LMFDB - Embedded newform 528.6.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,6,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("528.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	528.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:	$84.6826568613$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	528.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +57.7228 q^{5} -251.081 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +57.7228 q^{5} -251.081 q^{7} +81.0000 q^{9} +121.000 q^{11} -277.549 q^{13} -519.505 q^{15} +704.489 q^{17} +2861.18 q^{19} +2259.73 q^{21} +1066.85 q^{23} +206.923 q^{25} -729.000 q^{27} -3937.44 q^{29} +644.350 q^{31} -1089.00 q^{33} -14493.1 q^{35} -9042.34 q^{37} +2497.94 q^{39} +18219.0 q^{41} +4054.54 q^{43} +4675.55 q^{45} -20750.8 q^{47} +46234.9 q^{49} -6340.40 q^{51} -26485.9 q^{53} +6984.46 q^{55} -25750.7 q^{57} -4293.12 q^{59} -6831.76 q^{61} -20337.6 q^{63} -16020.9 q^{65} +56749.5 q^{67} -9601.68 q^{69} -3187.09 q^{71} -7397.14 q^{73} -1862.31 q^{75} -30380.9 q^{77} -24393.7 q^{79} +6561.00 q^{81} -102795. q^{83} +40665.1 q^{85} +35437.0 q^{87} +49599.4 q^{89} +69687.4 q^{91} -5799.15 q^{93} +165156. q^{95} -92279.5 q^{97} +9801.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{3} + 58 q^{5} - 146 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q - 18 * q^3 + 58 * q^5 - 146 * q^7 + 162 * q^9	$ 2 q - 18 q^{3} + 58 q^{5} - 146 q^{7} + 162 q^{9} + 242 q^{11} - 130 q^{13} - 522 q^{15} - 728 q^{17} + 828 q^{19} + 1314 q^{21} + 238 q^{23} - 2918 q^{25} - 1458 q^{27} + 696 q^{29} + 10480 q^{31} - 2178 q^{33} - 14464 q^{35} - 1908 q^{37} + 1170 q^{39} + 36484 q^{41} - 9768 q^{43} + 4698 q^{45} - 43742 q^{47} + 40470 q^{49} + 6552 q^{51} - 12174 q^{53} + 7018 q^{55} - 7452 q^{57} + 2788 q^{59} - 25302 q^{61} - 11826 q^{63} - 15980 q^{65} + 40520 q^{67} - 2142 q^{69} - 31386 q^{71} - 46780 q^{73} + 26262 q^{75} - 17666 q^{77} + 16850 q^{79} + 13122 q^{81} - 79440 q^{83} + 40268 q^{85} - 6264 q^{87} - 54204 q^{89} + 85192 q^{91} - 94320 q^{93} + 164592 q^{95} - 241568 q^{97} + 19602 q^{99}+O(q^{100}) $ 2 * q - 18 * q^3 + 58 * q^5 - 146 * q^7 + 162 * q^9 + 242 * q^11 - 130 * q^13 - 522 * q^15 - 728 * q^17 + 828 * q^19 + 1314 * q^21 + 238 * q^23 - 2918 * q^25 - 1458 * q^27 + 696 * q^29 + 10480 * q^31 - 2178 * q^33 - 14464 * q^35 - 1908 * q^37 + 1170 * q^39 + 36484 * q^41 - 9768 * q^43 + 4698 * q^45 - 43742 * q^47 + 40470 * q^49 + 6552 * q^51 - 12174 * q^53 + 7018 * q^55 - 7452 * q^57 + 2788 * q^59 - 25302 * q^61 - 11826 * q^63 - 15980 * q^65 + 40520 * q^67 - 2142 * q^69 - 31386 * q^71 - 46780 * q^73 + 26262 * q^75 - 17666 * q^77 + 16850 * q^79 + 13122 * q^81 - 79440 * q^83 + 40268 * q^85 - 6264 * q^87 - 54204 * q^89 + 85192 * q^91 - 94320 * q^93 + 164592 * q^95 - 241568 * q^97 + 19602 * q^99

