LMFDB - Embedded newform 60.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,6,Mod(1,60)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, 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("60.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	60.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:	$9.62302918878$
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	$\chi$	$=$	60.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-9.00000 q^{3} -25.0000 q^{5} +44.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q-9.00000 q^{3} -25.0000 q^{5} +44.0000 q^{7} +81.0000 q^{9} +216.000 q^{11} +770.000 q^{13} +225.000 q^{15} +534.000 q^{17} +1580.00 q^{19} -396.000 q^{21} +2904.00 q^{23} +625.000 q^{25} -729.000 q^{27} -4566.00 q^{29} +2744.00 q^{31} -1944.00 q^{33} -1100.00 q^{35} +1442.00 q^{37} -6930.00 q^{39} -13350.0 q^{41} +17204.0 q^{43} -2025.00 q^{45} -10824.0 q^{47} -14871.0 q^{49} -4806.00 q^{51} -9942.00 q^{53} -5400.00 q^{55} -14220.0 q^{57} -15576.0 q^{59} +39302.0 q^{61} +3564.00 q^{63} -19250.0 q^{65} +55796.0 q^{67} -26136.0 q^{69} +57120.0 q^{71} +50402.0 q^{73} -5625.00 q^{75} +9504.00 q^{77} -10552.0 q^{79} +6561.00 q^{81} +108564. q^{83} -13350.0 q^{85} +41094.0 q^{87} -116430. q^{89} +33880.0 q^{91} -24696.0 q^{93} -39500.0 q^{95} -2782.00 q^{97} +17496.0 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	44.0000	0.339397	0.169698	−	0.985496i	$-0.445721\pi$
$7$	44.0000	0.339397	0.169698	+	0.985496i	$0.445721\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	216.000	0.538235	0.269118	−	0.963107i	$-0.413268\pi$
$11$	216.000	0.538235	0.269118	+	0.963107i	$0.413268\pi$
$12$	0	0
$13$	770.000	1.26367	0.631833	−	0.775104i	$-0.282303\pi$
$13$	770.000	1.26367	0.631833	+	0.775104i	$0.282303\pi$
$14$	0	0
$15$	225.000	0.258199
$16$	0	0
$17$	534.000	0.448145	0.224073	−	0.974572i	$-0.428065\pi$
$17$	534.000	0.448145	0.224073	+	0.974572i	$0.428065\pi$
$18$	0	0
$19$	1580.00	1.00409	0.502046	−	0.864841i	$-0.332581\pi$
$19$	1580.00	1.00409	0.502046	+	0.864841i	$0.332581\pi$
$20$	0	0
$21$	−396.000	−0.195951
$22$	0	0
$23$	2904.00	1.14466	0.572331	−	0.820023i	$-0.306039\pi$
$23$	2904.00	1.14466	0.572331	+	0.820023i	$0.306039\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−4566.00	−1.00819	−0.504093	−	0.863649i	$-0.668173\pi$
$29$	−4566.00	−1.00819	−0.504093	+	0.863649i	$0.668173\pi$
$30$	0	0
$31$	2744.00	0.512838	0.256419	−	0.966566i	$-0.417457\pi$
$31$	2744.00	0.512838	0.256419	+	0.966566i	$0.417457\pi$
$32$	0	0
$33$	−1944.00	−0.310750
$34$	0	0
$35$	−1100.00	−0.151783
$36$	0	0
$37$	1442.00	0.173165	0.0865827	−	0.996245i	$-0.472405\pi$
$37$	1442.00	0.173165	0.0865827	+	0.996245i	$0.472405\pi$
$38$	0	0
$39$	−6930.00	−0.729578
$40$	0	0
$41$	−13350.0	−1.24029	−0.620143	−	0.784489i	$-0.712925\pi$
$41$	−13350.0	−1.24029	−0.620143	+	0.784489i	$0.712925\pi$
$42$	0	0
$43$	17204.0	1.41892	0.709461	−	0.704745i	$-0.248939\pi$
$43$	17204.0	1.41892	0.709461	+	0.704745i	$0.248939\pi$
$44$	0	0
$45$	−2025.00	−0.149071
$46$	0	0
$47$	−10824.0	−0.714732	−0.357366	−	0.933964i	$-0.616325\pi$
$47$	−10824.0	−0.714732	−0.357366	+	0.933964i	$0.616325\pi$
$48$	0	0
$49$	−14871.0	−0.884810
$50$	0	0
$51$	−4806.00	−0.258737
$52$	0	0
$53$	−9942.00	−0.486165	−0.243083	−	0.970006i	$-0.578159\pi$
$53$	−9942.00	−0.486165	−0.243083	+	0.970006i	$0.578159\pi$
$54$	0	0
$55$	−5400.00	−0.240706
$56$	0	0
$57$	−14220.0	−0.579712
$58$	0	0
$59$	−15576.0	−0.582540	−0.291270	−	0.956641i	$-0.594078\pi$
$59$	−15576.0	−0.582540	−0.291270	+	0.956641i	$0.594078\pi$
$60$	0	0
$61$	39302.0	1.35235	0.676176	−	0.736740i	$-0.263636\pi$
$61$	39302.0	1.35235	0.676176	+	0.736740i	$0.263636\pi$
$62$	0	0
$63$	3564.00	0.113132
$64$	0	0
$65$	−19250.0	−0.565129
$66$	0	0
$67$	55796.0	1.51850	0.759252	−	0.650797i	$-0.225565\pi$
$67$	55796.0	1.51850	0.759252	+	0.650797i	$0.225565\pi$
$68$	0	0
$69$	−26136.0	−0.660871
$70$	0	0
$71$	57120.0	1.34475	0.672376	−	0.740210i	$-0.265274\pi$
$71$	57120.0	1.34475	0.672376	+	0.740210i	$0.265274\pi$
$72$	0	0
$73$	50402.0	1.10698	0.553491	−	0.832855i	$-0.313295\pi$
$73$	50402.0	1.10698	0.553491	+	0.832855i	$0.313295\pi$
$74$	0	0
$75$	−5625.00	−0.115470
$76$	0	0
$77$	9504.00	0.182675
$78$	0	0
$79$	−10552.0	−0.190225	−0.0951124	−	0.995467i	$-0.530321\pi$
$79$	−10552.0	−0.190225	−0.0951124	+	0.995467i	$0.530321\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	108564.	1.72978	0.864889	−	0.501962i	$-0.167388\pi$
$83$	108564.	1.72978	0.864889	+	0.501962i	$0.167388\pi$
$84$	0	0
