LMFDB - Embedded newform 57.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [57,6,Mod(1,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("57.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 57 = 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	57.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:	$9.14187772934$
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	$\chi$	$=$	57.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+11.0000 q^{2} +9.00000 q^{3} +89.0000 q^{4} +6.00000 q^{5} +99.0000 q^{6} -176.000 q^{7} +627.000 q^{8} +81.0000 q^{9} +66.0000 q^{10} -496.000 q^{11} +801.000 q^{12} -178.000 q^{13} -1936.00 q^{14} +54.0000 q^{15} +4049.00 q^{16} +202.000 q^{17} +891.000 q^{18} -361.000 q^{19} +534.000 q^{20} -1584.00 q^{21} -5456.00 q^{22} +4396.00 q^{23} +5643.00 q^{24} -3089.00 q^{25} -1958.00 q^{26} +729.000 q^{27} -15664.0 q^{28} -5902.00 q^{29} +594.000 q^{30} +5760.00 q^{31} +24475.0 q^{32} -4464.00 q^{33} +2222.00 q^{34} -1056.00 q^{35} +7209.00 q^{36} -3906.00 q^{37} -3971.00 q^{38} -1602.00 q^{39} +3762.00 q^{40} +15774.0 q^{41} -17424.0 q^{42} -7492.00 q^{43} -44144.0 q^{44} +486.000 q^{45} +48356.0 q^{46} -7452.00 q^{47} +36441.0 q^{48} +14169.0 q^{49} -33979.0 q^{50} +1818.00 q^{51} -15842.0 q^{52} -29014.0 q^{53} +8019.00 q^{54} -2976.00 q^{55} -110352. q^{56} -3249.00 q^{57} -64922.0 q^{58} +13604.0 q^{59} +4806.00 q^{60} -12466.0 q^{61} +63360.0 q^{62} -14256.0 q^{63} +139657. q^{64} -1068.00 q^{65} -49104.0 q^{66} +43436.0 q^{67} +17978.0 q^{68} +39564.0 q^{69} -11616.0 q^{70} +28800.0 q^{71} +50787.0 q^{72} +80746.0 q^{73} -42966.0 q^{74} -27801.0 q^{75} -32129.0 q^{76} +87296.0 q^{77} -17622.0 q^{78} +76456.0 q^{79} +24294.0 q^{80} +6561.00 q^{81} +173514. q^{82} -56880.0 q^{83} -140976. q^{84} +1212.00 q^{85} -82412.0 q^{86} -53118.0 q^{87} -310992. q^{88} -103266. q^{89} +5346.00 q^{90} +31328.0 q^{91} +391244. q^{92} +51840.0 q^{93} -81972.0 q^{94} -2166.00 q^{95} +220275. q^{96} +82490.0 q^{97} +155859. q^{98} -40176.0 q^{99} +O(q^{100})\)

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 $

$2$	11.0000	1.94454	0.972272	−	0.233854i	$-0.0751336\pi$
$2$	11.0000	1.94454	0.972272	+	0.233854i	$0.0751336\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	89.0000	2.78125
$5$	6.00000	0.107331	0.0536656	−	0.998559i	$-0.482909\pi$
$5$	6.00000	0.107331	0.0536656	+	0.998559i	$0.482909\pi$
$6$	99.0000	1.12268
$7$	−176.000	−1.35759	−0.678793	−	0.734329i	$-0.737497\pi$
$7$	−176.000	−1.35759	−0.678793	+	0.734329i	$0.737497\pi$
$8$	627.000	3.46372
$9$	81.0000	0.333333
$10$	66.0000	0.208710
$11$	−496.000	−1.23595	−0.617974	−	0.786199i	$-0.712046\pi$
$11$	−496.000	−1.23595	−0.617974	+	0.786199i	$0.712046\pi$
$12$	801.000	1.60576
$13$	−178.000	−0.292120	−0.146060	−	0.989276i	$-0.546659\pi$
$13$	−178.000	−0.292120	−0.146060	+	0.989276i	$0.546659\pi$
$14$	−1936.00	−2.63989
$15$	54.0000	0.0619677
$16$	4049.00	3.95410
$17$	202.000	0.169523	0.0847616	−	0.996401i	$-0.472987\pi$
$17$	202.000	0.169523	0.0847616	+	0.996401i	$0.472987\pi$
$18$	891.000	0.648181
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	534.000	0.298515
$21$	−1584.00	−0.783803
$22$	−5456.00	−2.40335
$23$	4396.00	1.73276	0.866379	−	0.499386i	$-0.166441\pi$
$23$	4396.00	1.73276	0.866379	+	0.499386i	$0.166441\pi$
$24$	5643.00	1.99978
$25$	−3089.00	−0.988480
$26$	−1958.00	−0.568041
$27$	729.000	0.192450
$28$	−15664.0	−3.77579
$29$	−5902.00	−1.30318	−0.651590	−	0.758572i	$-0.725898\pi$
$29$	−5902.00	−1.30318	−0.651590	+	0.758572i	$0.725898\pi$
$30$	594.000	0.120499
$31$	5760.00	1.07651	0.538255	−	0.842782i	$-0.319084\pi$
$31$	5760.00	1.07651	0.538255	+	0.842782i	$0.319084\pi$
$32$	24475.0	4.22520
$33$	−4464.00	−0.713575
$34$	2222.00	0.329645
$35$	−1056.00	−0.145711
$36$	7209.00	0.927083
$37$	−3906.00	−0.469059	−0.234530	−	0.972109i	$-0.575355\pi$
$37$	−3906.00	−0.469059	−0.234530	+	0.972109i	$0.575355\pi$
$38$	−3971.00	−0.446109
$39$	−1602.00	−0.168656
$40$	3762.00	0.371765
$41$	15774.0	1.46549	0.732744	−	0.680505i	$-0.238239\pi$
$41$	15774.0	1.46549	0.732744	+	0.680505i	$0.238239\pi$
$42$	−17424.0	−1.52414
$43$	−7492.00	−0.617912	−0.308956	−	0.951076i	$-0.599980\pi$
$43$	−7492.00	−0.617912	−0.308956	+	0.951076i	$0.599980\pi$
$44$	−44144.0	−3.43748
$45$	486.000	0.0357771
$46$	48356.0	3.36942
$47$	−7452.00	−0.492071	−0.246036	−	0.969261i	$-0.579128\pi$
$47$	−7452.00	−0.492071	−0.246036	+	0.969261i	$0.579128\pi$
$48$	36441.0	2.28290
$49$	14169.0	0.843042
$50$	−33979.0	−1.92214
$51$	1818.00	0.0978742
$52$	−15842.0	−0.812459
$53$	−29014.0	−1.41879	−0.709395	−	0.704811i	$-0.751032\pi$
$53$	−29014.0	−1.41879	−0.709395	+	0.704811i	$0.751032\pi$
$54$	8019.00	0.374228
$55$	−2976.00	−0.132656
$56$	−110352.	−4.70230
$57$	−3249.00	−0.132453
$58$	−64922.0	−2.53409
$59$	13604.0	0.508788	0.254394	−	0.967101i	$-0.418124\pi$
$59$	13604.0	0.508788	0.254394	+	0.967101i	$0.418124\pi$
$60$	4806.00	0.172348
$61$	−12466.0	−0.428946	−0.214473	−	0.976730i	$-0.568803\pi$
$61$	−12466.0	−0.428946	−0.214473	+	0.976730i	$0.568803\pi$
$62$	63360.0	2.09332
$63$	−14256.0	−0.452529
$64$	139657.	4.26199
$65$	−1068.00	−0.0313536
$66$	−49104.0	−1.38758
$67$	43436.0	1.18212	0.591062	−	0.806626i	$-0.298709\pi$
$67$	43436.0	1.18212	0.591062	+	0.806626i	$0.298709\pi$
$68$	17978.0	0.471486
$69$	39564.0	1.00041
$70$	−11616.0	−0.283342
$71$	28800.0	0.678026	0.339013	−	0.940782i	$-0.389907\pi$
$71$	28800.0	0.678026	0.339013	+	0.940782i	$0.389907\pi$
$72$	50787.0	1.15457