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
$5$	57.7228	1.03258	0.516289	−	0.856415i	$-0.327313\pi$
$5$	57.7228	1.03258	0.516289	+	0.856415i	$0.327313\pi$
$6$	0	0
$7$	−251.081	−1.93673	−0.968366	−	0.249534i	$-0.919722\pi$
$7$	−251.081	−1.93673	−0.968366	+	0.249534i	$0.919722\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−277.549	−0.455492	−0.227746	−	0.973721i	$-0.573136\pi$
$13$	−277.549	−0.455492	−0.227746	+	0.973721i	$0.573136\pi$
$14$	0	0
$15$	−519.505	−0.596159
$16$	0	0
$17$	704.489	0.591224	0.295612	−	0.955308i	$-0.404477\pi$
$17$	704.489	0.591224	0.295612	+	0.955308i	$0.404477\pi$
$18$	0	0
$19$	2861.18	1.81828	0.909142	−	0.416486i	$-0.136739\pi$
$19$	2861.18	1.81828	0.909142	+	0.416486i	$0.136739\pi$
$20$	0	0
$21$	2259.73	1.11817
$22$	0	0
$23$	1066.85	0.420518	0.210259	−	0.977646i	$-0.432569\pi$
$23$	1066.85	0.420518	0.210259	+	0.977646i	$0.432569\pi$
$24$	0	0
$25$	206.923	0.0662154
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−3937.44	−0.869399	−0.434700	−	0.900575i	$-0.643145\pi$
$29$	−3937.44	−0.869399	−0.434700	+	0.900575i	$0.643145\pi$
$30$	0	0
$31$	644.350	0.120425	0.0602126	−	0.998186i	$-0.480822\pi$
$31$	644.350	0.120425	0.0602126	+	0.998186i	$0.480822\pi$
$32$	0	0
$33$	−1089.00	−0.174078
$34$	0	0
$35$	−14493.1	−1.99982
$36$	0	0
$37$	−9042.34	−1.08587	−0.542934	−	0.839776i	$-0.682686\pi$
$37$	−9042.34	−1.08587	−0.542934	+	0.839776i	$0.682686\pi$
$38$	0	0
$39$	2497.94	0.262979
$40$	0	0
$41$	18219.0	1.69264	0.846322	−	0.532672i	$-0.178812\pi$
$41$	18219.0	1.69264	0.846322	+	0.532672i	$0.178812\pi$
$42$	0	0
$43$	4054.54	0.334403	0.167202	−	0.985923i	$-0.446527\pi$
$43$	4054.54	0.334403	0.167202	+	0.985923i	$0.446527\pi$
$44$	0	0
$45$	4675.55	0.344192
$46$	0	0
$47$	−20750.8	−1.37022	−0.685110	−	0.728439i	$-0.740246\pi$
$47$	−20750.8	−1.37022	−0.685110	+	0.728439i	$0.740246\pi$
$48$	0	0
$49$	46234.9	2.75093
$50$	0	0
$51$	−6340.40	−0.341343
$52$	0	0
$53$	−26485.9	−1.29517	−0.647583	−	0.761995i	$-0.724220\pi$
$53$	−26485.9	−1.29517	−0.647583	+	0.761995i	$0.724220\pi$
$54$	0	0
$55$	6984.46	0.311334
$56$	0	0
$57$	−25750.7	−1.04979
$58$	0	0
$59$	−4293.12	−0.160562	−0.0802810	−	0.996772i	$-0.525582\pi$
$59$	−4293.12	−0.160562	−0.0802810	+	0.996772i	$0.525582\pi$
$60$	0	0
$61$	−6831.76	−0.235076	−0.117538	−	0.993068i	$-0.537500\pi$
$61$	−6831.76	−0.235076	−0.117538	+	0.993068i	$0.537500\pi$
$62$	0	0
$63$	−20337.6	−0.645577
$64$	0	0
$65$	−16020.9	−0.470331
$66$	0	0
$67$	56749.5	1.54445	0.772227	−	0.635347i	$-0.219143\pi$