$85$	−13350.0	−0.200417
$86$	0	0
$87$	41094.0	0.582077
$88$	0	0
$89$	−116430.	−1.55808	−0.779040	−	0.626974i	$-0.784293\pi$
$89$	−116430.	−1.55808	−0.779040	+	0.626974i	$0.784293\pi$
$90$	0	0
$91$	33880.0	0.428884
$92$	0	0
$93$	−24696.0	−0.296087
$94$	0	0
$95$	−39500.0	−0.449043
$96$	0	0
$97$	−2782.00	−0.0300212	−0.0150106	−	0.999887i	$-0.504778\pi$
$97$	−2782.00	−0.0300212	−0.0150106	+	0.999887i	$0.504778\pi$
$98$	0	0
$99$	17496.0	0.179412
$100$	0	0
$101$	27618.0	0.269395	0.134697	−	0.990887i	$-0.456994\pi$
$101$	27618.0	0.269395	0.134697	+	0.990887i	$0.456994\pi$
$102$	0	0
$103$	10556.0	0.0980407	0.0490203	−	0.998798i	$-0.484390\pi$
$103$	10556.0	0.0980407	0.0490203	+	0.998798i	$0.484390\pi$
$104$	0	0
$105$	9900.00	0.0876318
$106$	0	0
$107$	−67668.0	−0.571379	−0.285689	−	0.958322i	$-0.592223\pi$
$107$	−67668.0	−0.571379	−0.285689	+	0.958322i	$0.592223\pi$
$108$	0	0
$109$	−230362.	−1.85714	−0.928570	−	0.371158i	$-0.878961\pi$
$109$	−230362.	−1.85714	−0.928570	+	0.371158i	$0.878961\pi$
$110$	0	0
$111$	−12978.0	−0.0999771
$112$	0	0
$113$	73182.0	0.539148	0.269574	−	0.962980i	$-0.413117\pi$
$113$	73182.0	0.539148	0.269574	+	0.962980i	$0.413117\pi$
$114$	0	0
$115$	−72600.0	−0.511908
$116$	0	0
$117$	62370.0	0.421222
$118$	0	0
$119$	23496.0	0.152099
$120$	0	0
$121$	−114395.	−0.710303
$122$	0	0
$123$	120150.	0.716079
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−208636.	−1.14784	−0.573918	−	0.818913i	$-0.694577\pi$
$127$	−208636.	−1.14784	−0.573918	+	0.818913i	$0.694577\pi$
$128$	0	0
$129$	−154836.	−0.819215
$130$	0	0
$131$	18792.0	0.0956742	0.0478371	−	0.998855i	$-0.484767\pi$
$131$	18792.0	0.0956742	0.0478371	+	0.998855i	$0.484767\pi$
$132$	0	0
$133$	69520.0	0.340785
$134$	0	0
$135$	18225.0	0.0860663
$136$	0	0
$137$	−142218.	−0.647371	−0.323685	−	0.946165i	$-0.604922\pi$
$137$	−142218.	−0.647371	−0.323685	+	0.946165i	$0.604922\pi$
$138$	0	0
$139$	−297604.	−1.30648	−0.653238	−	0.757152i	$-0.726590\pi$
$139$	−297604.	−1.30648	−0.653238	+	0.757152i	$0.726590\pi$
$140$	0	0
$141$	97416.0	0.412651
$142$	0	0
$143$	166320.	0.680149
$144$	0	0
$145$	114150.	0.450875
$146$	0	0
$147$	133839.	0.510845
$148$	0	0
$149$	381738.	1.40864	0.704320	−	0.709883i	$-0.251252\pi$
$149$	381738.	1.40864	0.704320	+	0.709883i	$0.251252\pi$
$150$	0	0
$151$	−302680.	−1.08029	−0.540146	−	0.841571i	$-0.681631\pi$
$151$	−302680.	−1.08029	−0.540146	+	0.841571i	$0.681631\pi$
$152$	0	0
$153$	43254.0	0.149382
$154$	0	0
$155$	−68600.0	−0.229348
$156$	0	0
$157$	423458.	1.37107	0.685537	−	0.728037i	$-0.259567\pi$
$157$	423458.	1.37107	0.685537	+	0.728037i	$0.259567\pi$
$158$	0	0
$159$	89478.0	0.280688
$160$	0	0
$161$	127776.	0.388494
$162$	0	0
$163$	−358300.	−1.05628	−0.528138	−	0.849158i	$-0.677110\pi$
$163$	−358300.	−1.05628	−0.528138	+	0.849158i	$0.677110\pi$
$164$	0	0
$165$	48600.0	0.138972
$166$	0	0
$167$	4392.00	0.0121863	0.00609314	−	0.999981i	$-0.498060\pi$
$167$	4392.00	0.0121863	0.00609314	+	0.999981i	$0.498060\pi$
$168$	0	0
$169$	221607.	0.596852
$170$	0	0
$171$	127980.	0.334697
$172$	0	0
$173$	128226.	0.325732	0.162866	−	0.986648i	$-0.447926\pi$
$173$	128226.	0.325732	0.162866	+	0.986648i	$0.447926\pi$
$174$	0	0
$175$	27500.0	0.0678793
$176$	0	0
$177$	140184.	0.336330
$178$	0	0
$179$	559104.	1.30425	0.652124	−	0.758113i	$-0.273878\pi$
$179$	559104.	1.30425	0.652124	+	0.758113i	$0.273878\pi$
$180$	0	0
$181$	753470.	1.70950	0.854751	−	0.519039i	$-0.173710\pi$
$181$	753470.	1.70950	0.854751	+	0.519039i	$0.173710\pi$
$182$	0	0
$183$	−353718.	−0.780781
$184$	0	0
$185$	−36050.0	−0.0774419
$186$	0	0
$187$	115344.	0.241208
$188$	0	0
$189$	−32076.0	−0.0653169
$190$	0	0
$191$	−329016.	−0.652580	−0.326290	−	0.945270i	$-0.605799\pi$
$191$	−329016.	−0.652580	−0.326290	+	0.945270i	$0.605799\pi$
$192$	0	0
$193$	−262294.	−0.506868	−0.253434	−	0.967353i	$-0.581560\pi$
$193$	−262294.	−0.506868	−0.253434	+	0.967353i	$0.581560\pi$
$194$	0	0
$195$	173250.	0.326277
$196$	0	0
$197$	−977814.	−1.79511	−0.897554	−	0.440904i	$-0.854658\pi$
$197$	−977814.	−1.79511	−0.897554	+	0.440904i	$0.854658\pi$
$198$	0	0
$199$	172088.	0.308048	0.154024	−	0.988067i	$-0.450777\pi$
$199$	172088.	0.308048	0.154024	+	0.988067i	$0.450777\pi$
$200$	0	0
$201$	−502164.	−0.876709
$202$	0	0
$203$	−200904.	−0.342175
$204$	0	0
$205$	333750.	0.554672
$206$	0	0
$207$	235224.	0.381554
$208$	0	0
$209$	341280.	0.540437
$210$	0	0
$211$	−931900.	−1.44100	−0.720499	−	0.693456i	$-0.756087\pi$
$211$	−931900.	−1.44100	−0.720499	+	0.693456i	$0.756087\pi$