$73$	80746.0	1.77343	0.886715	−	0.462317i	$-0.152982\pi$
$73$	80746.0	1.77343	0.886715	+	0.462317i	$0.152982\pi$
$74$	−42966.0	−0.912107
$75$	−27801.0	−0.570699
$76$	−32129.0	−0.638063
$77$	87296.0	1.67791
$78$	−17622.0	−0.327958
$79$	76456.0	1.37830	0.689150	−	0.724619i	$-0.257984\pi$
$79$	76456.0	1.37830	0.689150	+	0.724619i	$0.257984\pi$
$80$	24294.0	0.424399
$81$	6561.00	0.111111
$82$	173514.	2.84970
$83$	−56880.0	−0.906284	−0.453142	−	0.891438i	$-0.649697\pi$
$83$	−56880.0	−0.906284	−0.453142	+	0.891438i	$0.649697\pi$
$84$	−140976.	−2.17995
$85$	1212.00	0.0181951
$86$	−82412.0	−1.20156
$87$	−53118.0	−0.752391
$88$	−310992.	−4.28097
$89$	−103266.	−1.38192	−0.690959	−	0.722894i	$-0.742812\pi$
$89$	−103266.	−1.38192	−0.690959	+	0.722894i	$0.742812\pi$
$90$	5346.00	0.0695701
$91$	31328.0	0.396579
$92$	391244.	4.81924
$93$	51840.0	0.621524
$94$	−81972.0	−0.956854
$95$	−2166.00	−0.0246235
$96$	220275.	2.43942
$97$	82490.0	0.890168	0.445084	−	0.895489i	$-0.353174\pi$
$97$	82490.0	0.890168	0.445084	+	0.895489i	$0.353174\pi$
$98$	155859.	1.63933
$99$	−40176.0	−0.411982
$100$	−274921.	−2.74921
$101$	47230.0	0.460696	0.230348	−	0.973108i	$-0.426014\pi$
$101$	47230.0	0.460696	0.230348	+	0.973108i	$0.426014\pi$
$102$	19998.0	0.190321
$103$	157456.	1.46240	0.731200	−	0.682163i	$-0.238961\pi$
$103$	157456.	1.46240	0.731200	+	0.682163i	$0.238961\pi$
$104$	−111606.	−1.01182
$105$	−9504.00	−0.0841266
$106$	−319154.	−2.75890
$107$	−62988.0	−0.531861	−0.265931	−	0.963992i	$-0.585679\pi$
$107$	−62988.0	−0.531861	−0.265931	+	0.963992i	$0.585679\pi$
$108$	64881.0	0.535252
$109$	38158.0	0.307623	0.153812	−	0.988100i	$-0.450845\pi$
$109$	38158.0	0.307623	0.153812	+	0.988100i	$0.450845\pi$
$110$	−32736.0	−0.257955
$111$	−35154.0	−0.270812
$112$	−712624.	−5.36804
$113$	9190.00	0.0677048	0.0338524	−	0.999427i	$-0.489222\pi$
$113$	9190.00	0.0677048	0.0338524	+	0.999427i	$0.489222\pi$
$114$	−35739.0	−0.257561
$115$	26376.0	0.185979
$116$	−525278.	−3.62447
$117$	−14418.0	−0.0973734
$118$	149644.	0.989360
$119$	−35552.0	−0.230142
$120$	33858.0	0.214639
$121$	84965.0	0.527566
$122$	−137126.	−0.834104
$123$	141966.	0.846100
$124$	512640.	2.99404
$125$	−37284.0	−0.213426
$126$	−156816.	−0.879962
$127$	−70448.0	−0.387578	−0.193789	−	0.981043i	$-0.562078\pi$
$127$	−70448.0	−0.387578	−0.193789	+	0.981043i	$0.562078\pi$
$128$	753027.	4.06243
$129$	−67428.0	−0.356752
$130$	−11748.0	−0.0609685
$131$	101864.	0.518612	0.259306	−	0.965795i	$-0.416506\pi$
$131$	101864.	0.518612	0.259306	+	0.965795i	$0.416506\pi$
$132$	−397296.	−1.98463
$133$	63536.0	0.311452
$134$	477796.	2.29869
$135$	4374.00	0.0206559
$136$	126654.	0.587181
$137$	−432126.	−1.96702	−0.983510	−	0.180851i	$-0.942115\pi$
$137$	−432126.	−1.96702	−0.983510	+	0.180851i	$0.942115\pi$
$138$	435204.	1.94534
$139$	−376684.	−1.65364	−0.826818	−	0.562469i	$-0.809852\pi$
$139$	−376684.	−1.65364	−0.826818	+	0.562469i	$0.809852\pi$
$140$	−93984.0	−0.405260
$141$	−67068.0	−0.284098
$142$	316800.	1.31845
$143$	88288.0	0.361045
$144$	327969.	1.31803
$145$	−35412.0	−0.139872
$146$	888206.	3.44851
$147$	127521.	0.486730
$148$	−347634.	−1.30457
$149$	−283554.	−1.04633	−0.523167	−	0.852230i	$-0.675249\pi$
$149$	−283554.	−1.04633	−0.523167	+	0.852230i	$0.675249\pi$
$150$	−305811.	−1.10975
$151$	−79200.0	−0.282672	−0.141336	−	0.989962i	$-0.545140\pi$
$151$	−79200.0	−0.282672	−0.141336	+	0.989962i	$0.545140\pi$
$152$	−226347.	−0.794631
$153$	16362.0	0.0565077
$154$	960256.	3.26276
$155$	34560.0	0.115543
$156$	−142578.	−0.469074
$157$	−129858.	−0.420455	−0.210228	−	0.977652i	$-0.567420\pi$
$157$	−129858.	−0.420455	−0.210228	+	0.977652i	$0.567420\pi$
$158$	841016.	2.68017
$159$	−261126.	−0.819138
$160$	146850.	0.453497
$161$	−773696.	−2.35237
$162$	72171.0	0.216060
$163$	57420.0	0.169276	0.0846378	−	0.996412i	$-0.473027\pi$
$163$	57420.0	0.169276	0.0846378	+	0.996412i	$0.473027\pi$
$164$	1.40389e6	4.07589
$165$	−26784.0	−0.0765889
$166$	−625680.	−1.76231
$167$	−254008.	−0.704784	−0.352392	−	0.935852i	$-0.614632\pi$
$167$	−254008.	−0.704784	−0.352392	+	0.935852i	$0.614632\pi$
$168$	−993168.	−2.71487
$169$	−339609.	−0.914666
$170$	13332.0	0.0353812
$171$	−29241.0	−0.0764719
$172$	−666788.	−1.71857
$173$	−177366.	−0.450563	−0.225281	−	0.974294i	$-0.572330\pi$
$173$	−177366.	−0.450563	−0.225281	+	0.974294i	$0.572330\pi$
$174$	−584298.	−1.46306
$175$	543664.	1.34195
$176$	−2.00830e6	−4.88706
$177$	122436.	0.293749
$178$	−1.13593e6	−2.68720
$179$	−25188.0	−0.0587572	−0.0293786	−	0.999568i	$-0.509353\pi$
$179$	−25188.0	−0.0587572	−0.0293786	+	0.999568i	$0.509353\pi$
$180$	43254.0	0.0995050
$181$	729382.	1.65485	0.827425	−	0.561576i	$-0.189805\pi$
$181$	729382.	1.65485	0.827425	+	0.561576i	$0.189805\pi$
$182$	344608.	0.771164
$183$	−112194.	−0.247652
$184$	2.75629e6	6.00179
$185$	−23436.0	−0.0503447
$186$	570240.	1.20858
$187$	−100192.	−0.209522
$188$	−663228.	−1.36857
$189$	−128304.	−0.261268
$190$	−23826.0	−0.0478814
$191$	−285060.	−0.565396	−0.282698	−	0.959209i	$-0.591229\pi$
$191$	−285060.	−0.565396	−0.282698	+	0.959209i	$0.591229\pi$
$192$	1.25691e6	2.46066
$193$	−457598.	−0.884282	−0.442141	−	0.896946i	$-0.645781\pi$
$193$	−457598.	−0.884282	−0.442141	+	0.896946i	$0.645781\pi$
$194$	907390.	1.73097
$195$	−9612.00	−0.0181020
$196$	1.26104e6	2.34471
$197$	−291178.	−0.534556	−0.267278	−	0.963620i	$-0.586124\pi$
$197$	−291178.	−0.534556	−0.267278	+	0.963620i	$0.586124\pi$
$198$	−441936.	−0.801118
$199$	364680.	0.652799	0.326399	−	0.945232i	$-0.394165\pi$