$67$	56749.5	1.54445	0.772227	+	0.635347i	$0.219143\pi$
$68$	0	0
$69$	−9601.68	−0.242786
$70$	0	0
$71$	−3187.09	−0.0750323	−0.0375161	−	0.999296i	$-0.511945\pi$
$71$	−3187.09	−0.0750323	−0.0375161	+	0.999296i	$0.511945\pi$
$72$	0	0
$73$	−7397.14	−0.162464	−0.0812319	−	0.996695i	$-0.525885\pi$
$73$	−7397.14	−0.162464	−0.0812319	+	0.996695i	$0.525885\pi$
$74$	0	0
$75$	−1862.31	−0.0382295
$76$	0	0
$77$	−30380.9	−0.583947
$78$	0	0
$79$	−24393.7	−0.439754	−0.219877	−	0.975528i	$-0.570566\pi$
$79$	−24393.7	−0.439754	−0.219877	+	0.975528i	$0.570566\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−102795.	−1.63786	−0.818932	−	0.573890i	$-0.805434\pi$
$83$	−102795.	−1.63786	−0.818932	+	0.573890i	$0.805434\pi$
$84$	0	0
$85$	40665.1	0.610484
$86$	0	0
$87$	35437.0	0.501948
$88$	0	0
$89$	49599.4	0.663745	0.331873	−	0.943324i	$-0.392320\pi$
$89$	49599.4	0.663745	0.331873	+	0.943324i	$0.392320\pi$
$90$	0	0
$91$	69687.4	0.882166
$92$	0	0
$93$	−5799.15	−0.0695275
$94$	0	0
$95$	165156.	1.87752
$96$	0	0
$97$	−92279.5	−0.995808	−0.497904	−	0.867232i	$-0.665897\pi$
$97$	−92279.5	−0.995808	−0.497904	+	0.867232i	$0.665897\pi$
$98$	0	0
$99$	9801.00	0.100504
$100$	0	0
$101$	−35546.1	−0.346728	−0.173364	−	0.984858i	$-0.555464\pi$
$101$	−35546.1	−0.346728	−0.173364	+	0.984858i	$0.555464\pi$
$102$	0	0
$103$	59876.8	0.556116	0.278058	−	0.960564i	$-0.410309\pi$
$103$	59876.8	0.556116	0.278058	+	0.960564i	$0.410309\pi$
$104$	0	0
$105$	130438.	1.15460
$106$	0	0
$107$	−89253.8	−0.753646	−0.376823	−	0.926285i	$-0.622983\pi$
$107$	−89253.8	−0.753646	−0.376823	+	0.926285i	$0.622983\pi$
$108$	0	0
$109$	22796.0	0.183777	0.0918887	−	0.995769i	$-0.470710\pi$
$109$	22796.0	0.183777	0.0918887	+	0.995769i	$0.470710\pi$
$110$	0	0
$111$	81381.1	0.626926
$112$	0	0
$113$	−166064.	−1.22343	−0.611717	−	0.791077i	$-0.709521\pi$
$113$	−166064.	−1.22343	−0.611717	+	0.791077i	$0.709521\pi$
$114$	0	0
$115$	61581.7	0.434218
$116$	0	0
$117$	−22481.5	−0.151831
$118$	0	0
$119$	−176884.	−1.14504
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	−163971.	−0.977248
$124$	0	0
$125$	−168440.	−0.964205
$126$	0	0
$127$	−159190.	−0.875802	−0.437901	−	0.899023i	$-0.644278\pi$
$127$	−159190.	−0.875802	−0.437901	+	0.899023i	$0.644278\pi$
$128$	0	0
$129$	−36490.9	−0.193068
$130$	0	0
$131$	−232174.	−1.18205	−0.591025	−	0.806654i	$-0.701276\pi$
$131$	−232174.	−1.18205	−0.591025	+	0.806654i	$0.701276\pi$
$132$	0	0
$133$	−718390.	−3.52153
$134$	0	0
$135$	−42079.9	−0.198720
$136$	0	0
$137$	68205.4	0.310468	0.155234	−	0.987878i	$-0.450387\pi$
$137$	68205.4	0.310468	0.155234	+	0.987878i	$0.450387\pi$
$138$	0	0
$139$	−298162.	−1.30893	−0.654464	−	0.756093i	$-0.727106\pi$