$212$	0	0
$213$	−514080.	−0.776393
$214$	0	0
$215$	−430100.	−0.634561
$216$	0	0
$217$	120736.	0.174055
$218$	0	0
$219$	−453618.	−0.639116
$220$	0	0
$221$	411180.	0.566306
$222$	0	0
$223$	−840460.	−1.13176	−0.565881	−	0.824487i	$-0.691464\pi$
$223$	−840460.	−1.13176	−0.565881	+	0.824487i	$0.691464\pi$
$224$	0	0
$225$	50625.0	0.0666667
$226$	0	0
$227$	−667140.	−0.859315	−0.429657	−	0.902992i	$-0.641366\pi$
$227$	−667140.	−0.859315	−0.429657	+	0.902992i	$0.641366\pi$
$228$	0	0
$229$	1.02945e6	1.29722	0.648612	−	0.761119i	$-0.275350\pi$
$229$	1.02945e6	1.29722	0.648612	+	0.761119i	$0.275350\pi$
$230$	0	0
$231$	−85536.0	−0.105468
$232$	0	0
$233$	−1.60845e6	−1.94097	−0.970483	−	0.241171i	$-0.922469\pi$
$233$	−1.60845e6	−1.94097	−0.970483	+	0.241171i	$0.922469\pi$
$234$	0	0
$235$	270600.	0.319638
$236$	0	0
$237$	94968.0	0.109826
$238$	0	0
$239$	1.62545e6	1.84068	0.920340	−	0.391119i	$-0.127912\pi$
$239$	1.62545e6	1.84068	0.920340	+	0.391119i	$0.127912\pi$
$240$	0	0
$241$	426818.	0.473369	0.236685	−	0.971587i	$-0.423939\pi$
$241$	426818.	0.473369	0.236685	+	0.971587i	$0.423939\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	371775.	0.395699
$246$	0	0
$247$	1.21660e6	1.26884
$248$	0	0
$249$	−977076.	−0.998688
$250$	0	0
$251$	−1.33531e6	−1.33782	−0.668911	−	0.743342i	$-0.733239\pi$
$251$	−1.33531e6	−1.33782	−0.668911	+	0.743342i	$0.733239\pi$
$252$	0	0
$253$	627264.	0.616097
$254$	0	0
$255$	120150.	0.115711
$256$	0	0
$257$	−2.08682e6	−1.97084	−0.985421	−	0.170134i	$-0.945580\pi$
$257$	−2.08682e6	−1.97084	−0.985421	+	0.170134i	$0.945580\pi$
$258$	0	0
$259$	63448.0	0.0587717
$260$	0	0
$261$	−369846.	−0.336062
$262$	0	0
$263$	940824.	0.838724	0.419362	−	0.907819i	$-0.362254\pi$
$263$	940824.	0.838724	0.419362	+	0.907819i	$0.362254\pi$
$264$	0	0
$265$	248550.	0.217420
$266$	0	0
$267$	1.04787e6	0.899558
$268$	0	0
$269$	1.01371e6	0.854144	0.427072	−	0.904218i	$-0.359545\pi$
$269$	1.01371e6	0.854144	0.427072	+	0.904218i	$0.359545\pi$
$270$	0	0
$271$	−288016.	−0.238228	−0.119114	−	0.992881i	$-0.538005\pi$
$271$	−288016.	−0.238228	−0.119114	+	0.992881i	$0.538005\pi$
$272$	0	0
$273$	−304920.	−0.247616
$274$	0	0
$275$	135000.	0.107647
$276$	0	0
$277$	860738.	0.674018	0.337009	−	0.941501i	$-0.390585\pi$
$277$	860738.	0.674018	0.337009	+	0.941501i	$0.390585\pi$
$278$	0	0
$279$	222264.	0.170946
$280$	0	0
$281$	420306.	0.317541	0.158770	−	0.987316i	$-0.449247\pi$
$281$	420306.	0.317541	0.158770	+	0.987316i	$0.449247\pi$
$282$	0	0
$283$	455372.	0.337987	0.168994	−	0.985617i	$-0.445948\pi$
$283$	455372.	0.337987	0.168994	+	0.985617i	$0.445948\pi$
$284$	0	0
$285$	355500.	0.259255
$286$	0	0
$287$	−587400.	−0.420949
$288$	0	0
$289$	−1.13470e6	−0.799166
$290$	0	0
$291$	25038.0	0.0173327
$292$	0	0
$293$	2.11540e6	1.43954	0.719770	−	0.694212i	$-0.244247\pi$
$293$	2.11540e6	1.43954	0.719770	+	0.694212i	$0.244247\pi$
$294$	0	0
$295$	389400.	0.260520
$296$	0	0
$297$	−157464.	−0.103583
$298$	0	0
$299$	2.23608e6	1.44647
$300$	0	0
$301$	756976.	0.481577
$302$	0	0
$303$	−248562.	−0.155535
$304$	0	0
$305$	−982550.	−0.604791
$306$	0	0
$307$	1.28313e6	0.777008	0.388504	−	0.921447i	$-0.372992\pi$
$307$	1.28313e6	0.777008	0.388504	+	0.921447i	$0.372992\pi$
$308$	0	0
$309$	−95004.0	−0.0566038
$310$	0	0
$311$	1.43270e6	0.839954	0.419977	−	0.907535i	$-0.362038\pi$
$311$	1.43270e6	0.839954	0.419977	+	0.907535i	$0.362038\pi$
$312$	0	0
$313$	−1.25049e6	−0.721474	−0.360737	−	0.932668i	$-0.617475\pi$
$313$	−1.25049e6	−0.721474	−0.360737	+	0.932668i	$0.617475\pi$
$314$	0	0
$315$	−89100.0	−0.0505943
$316$	0	0
$317$	−2.06461e6	−1.15396	−0.576978	−	0.816760i	$-0.695768\pi$
$317$	−2.06461e6	−1.15396	−0.576978	+	0.816760i	$0.695768\pi$
$318$	0	0
$319$	−986256.	−0.542641
$320$	0	0
$321$	609012.	0.329886
$322$	0	0
$323$	843720.	0.449979
$324$	0	0
$325$	481250.	0.252733
$326$	0	0
$327$	2.07326e6	1.07222
$328$	0	0
$329$	−476256.	−0.242578
$330$	0	0
$331$	−2.36211e6	−1.18503	−0.592516	−	0.805559i	$-0.701865\pi$
$331$	−2.36211e6	−1.18503	−0.592516	+	0.805559i	$0.701865\pi$
$332$	0	0
$333$	116802.	0.0577218
$334$	0	0
$335$	−1.39490e6	−0.679096
$336$	0	0
$337$	−3.09300e6	−1.48356	−0.741780	−	0.670644i	$-0.766018\pi$
$337$	−3.09300e6	−1.48356	−0.741780	+	0.670644i	$0.766018\pi$
$338$	0	0
$339$	−658638.	−0.311277
$340$	0	0
$341$	592704.	0.276027
$342$	0	0
$343$	−1.39383e6	−0.639698
$344$	0	0
$345$	653400.	0.295550
$346$	0	0
$347$	2.09461e6	0.933856	0.466928	−	0.884295i	$-0.345361\pi$
$347$	2.09461e6	0.933856	0.466928	+	0.884295i	$0.345361\pi$