$199$	364680.	0.652799	0.326399	+	0.945232i	$0.394165\pi$
$200$	−1.93680e6	−3.42382
$201$	390924.	0.682499
$202$	519530.	0.895844
$203$	1.03875e6	1.76918
$204$	161802.	0.272213
$205$	94644.0	0.157293
$206$	1.73202e6	2.84370
$207$	356076.	0.577586
$208$	−720722.	−1.15507
$209$	179056.	0.283546
$210$	−104544.	−0.163588
$211$	−452388.	−0.699528	−0.349764	−	0.936838i	$-0.613738\pi$
$211$	−452388.	−0.699528	−0.349764	+	0.936838i	$0.613738\pi$
$212$	−2.58225e6	−3.94601
$213$	259200.	0.391459
$214$	−692868.	−1.03423
$215$	−44952.0	−0.0663213
$216$	457083.	0.666593
$217$	−1.01376e6	−1.46146
$218$	419738.	0.598187
$219$	726714.	1.02389
$220$	−264864.	−0.368949
$221$	−35956.0	−0.0495211
$222$	−386694.	−0.526605
$223$	940296.	1.26620	0.633100	−	0.774070i	$-0.281782\pi$
$223$	940296.	1.26620	0.633100	+	0.774070i	$0.281782\pi$
$224$	−4.30760e6	−5.73608
$225$	−250209.	−0.329493
$226$	101090.	0.131655
$227$	796308.	1.02569	0.512845	−	0.858481i	$-0.328591\pi$
$227$	796308.	1.02569	0.512845	+	0.858481i	$0.328591\pi$
$228$	−289161.	−0.368386
$229$	153334.	0.193219	0.0966095	−	0.995322i	$-0.469200\pi$
$229$	153334.	0.193219	0.0966095	+	0.995322i	$0.469200\pi$
$230$	290136.	0.361645
$231$	785664.	0.968739
$232$	−3.70055e6	−4.51385
$233$	246858.	0.297891	0.148946	−	0.988845i	$-0.452412\pi$
$233$	246858.	0.297891	0.148946	+	0.988845i	$0.452412\pi$
$234$	−158598.	−0.189347
$235$	−44712.0	−0.0528147
$236$	1.21076e6	1.41507
$237$	688104.	0.795762
$238$	−391072.	−0.447522
$239$	105516.	0.119488	0.0597439	−	0.998214i	$-0.480972\pi$
$239$	105516.	0.119488	0.0597439	+	0.998214i	$0.480972\pi$
$240$	218646.	0.245027
$241$	41738.0	0.0462902	0.0231451	−	0.999732i	$-0.492632\pi$
$241$	41738.0	0.0462902	0.0231451	+	0.999732i	$0.492632\pi$
$242$	934615.	1.02587
$243$	59049.0	0.0641500
$244$	−1.10947e6	−1.19301
$245$	85014.0	0.0904847
$246$	1.56163e6	1.64528
$247$	64258.0	0.0670170
$248$	3.61152e6	3.72873
$249$	−511920.	−0.523243
$250$	−410124.	−0.415016
$251$	−362392.	−0.363073	−0.181537	−	0.983384i	$-0.558107\pi$
$251$	−362392.	−0.363073	−0.181537	+	0.983384i	$0.558107\pi$
$252$	−1.26878e6	−1.25860
$253$	−2.18042e6	−2.14160
$254$	−774928.	−0.753663
$255$	10908.0	0.0105050
$256$	3.81427e6	3.63757
$257$	−1.16120e6	−1.09667	−0.548334	−	0.836260i	$-0.684738\pi$
$257$	−1.16120e6	−1.09667	−0.548334	+	0.836260i	$0.684738\pi$
$258$	−741708.	−0.693719
$259$	687456.	0.636789
$260$	−95052.0	−0.0872023
$261$	−478062.	−0.434393
$262$	1.12050e6	1.00846
$263$	1.35860e6	1.21116	0.605579	−	0.795785i	$-0.292941\pi$
$263$	1.35860e6	1.21116	0.605579	+	0.795785i	$0.292941\pi$
$264$	−2.79893e6	−2.47162
$265$	−174084.	−0.152280
$266$	698896.	0.605632
$267$	−929394.	−0.797851
$268$	3.86580e6	3.28778
$269$	2.19871e6	1.85263	0.926314	−	0.376753i	$-0.122960\pi$
$269$	2.19871e6	1.85263	0.926314	+	0.376753i	$0.122960\pi$
$270$	48114.0	0.0401663
$271$	−512016.	−0.423507	−0.211753	−	0.977323i	$-0.567917\pi$
$271$	−512016.	−0.423507	−0.211753	+	0.977323i	$0.567917\pi$
$272$	817898.	0.670312
$273$	281952.	0.228965
$274$	−4.75339e6	−3.82496
$275$	1.53214e6	1.22171
$276$	3.52120e6	2.78239
$277$	857542.	0.671515	0.335758	−	0.941948i	$-0.391008\pi$
$277$	857542.	0.671515	0.335758	+	0.941948i	$0.391008\pi$
$278$	−4.14352e6	−3.21557
$279$	466560.	0.358837
$280$	−662112.	−0.504704
$281$	−375370.	−0.283592	−0.141796	−	0.989896i	$-0.545288\pi$
$281$	−375370.	−0.283592	−0.141796	+	0.989896i	$0.545288\pi$
$282$	−737748.	−0.552440
$283$	2204.00	0.00163586	0.000817929	−	1.00000i	$-0.499740\pi$
$283$	2204.00	0.00163586	0.000817929		1.00000i	$0.499740\pi$
$284$	2.56320e6	1.88576
$285$	−19494.0	−0.0142164
$286$	971168.	0.702068
$287$	−2.77622e6	−1.98953
$288$	1.98248e6	1.40840
$289$	−1.37905e6	−0.971262
$290$	−389532.	−0.271987
$291$	742410.	0.513939
$292$	7.18639e6	4.93235
$293$	−605198.	−0.411840	−0.205920	−	0.978569i	$-0.566019\pi$
$293$	−605198.	−0.411840	−0.205920	+	0.978569i	$0.566019\pi$
$294$	1.40273e6	0.946468
$295$	81624.0	0.0546088
$296$	−2.44906e6	−1.62469
$297$	−361584.	−0.237858
$298$	−3.11909e6	−2.03464
$299$	−782488.	−0.506174
$300$	−2.47429e6	−1.58726
$301$	1.31859e6	0.838869
$302$	−871200.	−0.549668
$303$	425070.	0.265983
$304$	−1.46169e6	−0.907133
$305$	−74796.0	−0.0460393
$306$	179982.	0.109882
$307$	−1.25593e6	−0.760537	−0.380268	−	0.924876i	$-0.624168\pi$
$307$	−1.25593e6	−0.760537	−0.380268	+	0.924876i	$0.624168\pi$
$308$	7.76934e6	4.66668
$309$	1.41710e6	0.844317
$310$	380160.	0.224679
$311$	−824580.	−0.483428	−0.241714	−	0.970348i	$-0.577710\pi$
$311$	−824580.	−0.483428	−0.241714	+	0.970348i	$0.577710\pi$
$312$	−1.00445e6	−0.584176
$313$	−1.23455e6	−0.712275	−0.356138	−	0.934434i	$-0.615907\pi$
$313$	−1.23455e6	−0.712275	−0.356138	+	0.934434i	$0.615907\pi$
$314$	−1.42844e6	−0.817593
$315$	−85536.0	−0.0485705
$316$	6.80458e6	3.83340
$317$	−1.19428e6	−0.667509	−0.333755	−	0.942660i	$-0.608316\pi$
$317$	−1.19428e6	−0.667509	−0.333755	+	0.942660i	$0.608316\pi$
$318$	−2.87239e6	−1.59285
$319$	2.92739e6	1.61066
$320$	837942.	0.457445
$321$	−566892.	−0.307070
$322$	−8.51066e6	−4.57429
$323$	−72922.0	−0.0388913
$324$	583929.	0.309028
$325$	549842.	0.288755
$326$	631620.	0.329164
$327$	343422.	0.177606
$328$	9.89030e6	5.07604
$329$	1.31155e6	0.668030
$330$	−294624.	−0.148930
$331$	1.99113e6	0.998919	0.499459	−	0.866337i	$-0.333532\pi$
$331$	1.99113e6	0.998919	0.499459	+	0.866337i	$0.333532\pi$
$332$	−5.06232e6	−2.52060
$333$	−316386.	−0.156353
$334$	−2.79409e6	−1.37048
$335$	260616.	0.126879
$336$	−6.41362e6	−3.09924