$139$	−298162.	−1.30893	−0.654464	+	0.756093i	$0.727106\pi$
$140$	0	0
$141$	186757.	0.791097
$142$	0	0
$143$	−33583.4	−0.137336
$144$	0	0
$145$	−227280.	−0.897722
$146$	0	0
$147$	−416114.	−1.58825
$148$	0	0
$149$	83131.5	0.306761	0.153380	−	0.988167i	$-0.450984\pi$
$149$	83131.5	0.306761	0.153380	+	0.988167i	$0.450984\pi$
$150$	0	0
$151$	−167598.	−0.598171	−0.299085	−	0.954226i	$-0.596682\pi$
$151$	−167598.	−0.598171	−0.299085	+	0.954226i	$0.596682\pi$
$152$	0	0
$153$	57063.6	0.197075
$154$	0	0
$155$	37193.7	0.124348
$156$	0	0
$157$	−63259.7	−0.204823	−0.102411	−	0.994742i	$-0.532656\pi$
$157$	−63259.7	−0.204823	−0.102411	+	0.994742i	$0.532656\pi$
$158$	0	0
$159$	238373.	0.747765
$160$	0	0
$161$	−267867.	−0.814431
$162$	0	0
$163$	−237727.	−0.700825	−0.350412	−	0.936596i	$-0.613959\pi$
$163$	−237727.	−0.700825	−0.350412	+	0.936596i	$0.613959\pi$
$164$	0	0
$165$	−62860.1	−0.179749
$166$	0	0
$167$	−487880.	−1.35370	−0.676849	−	0.736122i	$-0.736655\pi$
$167$	−487880.	−1.35370	−0.676849	+	0.736122i	$0.736655\pi$
$168$	0	0
$169$	−294260.	−0.792527
$170$	0	0
$171$	231756.	0.606095
$172$	0	0
$173$	325942.	0.827991	0.413996	−	0.910279i	$-0.364133\pi$
$173$	325942.	0.827991	0.413996	+	0.910279i	$0.364133\pi$
$174$	0	0
$175$	−51954.6	−0.128242
$176$	0	0
$177$	38638.1	0.0927005
$178$	0	0
$179$	−462906.	−1.07984	−0.539921	−	0.841716i	$-0.681546\pi$
$179$	−462906.	−1.07984	−0.539921	+	0.841716i	$0.681546\pi$
$180$	0	0
$181$	23901.3	0.0542281	0.0271141	−	0.999632i	$-0.491368\pi$
$181$	23901.3	0.0542281	0.0271141	+	0.999632i	$0.491368\pi$
$182$	0	0
$183$	61485.8	0.135721
$184$	0	0
$185$	−521950.	−1.12124
$186$	0	0
$187$	85243.1	0.178261
$188$	0	0
$189$	183038.	0.372724
$190$	0	0
$191$	565986.	1.12259	0.561297	−	0.827615i	$-0.310303\pi$
$191$	565986.	1.12259	0.561297	+	0.827615i	$0.310303\pi$
$192$	0	0
$193$	91762.3	0.177325	0.0886627	−	0.996062i	$-0.471741\pi$
$193$	91762.3	0.177325	0.0886627	+	0.996062i	$0.471741\pi$
$194$	0	0
$195$	144188.	0.271546
$196$	0	0
$197$	485247.	0.890836	0.445418	−	0.895323i	$-0.353055\pi$
$197$	485247.	0.890836	0.445418	+	0.895323i	$0.353055\pi$
$198$	0	0
$199$	−692632.	−1.23985	−0.619926	−	0.784660i	$-0.712838\pi$
$199$	−692632.	−1.23985	−0.619926	+	0.784660i	$0.712838\pi$
$200$	0	0
$201$	−510745.	−0.891690
$202$	0	0
$203$	988619.	1.68379
$204$	0	0
$205$	1.05165e6	1.74778
$206$	0	0
$207$	86415.1	0.140173
$208$	0	0
$209$	346203.	0.548233
$210$	0	0
$211$	441020.	0.681949	0.340975	−	0.940073i	$-0.389243\pi$
$211$	441020.	0.681949	0.340975	+	0.940073i	$0.389243\pi$
$212$	0	0
$213$	28683.8	0.0433199
$214$	0	0
$215$	234039.	0.345297
$216$	0	0
$217$	−161784.	−0.233231
$218$	0	0
$219$	66574.2	0.0937985