$348$	0	0
$349$	−3.06351e6	−1.34634	−0.673172	−	0.739486i	$-0.735069\pi$
$349$	−3.06351e6	−1.34634	−0.673172	+	0.739486i	$0.735069\pi$
$350$	0	0
$351$	−561330.	−0.243193
$352$	0	0
$353$	596502.	0.254786	0.127393	−	0.991852i	$-0.459339\pi$
$353$	596502.	0.254786	0.127393	+	0.991852i	$0.459339\pi$
$354$	0	0
$355$	−1.42800e6	−0.601392
$356$	0	0
$357$	−211464.	−0.0878144
$358$	0	0
$359$	−3.11254e6	−1.27461	−0.637306	−	0.770611i	$-0.719951\pi$
$359$	−3.11254e6	−1.27461	−0.637306	+	0.770611i	$0.719951\pi$
$360$	0	0
$361$	20301.0	0.00819878
$362$	0	0
$363$	1.02956e6	0.410094
$364$	0	0
$365$	−1.26005e6	−0.495057
$366$	0	0
$367$	2.05576e6	0.796724	0.398362	−	0.917228i	$-0.369579\pi$
$367$	2.05576e6	0.796724	0.398362	+	0.917228i	$0.369579\pi$
$368$	0	0
$369$	−1.08135e6	−0.413428
$370$	0	0
$371$	−437448.	−0.165003
$372$	0	0
$373$	−350566.	−0.130466	−0.0652331	−	0.997870i	$-0.520779\pi$
$373$	−350566.	−0.130466	−0.0652331	+	0.997870i	$0.520779\pi$
$374$	0	0
$375$	140625.	0.0516398
$376$	0	0
$377$	−3.51582e6	−1.27401
$378$	0	0
$379$	−3.86621e6	−1.38257	−0.691286	−	0.722581i	$-0.742955\pi$
$379$	−3.86621e6	−1.38257	−0.691286	+	0.722581i	$0.742955\pi$
$380$	0	0
$381$	1.87772e6	0.662704
$382$	0	0
$383$	2.70679e6	0.942883	0.471442	−	0.881897i	$-0.343734\pi$
$383$	2.70679e6	0.942883	0.471442	+	0.881897i	$0.343734\pi$
$384$	0	0
$385$	−237600.	−0.0816948
$386$	0	0
$387$	1.39352e6	0.472974
$388$	0	0
$389$	1.54411e6	0.517372	0.258686	−	0.965961i	$-0.416710\pi$
$389$	1.54411e6	0.517372	0.258686	+	0.965961i	$0.416710\pi$
$390$	0	0
$391$	1.55074e6	0.512975
$392$	0	0
$393$	−169128.	−0.0552375
$394$	0	0
$395$	263800.	0.0850711
$396$	0	0
$397$	−339478.	−0.108102	−0.0540512	−	0.998538i	$-0.517213\pi$
$397$	−339478.	−0.108102	−0.0540512	+	0.998538i	$0.517213\pi$
$398$	0	0
$399$	−625680.	−0.196752
$400$	0	0
$401$	4.70179e6	1.46016	0.730082	−	0.683359i	$-0.239482\pi$
$401$	4.70179e6	1.46016	0.730082	+	0.683359i	$0.239482\pi$
$402$	0	0
$403$	2.11288e6	0.648056
$404$	0	0
$405$	−164025.	−0.0496904
$406$	0	0
$407$	311472.	0.0932037
$408$	0	0
$409$	−3.98925e6	−1.17919	−0.589594	−	0.807699i	$-0.700712\pi$
$409$	−3.98925e6	−1.17919	−0.589594	+	0.807699i	$0.700712\pi$
$410$	0	0
$411$	1.27996e6	0.373760
$412$	0	0
$413$	−685344.	−0.197712
$414$	0	0
$415$	−2.71410e6	−0.773581
$416$	0	0
$417$	2.67844e6	0.754295
$418$	0	0
$419$	4.42015e6	1.22999	0.614996	−	0.788530i	$-0.289158\pi$
$419$	4.42015e6	1.22999	0.614996	+	0.788530i	$0.289158\pi$
$420$	0	0
$421$	3.74999e6	1.03116	0.515579	−	0.856842i	$-0.327577\pi$
$421$	3.74999e6	1.03116	0.515579	+	0.856842i	$0.327577\pi$
$422$	0	0
$423$	−876744.	−0.238244
$424$	0	0
$425$	333750.	0.0896291
$426$	0	0
$427$	1.72929e6	0.458984
$428$	0	0
$429$	−1.49688e6	−0.392684
$430$	0	0
$431$	−5.09659e6	−1.32156	−0.660780	−	0.750580i	$-0.729774\pi$
$431$	−5.09659e6	−1.32156	−0.660780	+	0.750580i	$0.729774\pi$
$432$	0	0
$433$	4.95267e6	1.26946	0.634730	−	0.772734i	$-0.281111\pi$
$433$	4.95267e6	1.26946	0.634730	+	0.772734i	$0.281111\pi$
$434$	0	0
$435$	−1.02735e6	−0.260313
$436$	0	0
$437$	4.58832e6	1.14934
$438$	0	0
$439$	4.14025e6	1.02533	0.512667	−	0.858588i	$-0.328658\pi$
$439$	4.14025e6	1.02533	0.512667	+	0.858588i	$0.328658\pi$
$440$	0	0
$441$	−1.20455e6	−0.294937
$442$	0	0
$443$	3.41812e6	0.827518	0.413759	−	0.910386i	$-0.364216\pi$
$443$	3.41812e6	0.827518	0.413759	+	0.910386i	$0.364216\pi$
$444$	0	0
$445$	2.91075e6	0.696795
$446$	0	0
$447$	−3.43564e6	−0.813278
$448$	0	0
$449$	−5.72410e6	−1.33996	−0.669980	−	0.742380i	$-0.733697\pi$
$449$	−5.72410e6	−1.33996	−0.669980	+	0.742380i	$0.733697\pi$
$450$	0	0
$451$	−2.88360e6	−0.667565
$452$	0	0
$453$	2.72412e6	0.623707
$454$	0	0
$455$	−847000.	−0.191803
$456$	0	0
$457$	−847750.	−0.189879	−0.0949396	−	0.995483i	$-0.530266\pi$
$457$	−847750.	−0.189879	−0.0949396	+	0.995483i	$0.530266\pi$
$458$	0	0
$459$	−389286.	−0.0862456
$460$	0	0
$461$	5.66575e6	1.24167	0.620833	−	0.783943i	$-0.286795\pi$
$461$	5.66575e6	1.24167	0.620833	+	0.783943i	$0.286795\pi$
$462$	0	0
$463$	2.52321e6	0.547018	0.273509	−	0.961870i	$-0.411816\pi$
$463$	2.52321e6	0.547018	0.273509	+	0.961870i	$0.411816\pi$
$464$	0	0
$465$	617400.	0.132414
$466$	0	0
$467$	−7.62251e6	−1.61736	−0.808678	−	0.588251i	$-0.799817\pi$
$467$	−7.62251e6	−1.61736	−0.808678	+	0.588251i	$0.799817\pi$
$468$	0	0
$469$	2.45502e6	0.515375
$470$	0	0
$471$	−3.81112e6	−0.791591
$472$	0	0
$473$	3.71606e6	0.763713
$474$	0	0
$475$	987500.	0.200818
$476$	0	0
$477$	−805302.	−0.162055
$478$	0	0