$337$	6442.00	0.00308991	0.00154496	−	0.999999i	$-0.499508\pi$
$337$	6442.00	0.00308991	0.00154496	+	0.999999i	$0.499508\pi$
$338$	−3.73570e6	−1.77861
$339$	82710.0	0.0390894
$340$	107868.	0.0506052
$341$	−2.85696e6	−1.33051
$342$	−321651.	−0.148703
$343$	464288.	0.213085
$344$	−4.69748e6	−2.14027
$345$	237384.	0.107375
$346$	−1.95103e6	−0.876139
$347$	−1.66693e6	−0.743179	−0.371589	−	0.928397i	$-0.621187\pi$
$347$	−1.66693e6	−0.743179	−0.371589	+	0.928397i	$0.621187\pi$
$348$	−4.72750e6	−2.09259
$349$	−3.89805e6	−1.71310	−0.856552	−	0.516060i	$-0.827398\pi$
$349$	−3.89805e6	−1.71310	−0.856552	+	0.516060i	$0.827398\pi$
$350$	5.98030e6	2.60948
$351$	−129762.	−0.0562186
$352$	−1.21396e7	−5.22213
$353$	407306.	0.173974	0.0869869	−	0.996209i	$-0.472276\pi$
$353$	407306.	0.173974	0.0869869	+	0.996209i	$0.472276\pi$
$354$	1.34680e6	0.571207
$355$	172800.	0.0727734
$356$	−9.19067e6	−3.84346
$357$	−319968.	−0.132873
$358$	−277068.	−0.114256
$359$	2.59413e6	1.06232	0.531161	−	0.847271i	$-0.321756\pi$
$359$	2.59413e6	1.06232	0.531161	+	0.847271i	$0.321756\pi$
$360$	304722.	0.123922
$361$	130321.	0.0526316
$362$	8.02320e6	3.21793
$363$	764685.	0.304590
$364$	2.78819e6	1.10298
$365$	484476.	0.190344
$366$	−1.23413e6	−0.481570
$367$	−761840.	−0.295256	−0.147628	−	0.989043i	$-0.547164\pi$
$367$	−761840.	−0.295256	−0.147628	+	0.989043i	$0.547164\pi$
$368$	1.77994e7	6.85150
$369$	1.27769e6	0.488496
$370$	−257796.	−0.0978976
$371$	5.10646e6	1.92613
$372$	4.61376e6	1.72861
$373$	−837506.	−0.311685	−0.155842	−	0.987782i	$-0.549809\pi$
$373$	−837506.	−0.311685	−0.155842	+	0.987782i	$0.549809\pi$
$374$	−1.10211e6	−0.407424
$375$	−335556.	−0.123222
$376$	−4.67240e6	−1.70440
$377$	1.05056e6	0.380685
$378$	−1.41134e6	−0.508046
$379$	−623876.	−0.223100	−0.111550	−	0.993759i	$-0.535582\pi$
$379$	−623876.	−0.223100	−0.111550	+	0.993759i	$0.535582\pi$
$380$	−192774.	−0.0684841
$381$	−634032.	−0.223768
$382$	−3.13566e6	−1.09944
$383$	97672.0	0.0340230	0.0170115	−	0.999855i	$-0.494585\pi$
$383$	97672.0	0.0340230	0.0170115	+	0.999855i	$0.494585\pi$
$384$	6.77724e6	2.34544
$385$	523776.	0.180092
$386$	−5.03358e6	−1.71953
$387$	−606852.	−0.205971
$388$	7.34161e6	2.47578
$389$	−2.23487e6	−0.748823	−0.374411	−	0.927263i	$-0.622155\pi$
$389$	−2.23487e6	−0.748823	−0.374411	+	0.927263i	$0.622155\pi$
$390$	−105732.	−0.0352002
$391$	887992.	0.293743
$392$	8.88396e6	2.92006
$393$	916776.	0.299421
$394$	−3.20296e6	−1.03947
$395$	458736.	0.147935
$396$	−3.57566e6	−1.14583
$397$	4.93416e6	1.57122	0.785610	−	0.618723i	$-0.212349\pi$
$397$	4.93416e6	1.57122	0.785610	+	0.618723i	$0.212349\pi$
$398$	4.01148e6	1.26940
$399$	571824.	0.179817
$400$	−1.25074e7	−3.90855
$401$	−3.73411e6	−1.15965	−0.579823	−	0.814742i	$-0.696878\pi$
$401$	−3.73411e6	−1.15965	−0.579823	+	0.814742i	$0.696878\pi$
$402$	4.30016e6	1.32715
$403$	−1.02528e6	−0.314470
$404$	4.20347e6	1.28131
$405$	39366.0	0.0119257
$406$	1.14263e7	3.44025
$407$	1.93738e6	0.579733
$408$	1.13989e6	0.339009
$409$	−2.46083e6	−0.727400	−0.363700	−	0.931516i	$-0.618487\pi$
$409$	−2.46083e6	−0.727400	−0.363700	+	0.931516i	$0.618487\pi$
$410$	1.04108e6	0.305862
$411$	−3.88913e6	−1.13566
$412$	1.40136e7	4.06730
$413$	−2.39430e6	−0.690723
$414$	3.91684e6	1.12314
$415$	−341280.	−0.0972726
$416$	−4.35655e6	−1.23427
$417$	−3.39016e6	−0.954728
$418$	1.96962e6	0.551367
$419$	−437512.	−0.121746	−0.0608730	−	0.998146i	$-0.519388\pi$
$419$	−437512.	−0.121746	−0.0608730	+	0.998146i	$0.519388\pi$
$420$	−845856.	−0.233977
$421$	2.91013e6	0.800217	0.400108	−	0.916468i	$-0.368972\pi$
$421$	2.91013e6	0.800217	0.400108	+	0.916468i	$0.368972\pi$
$422$	−4.97627e6	−1.36026
$423$	−603612.	−0.164024
$424$	−1.81918e7	−4.91429
$425$	−623978.	−0.167570
$426$	2.85120e6	0.761209
$427$	2.19402e6	0.582331
$428$	−5.60593e6	−1.47924
$429$	794592.	0.208450
$430$	−494472.	−0.128965
$431$	−4.64881e6	−1.20545	−0.602724	−	0.797950i	$-0.705918\pi$
$431$	−4.64881e6	−1.20545	−0.602724	+	0.797950i	$0.705918\pi$
$432$	2.95172e6	0.760967
$433$	6.90871e6	1.77083	0.885415	−	0.464801i	$-0.153874\pi$
$433$	6.90871e6	1.77083	0.885415	+	0.464801i	$0.153874\pi$
$434$	−1.11514e7	−2.84187
$435$	−318708.	−0.0807551
$436$	3.39606e6	0.855578
$437$	−1.58696e6	−0.397522
$438$	7.99385e6	1.99100
$439$	6.25621e6	1.54935	0.774676	−	0.632359i	$-0.217913\pi$
$439$	6.25621e6	1.54935	0.774676	+	0.632359i	$0.217913\pi$
$440$	−1.86595e6	−0.459482
$441$	1.14769e6	0.281014
$442$	−395516.	−0.0962960
$443$	2.14088e6	0.518302	0.259151	−	0.965837i	$-0.416557\pi$
$443$	2.14088e6	0.518302	0.259151	+	0.965837i	$0.416557\pi$
$444$	−3.12871e6	−0.753195
$445$	−619596.	−0.148323
$446$	1.03433e7	2.46218
$447$	−2.55199e6	−0.604101
$448$	−2.45796e7	−5.78603
$449$	4.92089e6	1.15194	0.575968	−	0.817472i	$-0.304625\pi$
$449$	4.92089e6	1.15194	0.575968	+	0.817472i	$0.304625\pi$
$450$	−2.75230e6	−0.640714
$451$	−7.82390e6	−1.81127
$452$	817910.	0.188304
$453$	−712800.	−0.163201
$454$	8.75939e6	1.99450
$455$	187968.	0.0425653
$456$	−2.03712e6	−0.458781
$457$	3.94009e6	0.882502	0.441251	−	0.897384i	$-0.354535\pi$
$457$	3.94009e6	0.882502	0.441251	+	0.897384i	$0.354535\pi$
$458$	1.68667e6	0.375723
$459$	147258.	0.0326247
$460$	2.34746e6	0.517255
$461$	−2.25945e6	−0.495166	−0.247583	−	0.968867i	$-0.579636\pi$
$461$	−2.25945e6	−0.495166	−0.247583	+	0.968867i	$0.579636\pi$
$462$	8.64230e6	1.88376
$463$	−3.31806e6	−0.719337	−0.359668	−	0.933080i	$-0.617110\pi$
$463$	−3.31806e6	−0.719337	−0.359668	+	0.933080i	$0.617110\pi$
$464$	−2.38972e7	−5.15290
$465$	311040.	0.0667089