$220$	0	0
$221$	−195530.	−0.269298
$222$	0	0
$223$	−1.13133e6	−1.52345	−0.761726	−	0.647899i	$-0.775648\pi$
$223$	−1.13133e6	−1.52345	−0.761726	+	0.647899i	$0.775648\pi$
$224$	0	0
$225$	16760.8	0.0220718
$226$	0	0
$227$	820354.	1.05666	0.528332	−	0.849038i	$-0.322818\pi$
$227$	820354.	1.05666	0.528332	+	0.849038i	$0.322818\pi$
$228$	0	0
$229$	−1.00301e6	−1.26392	−0.631958	−	0.775003i	$-0.717748\pi$
$229$	−1.00301e6	−1.26392	−0.631958	+	0.775003i	$0.717748\pi$
$230$	0	0
$231$	273428.	0.337142
$232$	0	0
$233$	734.569	0.000886427 0	0.000443214	−	1.00000i	$-0.499859\pi$
$233$	734.569	0.000886427 0	0.000443214		1.00000i	$0.499859\pi$
$234$	0	0
$235$	−1.19780e6	−1.41486
$236$	0	0
$237$	219543.	0.253892
$238$	0	0
$239$	1.06403e6	1.20492	0.602460	−	0.798149i	$-0.294187\pi$
$239$	1.06403e6	1.20492	0.602460	+	0.798149i	$0.294187\pi$
$240$	0	0
$241$	−661636.	−0.733798	−0.366899	−	0.930261i	$-0.619581\pi$
$241$	−661636.	−0.733798	−0.366899	+	0.930261i	$0.619581\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	2.66881e6	2.84055
$246$	0	0
$247$	−794118.	−0.828214
$248$	0	0
$249$	925158.	0.945622
$250$	0	0
$251$	−1.34864e6	−1.35118	−0.675590	−	0.737277i	$-0.736111\pi$
$251$	−1.34864e6	−1.35118	−0.675590	+	0.737277i	$0.736111\pi$
$252$	0	0
$253$	129089.	0.126791
$254$	0	0
$255$	−365986.	−0.352463
$256$	0	0
$257$	−1.65015e6	−1.55844	−0.779220	−	0.626751i	$-0.784384\pi$
$257$	−1.65015e6	−1.55844	−0.779220	+	0.626751i	$0.784384\pi$
$258$	0	0
$259$	2.27036e6	2.10303
$260$	0	0
$261$	−318933.	−0.289800
$262$	0	0
$263$	868163.	0.773948	0.386974	−	0.922091i	$-0.373520\pi$
$263$	868163.	0.773948	0.386974	+	0.922091i	$0.373520\pi$
$264$	0	0
$265$	−1.52884e6	−1.33736
$266$	0	0
$267$	−446395.	−0.383213
$268$	0	0
$269$	271547.	0.228804	0.114402	−	0.993435i	$-0.463505\pi$
$269$	271547.	0.228804	0.114402	+	0.993435i	$0.463505\pi$
$270$	0	0
$271$	752770.	0.622643	0.311322	−	0.950305i	$-0.399228\pi$
$271$	752770.	0.622643	0.311322	+	0.950305i	$0.399228\pi$
$272$	0	0
$273$	−627186.	−0.509319
$274$	0	0
$275$	25037.7	0.0199647
$276$	0	0
$277$	811445.	0.635418	0.317709	−	0.948188i	$-0.397086\pi$
$277$	811445.	0.635418	0.317709	+	0.948188i	$0.397086\pi$
$278$	0	0
$279$	52192.3	0.0401417
$280$	0	0
$281$	−1.72395e6	−1.30244	−0.651221	−	0.758888i	$-0.725743\pi$
$281$	−1.72395e6	−1.30244	−0.651221	+	0.758888i	$0.725743\pi$
$282$	0	0
$283$	−272979.	−0.202611	−0.101305	−	0.994855i	$-0.532302\pi$
$283$	−272979.	−0.202611	−0.101305	+	0.994855i	$0.532302\pi$
$284$	0	0
$285$	−1.48640e6	−1.08399
$286$	0	0
$287$	−4.57446e6	−3.27820
$288$	0	0
$289$	−923553.	−0.650455
$290$	0	0
$291$	830515.	0.574930
$292$	0	0
$293$	−713441.	−0.485500	−0.242750	−	0.970089i	$-0.578049\pi$