$479$	−5.17438e6	−1.03043	−0.515216	−	0.857060i	$-0.672288\pi$
$479$	−5.17438e6	−1.03043	−0.515216	+	0.857060i	$0.672288\pi$
$480$	0	0
$481$	1.11034e6	0.218823
$482$	0	0
$483$	−1.14998e6	−0.224297
$484$	0	0
$485$	69550.0	0.0134259
$486$	0	0
$487$	6.04711e6	1.15538	0.577691	−	0.816256i	$-0.303954\pi$
$487$	6.04711e6	1.15538	0.577691	+	0.816256i	$0.303954\pi$
$488$	0	0
$489$	3.22470e6	0.609842
$490$	0	0
$491$	−5.23066e6	−0.979157	−0.489579	−	0.871959i	$-0.662849\pi$
$491$	−5.23066e6	−0.979157	−0.489579	+	0.871959i	$0.662849\pi$
$492$	0	0
$493$	−2.43824e6	−0.451814
$494$	0	0
$495$	−437400.	−0.0802354
$496$	0	0
$497$	2.51328e6	0.456404
$498$	0	0
$499$	848204.	0.152493	0.0762463	−	0.997089i	$-0.475706\pi$
$499$	848204.	0.152493	0.0762463	+	0.997089i	$0.475706\pi$
$500$	0	0
$501$	−39528.0	−0.00703575
$502$	0	0
$503$	−7.60262e6	−1.33981	−0.669906	−	0.742446i	$-0.733665\pi$
$503$	−7.60262e6	−1.33981	−0.669906	+	0.742446i	$0.733665\pi$
$504$	0	0
$505$	−690450.	−0.120477
$506$	0	0
$507$	−1.99446e6	−0.344593
$508$	0	0
$509$	−423318.	−0.0724223	−0.0362111	−	0.999344i	$-0.511529\pi$
$509$	−423318.	−0.0724223	−0.0362111	+	0.999344i	$0.511529\pi$
$510$	0	0
$511$	2.21769e6	0.375706
$512$	0	0
$513$	−1.15182e6	−0.193237
$514$	0	0
$515$	−263900.	−0.0438451
$516$	0	0
$517$	−2.33798e6	−0.384694
$518$	0	0
$519$	−1.15403e6	−0.188062
$520$	0	0
$521$	−852558.	−0.137604	−0.0688018	−	0.997630i	$-0.521918\pi$
$521$	−852558.	−0.137604	−0.0688018	+	0.997630i	$0.521918\pi$
$522$	0	0
$523$	−1.10679e6	−0.176934	−0.0884668	−	0.996079i	$-0.528197\pi$
$523$	−1.10679e6	−0.176934	−0.0884668	+	0.996079i	$0.528197\pi$
$524$	0	0
$525$	−247500.	−0.0391902
$526$	0	0
$527$	1.46530e6	0.229826
$528$	0	0
$529$	1.99687e6	0.310250
$530$	0	0
$531$	−1.26166e6	−0.194180
$532$	0	0
$533$	−1.02795e7	−1.56731
$534$	0	0
$535$	1.69170e6	0.255528
$536$	0	0
$537$	−5.03194e6	−0.753008
$538$	0	0
$539$	−3.21214e6	−0.476236
$540$	0	0
$541$	−6.98348e6	−1.02584	−0.512919	−	0.858437i	$-0.671436\pi$
$541$	−6.98348e6	−1.02584	−0.512919	+	0.858437i	$0.671436\pi$
$542$	0	0
$543$	−6.78123e6	−0.986981
$544$	0	0
$545$	5.75905e6	0.830538
$546$	0	0
$547$	−6.69845e6	−0.957208	−0.478604	−	0.878031i	$-0.658857\pi$
$547$	−6.69845e6	−0.957208	−0.478604	+	0.878031i	$0.658857\pi$
$548$	0	0
$549$	3.18346e6	0.450784
$550$	0	0
$551$	−7.21428e6	−1.01231
$552$	0	0
$553$	−464288.	−0.0645617
$554$	0	0
$555$	324450.	0.0447111
$556$	0	0
$557$	−698142.	−0.0953467	−0.0476734	−	0.998863i	$-0.515181\pi$
$557$	−698142.	−0.0953467	−0.0476734	+	0.998863i	$0.515181\pi$
$558$	0	0
$559$	1.32471e7	1.79304
$560$	0	0
$561$	−1.03810e6	−0.139261
$562$	0	0
$563$	2.68254e6	0.356677	0.178339	−	0.983969i	$-0.442928\pi$
$563$	2.68254e6	0.356677	0.178339	+	0.983969i	$0.442928\pi$
$564$	0	0
$565$	−1.82955e6	−0.241114
$566$	0	0
$567$	288684.	0.0377107
$568$	0	0
$569$	3.80815e6	0.493098	0.246549	−	0.969130i	$-0.420703\pi$
$569$	3.80815e6	0.493098	0.246549	+	0.969130i	$0.420703\pi$
$570$	0	0
$571$	5.15378e6	0.661509	0.330754	−	0.943717i	$-0.392697\pi$
$571$	5.15378e6	0.661509	0.330754	+	0.943717i	$0.392697\pi$
$572$	0	0
$573$	2.96114e6	0.376767
$574$	0	0
$575$	1.81500e6	0.228932
$576$	0	0
$577$	−446782.	−0.0558671	−0.0279336	−	0.999610i	$-0.508893\pi$
$577$	−446782.	−0.0558671	−0.0279336	+	0.999610i	$0.508893\pi$
$578$	0	0
$579$	2.36065e6	0.292641
$580$	0	0
$581$	4.77682e6	0.587081
$582$	0	0
$583$	−2.14747e6	−0.261671
$584$	0	0
$585$	−1.55925e6	−0.188376
$586$	0	0
$587$	1.42485e7	1.70677	0.853384	−	0.521282i	$-0.174546\pi$
$587$	1.42485e7	1.70677	0.853384	+	0.521282i	$0.174546\pi$
$588$	0	0
$589$	4.33552e6	0.514936
$590$	0	0
$591$	8.80033e6	1.03641
$592$	0	0
$593$	−1.02387e7	−1.19566	−0.597831	−	0.801622i	$-0.703971\pi$
$593$	−1.02387e7	−1.19566	−0.597831	+	0.801622i	$0.703971\pi$
$594$	0	0
$595$	−587400.	−0.0680208
$596$	0	0
$597$	−1.54879e6	−0.177851
$598$	0	0
$599$	1.14115e7	1.29950	0.649749	−	0.760149i	$-0.274874\pi$
$599$	1.14115e7	1.29950	0.649749	+	0.760149i	$0.274874\pi$
$600$	0	0
$601$	−79222.0	−0.00894663	−0.00447332	−	0.999990i	$-0.501424\pi$
$601$	−79222.0	−0.00894663	−0.00447332	+	0.999990i	$0.501424\pi$
$602$	0	0
$603$	4.51948e6	0.506168
$604$	0	0
$605$	2.85987e6	0.317657
$606$	0	0
$607$	−947908.	−0.104423	−0.0522113	−	0.998636i	$-0.516627\pi$
$607$	−947908.	−0.104423	−0.0522113	+	0.998636i	$0.516627\pi$
$608$	0	0
$609$	1.80814e6	0.197555
$610$	0	0
$611$	−8.33448e6	−0.903182
$612$	0	0
$613$	−9.32400e6	−1.00219	−0.501096	−	0.865392i	$-0.667070\pi$
$613$	−9.32400e6	−1.00219	−0.501096	+	0.865392i	$0.667070\pi$