$466$	2.71544e6	0.579262
$467$	3.63171e6	0.770583	0.385291	−	0.922795i	$-0.374101\pi$
$467$	3.63171e6	0.770583	0.385291	+	0.922795i	$0.374101\pi$
$468$	−1.28320e6	−0.270820
$469$	−7.64474e6	−1.60483
$470$	−491832.	−0.102700
$471$	−1.16872e6	−0.242750
$472$	8.52971e6	1.76230
$473$	3.71603e6	0.763707
$474$	7.56914e6	1.54739
$475$	1.11513e6	0.226773
$476$	−3.16413e6	−0.640084
$477$	−2.35013e6	−0.472930
$478$	1.16068e6	0.232349
$479$	−1.45764e6	−0.290275	−0.145138	−	0.989411i	$-0.546363\pi$
$479$	−1.45764e6	−0.290275	−0.145138	+	0.989411i	$0.546363\pi$
$480$	1.32165e6	0.261826
$481$	695268.	0.137022
$482$	459118.	0.0900133
$483$	−6.96326e6	−1.35814
$484$	7.56188e6	1.46729
$485$	494940.	0.0955429
$486$	649539.	0.124743
$487$	6.73825e6	1.28743	0.643716	−	0.765264i	$-0.277392\pi$
$487$	6.73825e6	1.28743	0.643716	+	0.765264i	$0.277392\pi$
$488$	−7.81618e6	−1.48575
$489$	516780.	0.0977313
$490$	935154.	0.175951
$491$	3.47875e6	0.651208	0.325604	−	0.945506i	$-0.394432\pi$
$491$	3.47875e6	0.651208	0.325604	+	0.945506i	$0.394432\pi$
$492$	1.26350e7	2.35321
$493$	−1.19220e6	−0.220919
$494$	706838.	0.130317
$495$	−241056.	−0.0442186
$496$	2.33222e7	4.25663
$497$	−5.06880e6	−0.920480
$498$	−5.63112e6	−1.01747
$499$	−9.39514e6	−1.68909	−0.844543	−	0.535487i	$-0.820128\pi$
$499$	−9.39514e6	−1.68909	−0.844543	+	0.535487i	$0.820128\pi$
$500$	−3.31828e6	−0.593591
$501$	−2.28607e6	−0.406907
$502$	−3.98631e6	−0.706012
$503$	9.43514e6	1.66276	0.831378	−	0.555708i	$-0.187553\pi$
$503$	9.43514e6	1.66276	0.831378	+	0.555708i	$0.187553\pi$
$504$	−8.93851e6	−1.56743
$505$	283380.	0.0494471
$506$	−2.39846e7	−4.16443
$507$	−3.05648e6	−0.528083
$508$	−6.26987e6	−1.07795
$509$	6.92644e6	1.18499	0.592496	−	0.805573i	$-0.298142\pi$
$509$	6.92644e6	1.18499	0.592496	+	0.805573i	$0.298142\pi$
$510$	119988.	0.0204274
$511$	−1.42113e7	−2.40758
$512$	1.78601e7	3.01099
$513$	−263169.	−0.0441511
$514$	−1.27732e7	−2.13252
$515$	944736.	0.156961
$516$	−6.00109e6	−0.992216
$517$	3.69619e6	0.608174
$518$	7.56202e6	1.23826
$519$	−1.59629e6	−0.260132
$520$	−669636.	−0.108600
$521$	−6.53061e6	−1.05405	−0.527023	−	0.849851i	$-0.676692\pi$
$521$	−6.53061e6	−1.05405	−0.527023	+	0.849851i	$0.676692\pi$
$522$	−5.25868e6	−0.844696
$523$	−1.80276e6	−0.288194	−0.144097	−	0.989564i	$-0.546028\pi$
$523$	−1.80276e6	−0.288194	−0.144097	+	0.989564i	$0.546028\pi$
$524$	9.06590e6	1.44239
$525$	4.89298e6	0.774774
$526$	1.49446e7	2.35515
$527$	1.16352e6	0.182493
$528$	−1.80747e7	−2.82155
$529$	1.28885e7	2.00245
$530$	−1.91492e6	−0.296116
$531$	1.10192e6	0.169596
$532$	5.65470e6	0.866225
$533$	−2.80777e6	−0.428099
$534$	−1.02233e7	−1.55146
$535$	−377928.	−0.0570853
$536$	2.72344e7	4.09454
$537$	−226692.	−0.0339235
$538$	2.41859e7	3.60251
$539$	−7.02782e6	−1.04196
$540$	389286.	0.0574493
$541$	1.03740e7	1.52389	0.761946	−	0.647640i	$-0.224244\pi$
$541$	1.03740e7	1.52389	0.761946	+	0.647640i	$0.224244\pi$
$542$	−5.63218e6	−0.823527
$543$	6.56444e6	0.955428
$544$	4.94395e6	0.716270
$545$	228948.	0.0330176
$546$	3.10147e6	0.445232
$547$	−8.14488e6	−1.16390	−0.581951	−	0.813224i	$-0.697710\pi$
$547$	−8.14488e6	−1.16390	−0.581951	+	0.813224i	$0.697710\pi$
$548$	−3.84592e7	−5.47078
$549$	−1.00975e6	−0.142982
$550$	1.68536e7	2.37567
$551$	2.13062e6	0.298970
$552$	2.48066e7	3.46513
$553$	−1.34563e7	−1.87116
$554$	9.43296e6	1.30579
$555$	−210924.	−0.0290666
$556$	−3.35249e7	−4.59918
$557$	−5.47163e6	−0.747271	−0.373636	−	0.927576i	$-0.621889\pi$
$557$	−5.47163e6	−0.747271	−0.373636	+	0.927576i	$0.621889\pi$
$558$	5.13216e6	0.697774
$559$	1.33358e6	0.180505
$560$	−4.27574e6	−0.576158
$561$	−901728.	−0.120967
$562$	−4.12907e6	−0.551457
$563$	6.41428e6	0.852858	0.426429	−	0.904521i	$-0.359771\pi$
$563$	6.41428e6	0.852858	0.426429	+	0.904521i	$0.359771\pi$
$564$	−5.96905e6	−0.790146
$565$	55140.0	0.00726684
$566$	24244.0	0.00318100
$567$	−1.15474e6	−0.150843
$568$	1.80576e7	2.34849
$569$	−1.25705e7	−1.62769	−0.813847	−	0.581080i	$-0.802631\pi$
$569$	−1.25705e7	−1.62769	−0.813847	+	0.581080i	$0.802631\pi$
$570$	−214434.	−0.0276444
$571$	−7.51477e6	−0.964552	−0.482276	−	0.876019i	$-0.660190\pi$
$571$	−7.51477e6	−0.964552	−0.482276	+	0.876019i	$0.660190\pi$
$572$	7.85763e6	1.00416
$573$	−2.56554e6	−0.326432
$574$	−3.05385e7	−3.86872
$575$	−1.35792e7	−1.71280
$576$	1.13122e7	1.42066
$577$	1.96226e6	0.245367	0.122684	−	0.992446i	$-0.460850\pi$
$577$	1.96226e6	0.245367	0.122684	+	0.992446i	$0.460850\pi$
$578$	−1.51696e7	−1.88866
$579$	−4.11838e6	−0.510541
$580$	−3.15167e6	−0.389019
$581$	1.00109e7	1.23036
$582$	8.16651e6	0.999376
$583$	1.43909e7	1.75355
$584$	5.06277e7	6.14266
$585$	−86508.0	−0.0104512
$586$	−6.65718e6	−0.800841
$587$	7.03206e6	0.842339	0.421170	−	0.906982i	$-0.361620\pi$
$587$	7.03206e6	0.842339	0.421170	+	0.906982i	$0.361620\pi$
$588$	1.13494e7	1.35372
$589$	−2.07936e6	−0.246968
$590$	897864.	0.106189
$591$	−2.62060e6	−0.308626
$592$	−1.58154e7	−1.85471
$593$	−4.68493e6	−0.547099	−0.273550	−	0.961858i	$-0.588198\pi$
$593$	−4.68493e6	−0.547099	−0.273550	+	0.961858i	$0.588198\pi$
$594$	−3.97742e6	−0.462526
$595$	−213312.	−0.0247015
$596$	−2.52363e7	−2.91011
$597$	3.28212e6	0.376893
$598$	−8.60737e6	−0.984277
$599$	5.00460e6	0.569905	0.284952	−	0.958542i	$-0.408022\pi$
$599$	5.00460e6	0.569905	0.284952	+	0.958542i	$0.408022\pi$
$600$	−1.74312e7	−1.97674
$601$	−1.20527e7	−1.36112	−0.680561	−	0.732692i	$-0.738264\pi$
$601$	−1.20527e7	−1.36112	−0.680561	+	0.732692i	$0.738264\pi$
$602$	1.45045e7	1.63122