$293$	−713441.	−0.485500	−0.242750	+	0.970089i	$0.578049\pi$
$294$	0	0
$295$	−247811.	−0.165793
$296$	0	0
$297$	−88209.0	−0.0580259
$298$	0	0
$299$	−296104.	−0.191543
$300$	0	0
$301$	−1.01802e6	−0.647649
$302$	0	0
$303$	319915.	0.200184
$304$	0	0
$305$	−394348.	−0.242734
$306$	0	0
$307$	2.67734e6	1.62128	0.810640	−	0.585545i	$-0.199119\pi$
$307$	2.67734e6	1.62128	0.810640	+	0.585545i	$0.199119\pi$
$308$	0	0
$309$	−538891.	−0.321074
$310$	0	0
$311$	1.28078e6	0.750886	0.375443	−	0.926846i	$-0.377491\pi$
$311$	1.28078e6	0.750886	0.375443	+	0.926846i	$0.377491\pi$
$312$	0	0
$313$	1.36808e6	0.789313	0.394656	−	0.918829i	$-0.370864\pi$
$313$	1.36808e6	0.789313	0.394656	+	0.918829i	$0.370864\pi$
$314$	0	0
$315$	−1.17394e6	−0.666608
$316$	0	0
$317$	−7686.28	−0.00429604	−0.00214802	−	0.999998i	$-0.500684\pi$
$317$	−7686.28	−0.00429604	−0.00214802	+	0.999998i	$0.500684\pi$
$318$	0	0
$319$	−476431.	−0.262134
$320$	0	0
$321$	803284.	0.435117
$322$	0	0
$323$	2.01567e6	1.07501
$324$	0	0
$325$	−57431.3	−0.0301606
$326$	0	0
$327$	−205164.	−0.106104
$328$	0	0
$329$	5.21014e6	2.65375
$330$	0	0
$331$	−958347.	−0.480787	−0.240393	−	0.970676i	$-0.577276\pi$
$331$	−958347.	−0.480787	−0.240393	+	0.970676i	$0.577276\pi$
$332$	0	0
$333$	−732430.	−0.361956
$334$	0	0
$335$	3.27574e6	1.59477
$336$	0	0
$337$	4.08768e6	1.96066	0.980330	−	0.197365i	$-0.0632384\pi$
$337$	4.08768e6	1.96066	0.980330	+	0.197365i	$0.0632384\pi$
$338$	0	0
$339$	1.49458e6	0.706350
$340$	0	0
$341$	77966.3	0.0363096
$342$	0	0
$343$	−7.38880e6	−3.39108
$344$	0	0
$345$	−554236.	−0.250696
$346$	0	0
$347$	−84250.2	−0.0375619	−0.0187809	−	0.999824i	$-0.505979\pi$
$347$	−84250.2	−0.0375619	−0.0187809	+	0.999824i	$0.505979\pi$
$348$	0	0
$349$	1.11859e6	0.491597	0.245798	−	0.969321i	$-0.420950\pi$
$349$	1.11859e6	0.491597	0.245798	+	0.969321i	$0.420950\pi$
$350$	0	0
$351$	202333.	0.0876595
$352$	0	0
$353$	−3.73570e6	−1.59564	−0.797820	−	0.602896i	$-0.794013\pi$
$353$	−3.73570e6	−1.59564	−0.797820	+	0.602896i	$0.794013\pi$
$354$	0	0
$355$	−183968.	−0.0774766
$356$	0	0
$357$	1.59196e6	0.661090
$358$	0	0
$359$	−1.45342e6	−0.595189	−0.297595	−	0.954692i	$-0.596184\pi$
$359$	−1.45342e6	−0.595189	−0.297595	+	0.954692i	$0.596184\pi$
$360$	0	0
$361$	5.71027e6	2.30616
$362$	0	0
$363$	−131769.	−0.0524864
$364$	0	0
$365$	−426984.	−0.167756
$366$	0	0
$367$	−25363.4	−0.00982975	−0.00491487	−	0.999988i	$-0.501564\pi$
$367$	−25363.4	−0.00982975	−0.00491487	+	0.999988i	$0.501564\pi$
$368$	0	0
$369$	1.47574e6	0.564214
$370$	0	0
$371$	6.65013e6	2.50839
$372$	0	0
$373$	−1.72695e6	−0.642699	−0.321350	−	0.946961i	$-0.604136\pi$
$373$	−1.72695e6	−0.642699	−0.321350	+	0.946961i	$0.604136\pi$