$614$	0	0
$615$	−3.00375e6	−0.320240
$616$	0	0
$617$	1.36583e7	1.44439	0.722193	−	0.691692i	$-0.243134\pi$
$617$	1.36583e7	1.44439	0.722193	+	0.691692i	$0.243134\pi$
$618$	0	0
$619$	1.02615e7	1.07642	0.538212	−	0.842809i	$-0.319100\pi$
$619$	1.02615e7	1.07642	0.538212	+	0.842809i	$0.319100\pi$
$620$	0	0
$621$	−2.11702e6	−0.220290
$622$	0	0
$623$	−5.12292e6	−0.528807
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−3.07152e6	−0.312021
$628$	0	0
$629$	770028.	0.0776032
$630$	0	0
$631$	1.03462e7	1.03444	0.517222	−	0.855851i	$-0.326966\pi$
$631$	1.03462e7	1.03444	0.517222	+	0.855851i	$0.326966\pi$
$632$	0	0
$633$	8.38710e6	0.831960
$634$	0	0
$635$	5.21590e6	0.513328
$636$	0	0
$637$	−1.14507e7	−1.11810
$638$	0	0
$639$	4.62672e6	0.448251
$640$	0	0
$641$	6.76499e6	0.650313	0.325156	−	0.945660i	$-0.394583\pi$
$641$	6.76499e6	0.650313	0.325156	+	0.945660i	$0.394583\pi$
$642$	0	0
$643$	−4.13340e6	−0.394258	−0.197129	−	0.980378i	$-0.563162\pi$
$643$	−4.13340e6	−0.394258	−0.197129	+	0.980378i	$0.563162\pi$
$644$	0	0
$645$	3.87090e6	0.366364
$646$	0	0
$647$	7.18555e6	0.674838	0.337419	−	0.941355i	$-0.390446\pi$
$647$	7.18555e6	0.674838	0.337419	+	0.941355i	$0.390446\pi$
$648$	0	0
$649$	−3.36442e6	−0.313544
$650$	0	0
$651$	−1.08662e6	−0.100491
$652$	0	0
$653$	−1.34603e7	−1.23530	−0.617648	−	0.786454i	$-0.711915\pi$
$653$	−1.34603e7	−1.23530	−0.617648	+	0.786454i	$0.711915\pi$
$654$	0	0
$655$	−469800.	−0.0427868
$656$	0	0
$657$	4.08256e6	0.368994
$658$	0	0
$659$	−7.30303e6	−0.655073	−0.327536	−	0.944839i	$-0.606218\pi$
$659$	−7.30303e6	−0.655073	−0.327536	+	0.944839i	$0.606218\pi$
$660$	0	0
$661$	−1.97821e7	−1.76104	−0.880518	−	0.474012i	$-0.842805\pi$
$661$	−1.97821e7	−1.76104	−0.880518	+	0.474012i	$0.842805\pi$
$662$	0	0
$663$	−3.70062e6	−0.326957
$664$	0	0
$665$	−1.73800e6	−0.152404
$666$	0	0
$667$	−1.32597e7	−1.15403
$668$	0	0
$669$	7.56414e6	0.653423
$670$	0	0
$671$	8.48923e6	0.727884
$672$	0	0
$673$	536090.	0.0456247	0.0228124	−	0.999740i	$-0.492738\pi$
$673$	536090.	0.0456247	0.0228124	+	0.999740i	$0.492738\pi$
$674$	0	0
$675$	−455625.	−0.0384900
$676$	0	0
$677$	6.25127e6	0.524200	0.262100	−	0.965041i	$-0.415585\pi$
$677$	6.25127e6	0.524200	0.262100	+	0.965041i	$0.415585\pi$
$678$	0	0
$679$	−122408.	−0.0101891
$680$	0	0
$681$	6.00426e6	0.496126
$682$	0	0
$683$	9.83536e6	0.806749	0.403374	−	0.915035i	$-0.367837\pi$
$683$	9.83536e6	0.806749	0.403374	+	0.915035i	$0.367837\pi$
$684$	0	0
$685$	3.55545e6	0.289513
$686$	0	0
$687$	−9.26501e6	−0.748952
$688$	0	0
$689$	−7.65534e6	−0.614351
$690$	0	0
$691$	−1.31626e7	−1.04869	−0.524345	−	0.851506i	$-0.675690\pi$
$691$	−1.31626e7	−1.04869	−0.524345	+	0.851506i	$0.675690\pi$
$692$	0	0
$693$	769824.	0.0608917
$694$	0	0
$695$	7.44010e6	0.584274
$696$	0	0
$697$	−7.12890e6	−0.555828
$698$	0	0
$699$	1.44760e7	1.12062
$700$	0	0
$701$	−2.26820e7	−1.74335	−0.871677	−	0.490080i	$-0.836968\pi$
$701$	−2.26820e7	−1.74335	−0.871677	+	0.490080i	$0.836968\pi$
$702$	0	0
$703$	2.27836e6	0.173874
$704$	0	0
$705$	−2.43540e6	−0.184543
$706$	0	0
$707$	1.21519e6	0.0914316
$708$	0	0
$709$	−8.77883e6	−0.655875	−0.327937	−	0.944699i	$-0.606353\pi$
$709$	−8.77883e6	−0.655875	−0.327937	+	0.944699i	$0.606353\pi$
$710$	0	0
$711$	−854712.	−0.0634083
$712$	0	0
$713$	7.96858e6	0.587025
$714$	0	0
$715$	−4.15800e6	−0.304172
$716$	0	0
$717$	−1.46290e7	−1.06272
$718$	0	0
$719$	−2.50221e7	−1.80510	−0.902551	−	0.430583i	$-0.858308\pi$
$719$	−2.50221e7	−1.80510	−0.902551	+	0.430583i	$0.858308\pi$
$720$	0	0
$721$	464464.	0.0332747
$722$	0	0
$723$	−3.84136e6	−0.273300
$724$	0	0
$725$	−2.85375e6	−0.201637
$726$	0	0
$727$	1.06215e7	0.745333	0.372666	−	0.927965i	$-0.378444\pi$
$727$	1.06215e7	0.745333	0.372666	+	0.927965i	$0.378444\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	9.18694e6	0.635883
$732$	0	0
$733$	2.22439e6	0.152916	0.0764578	−	0.997073i	$-0.475639\pi$
$733$	2.22439e6	0.152916	0.0764578	+	0.997073i	$0.475639\pi$
$734$	0	0
$735$	−3.34598e6	−0.228457
$736$	0	0
$737$	1.20519e7	0.817312
$738$	0	0
$739$	−9.33082e6	−0.628505	−0.314252	−	0.949339i	$-0.601754\pi$
$739$	−9.33082e6	−0.628505	−0.314252	+	0.949339i	$0.601754\pi$
$740$	0	0
$741$	−1.09494e7	−0.732563
$742$	0	0
$743$	9.45010e6	0.628007	0.314003	−	0.949422i	$-0.398330\pi$
$743$	9.45010e6	0.628007	0.314003	+	0.949422i	$0.398330\pi$
$744$	0	0
$745$	−9.54345e6	−0.629963
$746$	0	0
$747$	8.79368e6	0.576593
$748$	0	0
$749$	−2.97739e6	−0.193924
$750$	0	0
$751$	549512.	0.0355531	0.0177765	−	0.999842i	$-0.494341\pi$