$603$	3.51832e6	0.394041
$604$	−7.04880e6	−0.786182
$605$	509790.	0.0566243
$606$	4.67577e6	0.517216
$607$	−9.62474e6	−1.06027	−0.530136	−	0.847913i	$-0.677859\pi$
$607$	−9.62474e6	−1.06027	−0.530136	+	0.847913i	$0.677859\pi$
$608$	−8.83548e6	−0.969328
$609$	9.34877e6	1.02144
$610$	−822756.	−0.0895254
$611$	1.32646e6	0.143744
$612$	1.45622e6	0.157162
$613$	2.45060e6	0.263403	0.131702	−	0.991289i	$-0.457956\pi$
$613$	2.45060e6	0.263403	0.131702	+	0.991289i	$0.457956\pi$
$614$	−1.38153e7	−1.47890
$615$	851796.	0.0908130
$616$	5.47346e7	5.81179
$617$	4.17808e6	0.441839	0.220920	−	0.975292i	$-0.429094\pi$
$617$	4.17808e6	0.441839	0.220920	+	0.975292i	$0.429094\pi$
$618$	1.55881e7	1.64181
$619$	1.80615e7	1.89465	0.947323	−	0.320279i	$-0.103777\pi$
$619$	1.80615e7	1.89465	0.947323	+	0.320279i	$0.103777\pi$
$620$	3.07584e6	0.321355
$621$	3.20468e6	0.333470
$622$	−9.07038e6	−0.940047
$623$	1.81748e7	1.87607
$624$	−6.48650e6	−0.666882
$625$	9.42942e6	0.965573
$626$	−1.35800e7	−1.38505
$627$	1.61150e6	0.163705
$628$	−1.15574e7	−1.16939
$629$	−789012.	−0.0795165
$630$	−940896.	−0.0944475
$631$	1.35029e7	1.35006	0.675030	−	0.737790i	$-0.264131\pi$
$631$	1.35029e7	1.35006	0.675030	+	0.737790i	$0.264131\pi$
$632$	4.79379e7	4.77404
$633$	−4.07149e6	−0.403873
$634$	−1.31371e7	−1.29800
$635$	−422688.	−0.0415993
$636$	−2.32402e7	−2.27823
$637$	−2.52208e6	−0.246269
$638$	3.22013e7	3.13200
$639$	2.33280e6	0.226009
$640$	4.51816e6	0.436025
$641$	−8.29497e6	−0.797388	−0.398694	−	0.917084i	$-0.630536\pi$
$641$	−8.29497e6	−0.797388	−0.398694	+	0.917084i	$0.630536\pi$
$642$	−6.23581e6	−0.597112
$643$	−7.22854e6	−0.689482	−0.344741	−	0.938698i	$-0.612033\pi$
$643$	−7.22854e6	−0.689482	−0.344741	+	0.938698i	$0.612033\pi$
$644$	−6.88589e7	−6.54253
$645$	−404568.	−0.0382906
$646$	−802142.	−0.0756258
$647$	6.59915e6	0.619765	0.309883	−	0.950775i	$-0.399710\pi$
$647$	6.59915e6	0.619765	0.309883	+	0.950775i	$0.399710\pi$
$648$	4.11375e6	0.384858
$649$	−6.74758e6	−0.628835
$650$	6.04826e6	0.561497
$651$	−9.12384e6	−0.843772
$652$	5.11038e6	0.470798
$653$	1.68576e7	1.54708	0.773540	−	0.633747i	$-0.218484\pi$
$653$	1.68576e7	1.54708	0.773540	+	0.633747i	$0.218484\pi$
$654$	3.77764e6	0.345363
$655$	611184.	0.0556633
$656$	6.38689e7	5.79469
$657$	6.54043e6	0.591143
$658$	1.44271e7	1.29901
$659$	−2.13425e6	−0.191440	−0.0957199	−	0.995408i	$-0.530515\pi$
$659$	−2.13425e6	−0.191440	−0.0957199	+	0.995408i	$0.530515\pi$
$660$	−2.38378e6	−0.213013
$661$	1.65547e7	1.47373	0.736865	−	0.676040i	$-0.236305\pi$
$661$	1.65547e7	1.47373	0.736865	+	0.676040i	$0.236305\pi$
$662$	2.19025e7	1.94244
$663$	−323604.	−0.0285910
$664$	−3.56638e7	−3.13911
$665$	381216.	0.0334285
$666$	−3.48025e6	−0.304036
$667$	−2.59452e7	−2.25810
$668$	−2.26067e7	−1.96018
$669$	8.46266e6	0.731041
$670$	2.86678e6	0.246721
$671$	6.18314e6	0.530155
$672$	−3.87684e7	−3.31173
$673$	−704670.	−0.0599719	−0.0299860	−	0.999550i	$-0.509546\pi$
$673$	−704670.	−0.0599719	−0.0299860	+	0.999550i	$0.509546\pi$
$674$	70862.0	0.00600847
$675$	−2.25188e6	−0.190233
$676$	−3.02252e7	−2.54391
$677$	6.83619e6	0.573248	0.286624	−	0.958043i	$-0.407467\pi$
$677$	6.83619e6	0.573248	0.286624	+	0.958043i	$0.407467\pi$
$678$	909810.	0.0760110
$679$	−1.45182e7	−1.20848
$680$	759924.	0.0630228
$681$	7.16677e6	0.592183
$682$	−3.14266e7	−2.58724
$683$	1.24211e7	1.01884	0.509422	−	0.860517i	$-0.329859\pi$
$683$	1.24211e7	1.01884	0.509422	+	0.860517i	$0.329859\pi$
$684$	−2.60245e6	−0.212688
$685$	−2.59276e6	−0.211123
$686$	5.10717e6	0.414352
$687$	1.38001e6	0.111555
$688$	−3.03351e7	−2.44329
$689$	5.16449e6	0.414457
$690$	2.61122e6	0.208796
$691$	−8.00772e6	−0.637990	−0.318995	−	0.947756i	$-0.603345\pi$
$691$	−8.00772e6	−0.637990	−0.318995	+	0.947756i	$0.603345\pi$
$692$	−1.57856e7	−1.25313
$693$	7.07098e6	0.559302
$694$	−1.83362e7	−1.44514
$695$	−2.26010e6	−0.177487
$696$	−3.33050e7	−2.60607
$697$	3.18635e6	0.248434
$698$	−4.28786e7	−3.33121
$699$	2.22172e6	0.171987
$700$	4.83861e7	3.73229
$701$	−1.18657e7	−0.912005	−0.456003	−	0.889978i	$-0.650719\pi$
$701$	−1.18657e7	−0.912005	−0.456003	+	0.889978i	$0.650719\pi$
$702$	−1.42738e6	−0.109319
$703$	1.41007e6	0.107610
$704$	−6.92699e7	−5.26760
$705$	−402408.	−0.0304926
$706$	4.48037e6	0.338300
$707$	−8.31248e6	−0.625435
$708$	1.08968e7	0.816989
$709$	8.30329e6	0.620347	0.310173	−	0.950680i	$-0.399613\pi$
$709$	8.30329e6	0.620347	0.310173	+	0.950680i	$0.399613\pi$
$710$	1.90080e6	0.141511
$711$	6.19294e6	0.459433
$712$	−6.47478e7	−4.78658
$713$	2.53210e7	1.86533
$714$	−3.51965e6	−0.258377
$715$	529728.	0.0387514
$716$	−2.24173e6	−0.163418
$717$	949644.	0.0689863
$718$	2.85355e7	2.06573
$719$	−1.18495e7	−0.854828	−0.427414	−	0.904056i	$-0.640575\pi$
$719$	−1.18495e7	−0.854828	−0.427414	+	0.904056i	$0.640575\pi$
$720$	1.96781e6	0.141466
$721$	−2.77123e7	−1.98533
$722$	1.43353e6	0.102344
$723$	375642.	0.0267257
$724$	6.49150e7	4.60255
$725$	1.82313e7	1.28817
$726$	8.41154e6	0.592289
$727$	4.81482e6	0.337866	0.168933	−	0.985628i	$-0.445968\pi$
$727$	4.81482e6	0.337866	0.168933	+	0.985628i	$0.445968\pi$
$728$	1.96427e7	1.37364
$729$	531441.	0.0370370
$730$	5.32924e6	0.370133
$731$	−1.51338e6	−0.104750
$732$	−9.98527e6	−0.688782
$733$	1.10193e6	0.0757523	0.0378761	−	0.999282i	$-0.487941\pi$
$733$	1.10193e6	0.0757523	0.0378761	+	0.999282i	$0.487941\pi$
$734$	−8.38024e6	−0.574138
$735$	765126.	0.0522414
$736$	1.07592e8	7.32126
$737$	−2.15443e7	−1.46104
$738$	1.40546e7	0.949902
$739$	2.45211e7	1.65169	0.825845	−	0.563898i	$-0.190699\pi$
