	−0.128120
$910$	−2.02056e6	−0.0808852
$911$	−3.93831e7	−1.57222	−0.786111	−	0.618086i	$-0.787908\pi$
$911$	−3.93831e7	−1.57222	−0.786111	+	0.618086i	$0.787908\pi$
$912$	−5.05267e6	−0.201156
$913$	1.45991e7	0.579628
$914$	3.11992e7	1.23532
$915$	1.18458e7	0.467747
$916$	1.43941e7	0.566823
$917$	932617.	0.0366252
$918$	5.72994e6	0.224411
$919$	−3.39188e7	−1.32480	−0.662401	−	0.749149i	$-0.730463\pi$
$919$	−3.39188e7	−1.32480	−0.662401	+	0.749149i	$0.730463\pi$
$920$	1.56746e7	0.610556
$921$	−1.69857e7	−0.659832
$922$	−1.24369e7	−0.481818
$923$	−9.04150e6	−0.349330
$924$	1.27714e6	0.0492104
$925$	−5.81279e6	−0.223373
$926$	−717284.	−0.0274893
$927$	1.08991e7	0.416574
$928$	−7.00518e6	−0.267024
$929$	3.65914e7	1.39104	0.695520	−	0.718506i	$-0.255174\pi$
$929$	3.65914e7	1.39104	0.695520	+	0.718506i	$0.255174\pi$
$930$	1.97464e7	0.748654
$931$	5.26539e6	0.199093
$932$	−1.85901e7	−0.701040
$933$	−4.20633e6	−0.158197
$934$	8.20113e6	0.307614
$935$	2.16956e7	0.811600
$936$	−876096.	−0.0326860
$937$	2.25358e7	0.838540	0.419270	−	0.907862i	$-0.362286\pi$
$937$	2.25358e7	0.838540	0.419270	+	0.907862i	$0.362286\pi$
$938$	9.74002e6	0.361454
$939$	1.05135e7	0.389120
$940$	−1.83566e7	−0.677599
$941$	−3.43872e7	−1.26597	−0.632984	−	0.774165i	$-0.718170\pi$
$941$	−3.43872e7	−1.26597	−0.632984	+	0.774165i	$0.718170\pi$
$942$	1.14815e7	0.421572
$943$	3.97485e7	1.45560
$944$	8.11520e6	0.296394
$945$	−2.17898e6	−0.0793732
$946$	1.01179e7	0.367589
$947$	2.62050e7	0.949531	0.474765	−	0.880112i	$-0.342533\pi$
$947$	2.62050e7	0.949531	0.474765	+	0.880112i	$0.342533\pi$
$948$	−2.54563e6	−0.0919972
$949$	−1.01632e7	−0.366322
$950$	5.22811e6	0.187947
$951$	−1.39900e7	−0.501610
$952$	6.16224e6	0.220367
$953$	−3.53081e7	−1.25934	−0.629668	−	0.776864i	$-0.716809\pi$
$953$	−3.53081e7	−1.25934	−0.629668	+	0.776864i	$0.716809\pi$
$954$	674568.	0.0239969
$955$	−2.30522e7	−0.817907
$956$	2.30705e7	0.816418
$957$	1.11440e7	0.393333
$958$	−3.23359e7	−1.13834
$959$	−3.33587e6	−0.117129
$960$	2.24870e6	0.0787507
$961$	5.22269e7	1.82426
$962$	6.59303e6	0.229693
$963$	−9.06763e6	−0.315085
$964$	1.36025e7	0.471441
$965$	−2.22167e7	−0.768000
$966$	−7.08246e6	−0.244198
$967$	2.77439e7	0.954116	0.477058	−	0.878872i	$-0.341703\pi$
$967$	2.77439e7	0.954116	0.477058	+	0.878872i	$0.341703\pi$
$968$	−8.21056e6	−0.281634
$969$	3.87832e7	1.32689
$970$	−9.28518e6	−0.316855
$971$	1.66754e7	0.567581	0.283791	−	0.958886i	$-0.408408\pi$
$971$	1.66754e7	0.567581	0.283791	+	0.958886i	$0.408408\pi$
$972$	−944784.	−0.0320750
$973$	−6.02602e6	−0.204056
$974$	−3.28096e7	−1.10816
$975$	906516.	0.0305396
$976$	5.52371e6	0.185612
$977$	393.000	1.31721e−5 0	6.58607e−6	−	1.00000i	$-0.499998\pi$
$977$	393.000	1.31721e−5 0	6.58607e−6		1.00000i	$0.499998\pi$
$978$	−1.94288e7	−0.649531
$979$	1.47859e7	0.493050
$980$	−2.34338e6	−0.0779429
$981$	−3.48454e6	−0.115604
$982$	1.75810e7	0.581788
$983$	3.59949e7	1.18811	0.594055	−	0.804424i	$-0.297526\pi$
$983$	3.59949e7	1.18811	0.594055	+	0.804424i	$0.297526\pi$
$984$	5.70240e6	0.187746
$985$	−3.52392e7	−1.15727
$986$	5.37703e7	1.76137
$987$	8.29433e6	0.271012
$988$	−5.92987e6	−0.193265
$989$	−5.61096e7	−1.82409
$990$	−3.57728e6	−0.116002
$991$	−4.58806e7	−1.48404	−0.742019	−	0.670379i	$-0.766131\pi$
$991$	−4.58806e7	−1.48404	−0.742019	+	0.670379i	$0.766131\pi$
$992$	9.20781e6	0.297083
$993$	−308466.	−0.00992737
$994$	−1.04860e7	−0.336623
$995$	−2.23333e7	−0.715145
$996$	−1.16148e7	−0.370989
$997$	−6.92934e6	−0.220777	−0.110389	−	0.993889i	$-0.535210\pi$
$997$	−6.92934e6	−0.220777	−0.110389	+	0.993889i	$0.535210\pi$
$998$	−4.39569e7	−1.39701
$999$	7.10994e6	0.225399

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.6.a.c.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

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