$751$	549512.	0.0355531	0.0177765	+	0.999842i	$0.494341\pi$
$752$	0	0
$753$	1.20178e7	0.772392
$754$	0	0
$755$	7.56700e6	0.483121
$756$	0	0
$757$	−1.57436e7	−0.998538	−0.499269	−	0.866447i	$-0.666398\pi$
$757$	−1.57436e7	−0.998538	−0.499269	+	0.866447i	$0.666398\pi$
$758$	0	0
$759$	−5.64538e6	−0.355704
$760$	0	0
$761$	4.00504e6	0.250695	0.125347	−	0.992113i	$-0.459995\pi$
$761$	4.00504e6	0.250695	0.125347	+	0.992113i	$0.459995\pi$
$762$	0	0
$763$	−1.01359e7	−0.630307
$764$	0	0
$765$	−1.08135e6	−0.0668056
$766$	0	0
$767$	−1.19935e7	−0.736136
$768$	0	0
$769$	1.87997e7	1.14640	0.573198	−	0.819417i	$-0.305703\pi$
$769$	1.87997e7	1.14640	0.573198	+	0.819417i	$0.305703\pi$
$770$	0	0
$771$	1.87814e7	1.13787
$772$	0	0
$773$	−9.13834e6	−0.550071	−0.275035	−	0.961434i	$-0.588690\pi$
$773$	−9.13834e6	−0.550071	−0.275035	+	0.961434i	$0.588690\pi$
$774$	0	0
$775$	1.71500e6	0.102568
$776$	0	0
$777$	−571032.	−0.0339319
$778$	0	0
$779$	−2.10930e7	−1.24536
$780$	0	0
$781$	1.23379e7	0.723793
$782$	0	0
$783$	3.32861e6	0.194026
$784$	0	0
$785$	−1.05865e7	−0.613163
$786$	0	0
$787$	−2.17265e6	−0.125041	−0.0625206	−	0.998044i	$-0.519914\pi$
$787$	−2.17265e6	−0.125041	−0.0625206	+	0.998044i	$0.519914\pi$
$788$	0	0
$789$	−8.46742e6	−0.484237
$790$	0	0
$791$	3.22001e6	0.182985
$792$	0	0
$793$	3.02625e7	1.70892
$794$	0	0
$795$	−2.23695e6	−0.125527
$796$	0	0
$797$	1.82343e7	1.01682	0.508408	−	0.861116i	$-0.330234\pi$
$797$	1.82343e7	1.01682	0.508408	+	0.861116i	$0.330234\pi$
$798$	0	0
$799$	−5.78002e6	−0.320304
$800$	0	0
$801$	−9.43083e6	−0.519360
$802$	0	0
$803$	1.08868e7	0.595817
$804$	0	0
$805$	−3.19440e6	−0.173740
$806$	0	0
$807$	−9.12335e6	−0.493141
$808$	0	0
$809$	−2.88965e6	−0.155230	−0.0776148	−	0.996983i	$-0.524730\pi$
$809$	−2.88965e6	−0.155230	−0.0776148	+	0.996983i	$0.524730\pi$
$810$	0	0
$811$	2.33838e7	1.24843	0.624214	−	0.781254i	$-0.285419\pi$
$811$	2.33838e7	1.24843	0.624214	+	0.781254i	$0.285419\pi$
$812$	0	0
$813$	2.59214e6	0.137541
$814$	0	0
$815$	8.95750e6	0.472381
$816$	0	0
$817$	2.71823e7	1.42473
$818$	0	0
$819$	2.74428e6	0.142961
$820$	0	0
$821$	−949374.	−0.0491563	−0.0245782	−	0.999698i	$-0.507824\pi$
$821$	−949374.	−0.0491563	−0.0245782	+	0.999698i	$0.507824\pi$
$822$	0	0
$823$	3.55430e7	1.82917	0.914586	−	0.404391i	$-0.132516\pi$
$823$	3.55430e7	1.82917	0.914586	+	0.404391i	$0.132516\pi$
$824$	0	0
$825$	−1.21500e6	−0.0621500
$826$	0	0
$827$	7.48799e6	0.380716	0.190358	−	0.981715i	$-0.439035\pi$
$827$	7.48799e6	0.380716	0.190358	+	0.981715i	$0.439035\pi$
$828$	0	0
$829$	1.05885e7	0.535116	0.267558	−	0.963542i	$-0.413783\pi$
$829$	1.05885e7	0.535116	0.267558	+	0.963542i	$0.413783\pi$
$830$	0	0
$831$	−7.74664e6	−0.389145
$832$	0	0
$833$	−7.94111e6	−0.396523
$834$	0	0
$835$	−109800.	−0.00544987
$836$	0	0
$837$	−2.00038e6	−0.0986956
$838$	0	0
$839$	−7.27486e6	−0.356796	−0.178398	−	0.983958i	$-0.557091\pi$
$839$	−7.27486e6	−0.356796	−0.178398	+	0.983958i	$0.557091\pi$
$840$	0	0
$841$	337207.	0.0164402
$842$	0	0
$843$	−3.78275e6	−0.183332
$844$	0	0
$845$	−5.54017e6	−0.266920
$846$	0	0
$847$	−5.03338e6	−0.241074
$848$	0	0
$849$	−4.09835e6	−0.195137
$850$	0	0
$851$	4.18757e6	0.198216
$852$	0	0
$853$	3.40904e7	1.60420	0.802102	−	0.597187i	$-0.203715\pi$
$853$	3.40904e7	1.60420	0.802102	+	0.597187i	$0.203715\pi$
$854$	0	0
$855$	−3.19950e6	−0.149681
$856$	0	0
$857$	1.97494e7	0.918550	0.459275	−	0.888294i	$-0.348109\pi$
$857$	1.97494e7	0.918550	0.459275	+	0.888294i	$0.348109\pi$
$858$	0	0
$859$	−2.64947e7	−1.22511	−0.612556	−	0.790428i	$-0.709858\pi$
$859$	−2.64947e7	−1.22511	−0.612556	+	0.790428i	$0.709858\pi$
$860$	0	0
$861$	5.28660e6	0.243035
$862$	0	0
$863$	−1.37256e7	−0.627340	−0.313670	−	0.949532i	$-0.601559\pi$
$863$	−1.37256e7	−0.627340	−0.313670	+	0.949532i	$0.601559\pi$
$864$	0	0
$865$	−3.20565e6	−0.145672
$866$	0	0
$867$	1.02123e7	0.461399
$868$	0	0
$869$	−2.27923e6	−0.102386
$870$	0	0
$871$	4.29629e7	1.91888
$872$	0	0
$873$	−225342.	−0.0100071
$874$	0	0
$875$	−687500.	−0.0303566
$876$	0	0
$877$	1.67999e7	0.737578	0.368789	−	0.929513i	$-0.379773\pi$
$877$	1.67999e7	0.737578	0.368789	+	0.929513i	$0.379773\pi$
$878$	0	0
$879$	−1.90386e7	−0.831119
$880$	0	0
$881$	3.52966e7	1.53212	0.766062	−	0.642767i	$-0.222214\pi$
$881$	3.52966e7	1.53212	0.766062	+	0.642767i	$0.222214\pi$
$882$	0	0
$883$	−4.12228e7	−1.77925	−0.889623	−	0.456696i	$-0.849033\pi$
$883$	−4.12228e7	−1.77925	−0.889623	+	0.456696i	$0.849033\pi$
$884$	0	0
$885$	−3.50460e6	−0.150411
$886$	0	0