$739$	2.45211e7	1.65169	0.825845	+	0.563898i	$0.190699\pi$
$740$	−2.08580e6	−0.140021
$741$	578322.	0.0386923
$742$	5.61711e7	3.74544
$743$	−8.14072e6	−0.540992	−0.270496	−	0.962721i	$-0.587188\pi$
$743$	−8.14072e6	−0.540992	−0.270496	+	0.962721i	$0.587188\pi$
$744$	3.25037e7	2.15278
$745$	−1.70132e6	−0.112304
$746$	−9.21257e6	−0.606085
$747$	−4.60728e6	−0.302095
$748$	−8.91709e6	−0.582732
$749$	1.10859e7	0.722048
$750$	−3.69112e6	−0.239610
$751$	1.07734e6	0.0697030	0.0348515	−	0.999393i	$-0.488904\pi$
$751$	1.07734e6	0.0697030	0.0348515	+	0.999393i	$0.488904\pi$
$752$	−3.01731e7	−1.94570
$753$	−3.26153e6	−0.209620
$754$	1.15561e7	0.740259
$755$	−475200.	−0.0303395
$756$	−1.14191e7	−0.726651
$757$	1.90412e7	1.20769	0.603843	−	0.797103i	$-0.293635\pi$
$757$	1.90412e7	1.20769	0.603843	+	0.797103i	$0.293635\pi$
$758$	−6.86264e6	−0.433828
$759$	−1.96237e7	−1.23645
$760$	−1.35808e6	−0.0852888
$761$	2.74948e7	1.72103	0.860515	−	0.509426i	$-0.170142\pi$
$761$	2.74948e7	1.72103	0.860515	+	0.509426i	$0.170142\pi$
$762$	−6.97435e6	−0.435127
$763$	−6.71581e6	−0.417625
$764$	−2.53703e7	−1.57251
$765$	98172.0	0.00606505
$766$	1.07439e6	0.0661593
$767$	−2.42151e6	−0.148627
$768$	3.43285e7	2.10015
$769$	2.51266e7	1.53221	0.766105	−	0.642716i	$-0.222192\pi$
$769$	2.51266e7	1.53221	0.766105	+	0.642716i	$0.222192\pi$
$770$	5.76154e6	0.350196
$771$	−1.04508e7	−0.633161
$772$	−4.07262e7	−2.45941
$773$	−1.50927e7	−0.908483	−0.454242	−	0.890879i	$-0.650090\pi$
$773$	−1.50927e7	−0.908483	−0.454242	+	0.890879i	$0.650090\pi$
$774$	−6.67537e6	−0.400519
$775$	−1.77926e7	−1.06411
$776$	5.17212e7	3.08329
$777$	6.18710e6	0.367650
$778$	−2.45836e7	−1.45612
$779$	−5.69441e6	−0.336206
$780$	−855468.	−0.0503463
$781$	−1.42848e7	−0.838005
$782$	9.76791e6	0.571196
$783$	−4.30256e6	−0.250797
$784$	5.73703e7	3.33347
$785$	−779148.	−0.0451280
$786$	1.00845e7	0.582237
$787$	−2.71612e7	−1.56319	−0.781596	−	0.623785i	$-0.785594\pi$
$787$	−2.71612e7	−1.56319	−0.781596	+	0.623785i	$0.785594\pi$
$788$	−2.59148e7	−1.48673
$789$	1.22274e7	0.699263
$790$	5.04610e6	0.287666
$791$	−1.61744e6	−0.0919151
$792$	−2.51904e7	−1.42699
$793$	2.21895e6	0.125304
$794$	5.42757e7	3.05530
$795$	−1.56676e6	−0.0879192
$796$	3.24565e7	1.81560
$797$	−2.12149e7	−1.18303	−0.591515	−	0.806294i	$-0.701470\pi$
$797$	−2.12149e7	−1.18303	−0.591515	+	0.806294i	$0.701470\pi$
$798$	6.29006e6	0.349662
$799$	−1.50530e6	−0.0834175
$800$	−7.56033e7	−4.17653
$801$	−8.36455e6	−0.460639
$802$	−4.10752e7	−2.25498
$803$	−4.00500e7	−2.19187
$804$	3.47922e7	1.89820
$805$	−4.64218e6	−0.252483
$806$	−1.12781e7	−0.611502
$807$	1.97884e7	1.06961
$808$	2.96132e7	1.59572
$809$	−6.93973e6	−0.372796	−0.186398	−	0.982474i	$-0.559681\pi$
$809$	−6.93973e6	−0.372796	−0.186398	+	0.982474i	$0.559681\pi$
$810$	433026.	0.0231900
$811$	2.52867e6	0.135002	0.0675009	−	0.997719i	$-0.478497\pi$
$811$	2.52867e6	0.135002	0.0675009	+	0.997719i	$0.478497\pi$
$812$	9.24489e7	4.92053
$813$	−4.60814e6	−0.244512
$814$	2.13111e7	1.12732
$815$	344520.	0.0181686
$816$	7.36108e6	0.387005
$817$	2.70461e6	0.141759
$818$	−2.70691e7	−1.41446
$819$	2.53757e6	0.132193
$820$	8.42332e6	0.437470
$821$	−2.31034e7	−1.19624	−0.598120	−	0.801406i	$-0.704085\pi$
$821$	−2.31034e7	−1.19624	−0.598120	+	0.801406i	$0.704085\pi$
$822$	−4.27805e7	−2.20834
$823$	103360.	0.00531928	0.00265964	−	0.999996i	$-0.499153\pi$
$823$	103360.	0.00531928	0.00265964	+	0.999996i	$0.499153\pi$
$824$	9.87249e7	5.06534
$825$	1.37893e7	0.705354
$826$	−2.63373e7	−1.34314
$827$	3.23111e6	0.164281	0.0821406	−	0.996621i	$-0.473824\pi$
$827$	3.23111e6	0.164281	0.0821406	+	0.996621i	$0.473824\pi$
$828$	3.16908e7	1.60641
$829$	−1.24466e7	−0.629022	−0.314511	−	0.949254i	$-0.601841\pi$
$829$	−1.24466e7	−0.629022	−0.314511	+	0.949254i	$0.601841\pi$
$830$	−3.75408e6	−0.189151
$831$	7.71788e6	0.387700
$832$	−2.48589e7	−1.24501
$833$	2.86214e6	0.142915
$834$	−3.72917e7	−1.85651
$835$	−1.52405e6	−0.0756454
$836$	1.59360e7	0.788612
$837$	4.19904e6	0.207175
$838$	−4.81263e6	−0.236741
$839$	1.20780e7	0.592366	0.296183	−	0.955131i	$-0.404286\pi$
$839$	1.20780e7	0.592366	0.296183	+	0.955131i	$0.404286\pi$
$840$	−5.95901e6	−0.291391
$841$	1.43225e7	0.698277
$842$	3.20115e7	1.55606
$843$	−3.37833e6	−0.163732
$844$	−4.02625e7	−1.94556
$845$	−2.03765e6	−0.0981722
$846$	−6.63973e6	−0.318951
$847$	−1.49538e7	−0.716216
$848$	−1.17478e8	−5.61004
$849$	19836.0	0.000944463 0
$850$	−6.86376e6	−0.325848
$851$	−1.71708e7	−0.812767
$852$	2.30688e7	1.08874
$853$	−7.24764e6	−0.341055	−0.170527	−	0.985353i	$-0.554547\pi$
$853$	−7.24764e6	−0.341055	−0.170527	+	0.985353i	$0.554547\pi$
$854$	2.41342e7	1.13237
$855$	−175446.	−0.00820783
$856$	−3.94935e7	−1.84222
$857$	2.29801e6	0.106881	0.0534405	−	0.998571i	$-0.482981\pi$
$857$	2.29801e6	0.106881	0.0534405	+	0.998571i	$0.482981\pi$
$858$	8.74051e6	0.405339
$859$	−1.60848e7	−0.743758	−0.371879	−	0.928281i	$-0.621286\pi$
$859$	−1.60848e7	−0.743758	−0.371879	+	0.928281i	$0.621286\pi$
$860$	−4.00073e6	−0.184456
$861$	−2.49860e7	−1.14865
$862$	−5.11369e7	−2.34405
$863$	6.42193e6	0.293521	0.146760	−	0.989172i	$-0.453115\pi$
$863$	6.42193e6	0.293521	0.146760	+	0.989172i	$0.453115\pi$
$864$	1.78423e7	0.813141
$865$	−1.06420e6	−0.0483595
$866$	7.59958e7	3.44346
$867$	−1.24115e7	−0.560758
$868$	−9.02246e7	−4.06468
$869$	−3.79222e7	−1.70351
$870$	−3.50579e6	−0.157032
$871$	−7.73161e6	−0.345322
$872$	2.39251e7	1.06552
$873$	6.68169e6	0.296723
$874$	−1.74565e7	−0.772999
$875$	6.56198e6	0.289744
$876$	6.46775e7	2.84769
$877$	2.82068e7	1.23838	0.619192	−	0.785240i	$-0.287460\pi$
$877$	2.82068e7	1.23838	0.619192	+	0.785240i	$0.287460\pi$
$878$	6.88183e7	3.01278
$879$	−5.44678e6	−0.237776
$880$	−1.20498e7	−0.524534
$881$	−3.93965e7	−1.71009	−0.855043	−	0.518557i	$-0.826470\pi$
$881$	−3.93965e7	−1.71009	−0.855043	+	0.518557i	$0.826470\pi$
$882$	1.26246e7	0.546444
$883$	3.38638e7	1.46162	0.730809	−	0.682582i	$-0.239143\pi$
$883$	3.38638e7	1.46162	0.730809	+	0.682582i	$0.239143\pi$
$884$	−3.20008e6	−0.137731
$885$	734616.	0.0315284
$886$	2.35497e7	1.00786
$887$	2.08861e7	0.891351	0.445675	−	0.895195i	$-0.352964\pi$
$887$	2.08861e7	0.891351	0.445675	+	0.895195i	$0.352964\pi$
$888$	−2.20416e7	−0.938015
$889$	1.23988e7	0.526171
$890$	−6.81556e6	−0.288421
$891$	−3.25426e6	−0.137327
$892$	8.36863e7	3.52162
$893$	2.69017e6	0.112889
$894$	−2.80718e7	−1.17470
$895$	−151128.	−0.00630648
$896$	−1.32533e8	−5.51510
$897$	−7.04239e6	−0.292240
$898$	5.41298e7	2.23999
$899$	−3.39955e7	−1.40289
$900$	−2.22686e7	−0.916403
$901$	−5.86083e6	−0.240518
$902$	−8.60629e7	−3.52209
$903$	1.18673e7	0.484321
$904$	5.76213e6	0.234510
$905$	4.37629e6	0.177617
$906$	−7.84080e6	−0.317351
$907$	−3.63949e7	−1.46900	−0.734501	−	0.678608i	$-0.762584\pi$
$907$	−3.63949e7	−1.46900	−0.734501	+	0.678608i	$0.762584\pi$
$908$	7.08714e7	2.85270
$909$	3.82563e6	0.153565
$910$	2.06765e6	0.0827700
$911$	1.83331e6	0.0731881	0.0365940	−	0.999330i	$-0.488349\pi$
$911$	1.83331e6	0.0731881	0.0365940	+	0.999330i	$0.488349\pi$
$912$	−1.31552e7	−0.523734
$913$	2.82125e7	1.12012
$914$	4.33410e7	1.71606
$915$	−673164.	−0.0265808
$916$	1.36467e7	0.537390
$917$	−1.79281e7	−0.704061
$918$	1.61984e6	0.0634402
$919$	−1.62178e7	−0.633436	−0.316718	−	0.948520i	$-0.602581\pi$
$919$	−1.62178e7	−0.633436	−0.316718	+	0.948520i	$0.602581\pi$
$920$	1.65378e7	0.644180
$921$	−1.13034e7	−0.439096
$922$	−2.48540e7	−0.962871
$923$	−5.12640e6	−0.198065
$924$	6.99241e7	2.69431
$925$	1.20656e7	0.463656
$926$	−3.64987e7	−1.39878
$927$	1.27539e7	0.487467
$928$	−1.44451e8	−5.50620
$929$	2.06857e7	0.786379	0.393189	−	0.919457i	$-0.371372\pi$
$929$	2.06857e7	0.786379	0.393189	+	0.919457i	$0.371372\pi$
$930$	3.42144e6	0.129718
$931$	−5.11501e6	−0.193407
$932$	2.19704e7	0.828509
$933$	−7.42122e6	−0.279107
$934$	3.99488e7	1.49843
$935$	−601152.	−0.0224882
$936$	−9.04009e6	−0.337274
$937$	7.79438e6	0.290023	0.145012	−	0.989430i	$-0.453678\pi$
$937$	7.79438e6	0.290023	0.145012	+	0.989430i	$0.453678\pi$
$938$	−8.40921e7	−3.12067
$939$	−1.11110e7	−0.411232
$940$	−3.97937e6	−0.146891
$941$	−2.20151e7	−0.810487	−0.405243	−	0.914209i	$-0.632813\pi$
$941$	−2.20151e7	−0.810487	−0.405243	+	0.914209i	$0.632813\pi$
$942$	−1.28559e7	−0.472038
$943$	6.93425e7	2.53934
$944$	5.50826e7	2.01180
$945$	−769824.	−0.0280422
$946$	4.08764e7	1.48506
$947$	3.67660e6	0.133221	0.0666103	−	0.997779i	$-0.478782\pi$
$947$	3.67660e6	0.133221	0.0666103	+	0.997779i	$0.478782\pi$
$948$	6.12413e7	2.21321
$949$	−1.43728e7	−0.518055
$950$	1.22664e7	0.440970
$951$	−1.07485e7	−0.385387
$952$	−2.22911e7	−0.797148
$953$	−6.27011e6	−0.223636	−0.111818	−	0.993729i	$-0.535667\pi$
$953$	−6.27011e6	−0.223636	−0.111818	+	0.993729i	$0.535667\pi$
$954$	−2.58515e7	−0.919633
$955$	−1.71036e6	−0.0606847
$956$	9.39092e6	0.332325
$957$	2.63465e7	0.929916
$958$	−1.60340e7	−0.564453
$959$	7.60542e7	2.67040
$960$	7.54148e6	0.264106
$961$	4.54845e6	0.158875
$962$	7.64795e6	0.266445
$963$	−5.10203e6	−0.177287
$964$	3.71468e6	0.128745
$965$	−2.74559e6	−0.0949111
$966$	−7.65959e7	−2.64097
$967$	3.19646e7	1.09927	0.549633	−	0.835406i	$-0.314768\pi$
$967$	3.19646e7	1.09927	0.549633	+	0.835406i	$0.314768\pi$
$968$	5.32731e7	1.82734
$969$	−656298.	−0.0224539
$970$	5.44434e6	0.185787
$971$	−3.00150e7	−1.02162	−0.510811	−	0.859693i	$-0.670655\pi$
$971$	−3.00150e7	−1.02162	−0.510811	+	0.859693i	$0.670655\pi$
$972$	5.25536e6	0.178417
$973$	6.62964e7	2.24496
$974$	7.41207e7	2.50347
$975$	4.94858e6	0.166713
$976$	−5.04748e7	−1.69610
$977$	2.19294e7	0.735006	0.367503	−	0.930022i	$-0.380213\pi$
$977$	2.19294e7	0.735006	0.367503	+	0.930022i	$0.380213\pi$
$978$	5.68458e6	0.190043
$979$	5.12199e7	1.70798
$980$	7.56625e6	0.251661
$981$	3.09080e6	0.102541
$982$	3.82663e7	1.26630
$983$	3.32485e7	1.09746	0.548730	−	0.836000i	$-0.315112\pi$
$983$	3.32485e7	1.09746	0.548730	+	0.836000i	$0.315112\pi$
$984$	8.90127e7	2.93065
$985$	−1.74707e6	−0.0573745
$986$	−1.31142e7	−0.429587
$987$	1.18040e7	0.385687
$988$	5.71896e6	0.186391
$989$	−3.29348e7	−1.07069
$990$	−2.65162e6	−0.0859850
$991$	−5.46424e7	−1.76744	−0.883722	−	0.468012i	$-0.844970\pi$
$991$	−5.46424e7	−1.76744	−0.883722	+	0.468012i	$0.844970\pi$
$992$	1.40976e8	4.54848
$993$	1.79202e7	0.576726
$994$	−5.57568e7	−1.78991
$995$	2.18808e6	0.0700657
$996$	−4.55609e7	−1.45527
$997$	−2.89095e7	−0.921090	−0.460545	−	0.887636i	$-0.652346\pi$
$997$	−2.89095e7	−0.921090	−0.460545	+	0.887636i	$0.652346\pi$
$998$	−1.03347e8	−3.28450
$999$	−2.84747e6	−0.0902705

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.6.a.b.1.1	✓	1
3.2	odd	2		171.6.a.a.1.1		1
4.3	odd	2		912.6.a.d.1.1		1
19.18	odd	2		1083.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.6.a.b.1.1	✓	1	1.1	even	1	trivial
171.6.a.a.1.1		1	3.2	odd	2
912.6.a.d.1.1		1	4.3	odd	2
1083.6.a.a.1.1		1	19.18	odd	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:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	57.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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