$887$	−1.56703e7	−0.668757	−0.334378	−	0.942439i	$-0.608526\pi$
$887$	−1.56703e7	−0.668757	−0.334378	+	0.942439i	$0.608526\pi$
$888$	0	0
$889$	−9.17998e6	−0.389572
$890$	0	0
$891$	1.41718e6	0.0598039
$892$	0	0
$893$	−1.71019e7	−0.717656
$894$	0	0
$895$	−1.39776e7	−0.583277
$896$	0	0
$897$	−2.01247e7	−0.835120
$898$	0	0
$899$	−1.25291e7	−0.517036
$900$	0	0
$901$	−5.30903e6	−0.217873
$902$	0	0
$903$	−6.81278e6	−0.278039
$904$	0	0
$905$	−1.88368e7	−0.764512
$906$	0	0
$907$	−1.79580e7	−0.724837	−0.362419	−	0.932015i	$-0.618049\pi$
$907$	−1.79580e7	−0.724837	−0.362419	+	0.932015i	$0.618049\pi$
$908$	0	0
$909$	2.23706e6	0.0897982
$910$	0	0
$911$	−4.06576e7	−1.62310	−0.811552	−	0.584281i	$-0.801377\pi$
$911$	−4.06576e7	−1.62310	−0.811552	+	0.584281i	$0.801377\pi$
$912$	0	0
$913$	2.34498e7	0.931028
$914$	0	0
$915$	8.84295e6	0.349176
$916$	0	0
$917$	826848.	0.0324715
$918$	0	0
$919$	−3.01019e6	−0.117572	−0.0587862	−	0.998271i	$-0.518723\pi$
$919$	−3.01019e6	−0.117572	−0.0587862	+	0.998271i	$0.518723\pi$
$920$	0	0
$921$	−1.15482e7	−0.448606
$922$	0	0
$923$	4.39824e7	1.69932
$924$	0	0
$925$	901250.	0.0346331
$926$	0	0
$927$	855036.	0.0326802
$928$	0	0
$929$	−3.02867e7	−1.15136	−0.575681	−	0.817674i	$-0.695263\pi$
$929$	−3.02867e7	−1.15136	−0.575681	+	0.817674i	$0.695263\pi$
$930$	0	0
$931$	−2.34962e7	−0.888430
$932$	0	0
$933$	−1.28943e7	−0.484948
$934$	0	0
$935$	−2.88360e6	−0.107871
$936$	0	0
$937$	2.65868e7	0.989277	0.494638	−	0.869099i	$-0.335301\pi$
$937$	2.65868e7	0.989277	0.494638	+	0.869099i	$0.335301\pi$
$938$	0	0
$939$	1.12544e7	0.416543
$940$	0	0
$941$	−2.76447e7	−1.01774	−0.508871	−	0.860843i	$-0.669937\pi$
$941$	−2.76447e7	−1.01774	−0.508871	+	0.860843i	$0.669937\pi$
$942$	0	0
$943$	−3.87684e7	−1.41971
$944$	0	0
$945$	801900.	0.0292106
$946$	0	0
$947$	−3.84009e7	−1.39145	−0.695724	−	0.718309i	$-0.744916\pi$
$947$	−3.84009e7	−1.39145	−0.695724	+	0.718309i	$0.744916\pi$
$948$	0	0
$949$	3.88095e7	1.39886
$950$	0	0
$951$	1.85815e7	0.666236
$952$	0	0
$953$	2.25535e7	0.804419	0.402209	−	0.915548i	$-0.368242\pi$
$953$	2.25535e7	0.804419	0.402209	+	0.915548i	$0.368242\pi$
$954$	0	0
$955$	8.22540e6	0.291843
$956$	0	0
$957$	8.87630e6	0.313294
$958$	0	0
$959$	−6.25759e6	−0.219716
$960$	0	0
$961$	−2.10996e7	−0.736998
$962$	0	0
$963$	−5.48111e6	−0.190460
$964$	0	0
$965$	6.55735e6	0.226678
$966$	0	0
$967$	−2.29919e7	−0.790693	−0.395347	−	0.918532i	$-0.629375\pi$
$967$	−2.29919e7	−0.790693	−0.395347	+	0.918532i	$0.629375\pi$
$968$	0	0
$969$	−7.59348e6	−0.259795
$970$	0	0
$971$	2.57163e7	0.875307	0.437653	−	0.899144i	$-0.355810\pi$
$971$	2.57163e7	0.875307	0.437653	+	0.899144i	$0.355810\pi$
$972$	0	0
$973$	−1.30946e7	−0.443414
$974$	0	0
$975$	−4.33125e6	−0.145916
$976$	0	0
$977$	−1.54467e7	−0.517725	−0.258862	−	0.965914i	$-0.583348\pi$
$977$	−1.54467e7	−0.517725	−0.258862	+	0.965914i	$0.583348\pi$
$978$	0	0
$979$	−2.51489e7	−0.838614
$980$	0	0
$981$	−1.86593e7	−0.619047
$982$	0	0
$983$	2.78702e7	0.919933	0.459966	−	0.887936i	$-0.347861\pi$
$983$	2.78702e7	0.919933	0.459966	+	0.887936i	$0.347861\pi$
$984$	0	0
$985$	2.44454e7	0.802797
$986$	0	0
$987$	4.28630e6	0.140052
$988$	0	0
$989$	4.99604e7	1.62418
$990$	0	0
$991$	1.32911e7	0.429909	0.214954	−	0.976624i	$-0.431040\pi$
$991$	1.32911e7	0.429909	0.214954	+	0.976624i	$0.431040\pi$
$992$	0	0
$993$	2.12590e7	0.684178
$994$	0	0
$995$	−4.30220e6	−0.137763
$996$	0	0
$997$	1.67943e7	0.535085	0.267543	−	0.963546i	$-0.413788\pi$
$997$	1.67943e7	0.535085	0.267543	+	0.963546i	$0.413788\pi$
$998$	0	0
$999$	−1.05122e6	−0.0333257

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.6.a.a.1.1	✓	1
3.2	odd	2		180.6.a.d.1.1		1
4.3	odd	2		240.6.a.i.1.1		1
5.2	odd	4		300.6.d.e.49.2		2
5.3	odd	4		300.6.d.e.49.1		2
5.4	even	2		300.6.a.d.1.1		1
8.3	odd	2		960.6.a.i.1.1		1
8.5	even	2		960.6.a.z.1.1		1
12.11	even	2		720.6.a.p.1.1		1
15.2	even	4		900.6.d.b.649.2		2
15.8	even	4		900.6.d.b.649.1		2
15.14	odd	2		900.6.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.6.a.a.1.1	✓	1	1.1	even	1	trivial
180.6.a.d.1.1		1	3.2	odd	2
240.6.a.i.1.1		1	4.3	odd	2
300.6.a.d.1.1		1	5.4	even	2
300.6.d.e.49.1		2	5.3	odd	4
300.6.d.e.49.2		2	5.2	odd	4
720.6.a.p.1.1		1	12.11	even	2
900.6.a.f.1.1		1	15.14	odd	2
900.6.d.b.649.1		2	15.8	even	4
900.6.d.b.649.2		2	15.2	even	4
960.6.a.i.1.1		1	8.3	odd	2
960.6.a.z.1.1		1	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	60.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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