$374$	0	0
$375$	1.51596e6	0.556684
$376$	0	0
$377$	1.09283e6	0.396005
$378$	0	0
$379$	4.29401e6	1.53555	0.767777	−	0.640718i	$-0.221363\pi$
$379$	4.29401e6	1.53555	0.767777	+	0.640718i	$0.221363\pi$
$380$	0	0
$381$	1.43271e6	0.505644
$382$	0	0
$383$	−1.29297e6	−0.450392	−0.225196	−	0.974313i	$-0.572302\pi$
$383$	−1.29297e6	−0.450392	−0.225196	+	0.974313i	$0.572302\pi$
$384$	0	0
$385$	−1.75367e6	−0.602970
$386$	0	0
$387$	328418.	0.111468
$388$	0	0
$389$	1.55257e6	0.520209	0.260104	−	0.965581i	$-0.416243\pi$
$389$	1.55257e6	0.520209	0.260104	+	0.965581i	$0.416243\pi$
$390$	0	0
$391$	751586.	0.248620
$392$	0	0
$393$	2.08957e6	0.682456
$394$	0	0
$395$	−1.40807e6	−0.454080
$396$	0	0
$397$	819569.	0.260981	0.130491	−	0.991450i	$-0.458345\pi$
$397$	819569.	0.260981	0.130491	+	0.991450i	$0.458345\pi$
$398$	0	0
$399$	6.46551e6	2.03316
$400$	0	0
$401$	1.38956e6	0.431536	0.215768	−	0.976445i	$-0.430774\pi$
$401$	1.38956e6	0.431536	0.215768	+	0.976445i	$0.430774\pi$
$402$	0	0
$403$	−178839.	−0.0548528
$404$	0	0
$405$	378719.	0.114731
$406$	0	0
$407$	−1.09412e6	−0.327401
$408$	0	0
$409$	−3.53091e6	−1.04371	−0.521854	−	0.853035i	$-0.674759\pi$
$409$	−3.53091e6	−1.04371	−0.521854	+	0.853035i	$0.674759\pi$
$410$	0	0
$411$	−613849.	−0.179249
$412$	0	0
$413$	1.07792e6	0.310966
$414$	0	0
$415$	−5.93363e6	−1.69122
$416$	0	0
$417$	2.68346e6	0.755710
$418$	0	0
$419$	6.69977e6	1.86434	0.932170	−	0.362021i	$-0.117913\pi$
$419$	6.69977e6	1.86434	0.932170	+	0.362021i	$0.117913\pi$
$420$	0	0
$421$	−5.01124e6	−1.37797	−0.688986	−	0.724775i	$-0.741944\pi$
$421$	−5.01124e6	−1.37797	−0.688986	+	0.724775i	$0.741944\pi$
$422$	0	0
$423$	−1.68082e6	−0.456740
$424$	0	0
$425$	145775.	0.0391481
$426$	0	0
$427$	1.71533e6	0.455279
$428$	0	0
$429$	302251.	0.0792910
$430$	0	0
$431$	1.14741e6	0.297526	0.148763	−	0.988873i	$-0.452471\pi$
$431$	1.14741e6	0.297526	0.148763	+	0.988873i	$0.452471\pi$
$432$	0	0
$433$	−1.13393e6	−0.290649	−0.145324	−	0.989384i	$-0.546423\pi$
$433$	−1.13393e6	−0.290649	−0.145324	+	0.989384i	$0.546423\pi$
$434$	0	0
$435$	2.04552e6	0.518300
$436$	0	0
$437$	3.05246e6	0.764622
$438$	0	0
$439$	−7.44692e6	−1.84423	−0.922115	−	0.386915i	$-0.873541\pi$
$439$	−7.44692e6	−1.84423	−0.922115	+	0.386915i	$0.873541\pi$
$440$	0	0
$441$	3.74503e6	0.916977
$442$	0	0
$443$	−6.72521e6	−1.62816	−0.814079	−	0.580754i	$-0.802758\pi$
$443$	−6.72521e6	−1.62816	−0.814079	+	0.580754i	$0.802758\pi$
$444$	0	0
$445$	2.86302e6	0.685368
$446$	0	0
$447$	−748183.	−0.177108
$448$	0	0
$449$	2.17793e6	0.509834	0.254917	−	0.966963i	$-0.417952\pi$
$449$	2.17793e6	0.509834	0.254917	+	0.966963i	$0.417952\pi$
$450$	0	0
$451$	2.20450e6	0.510351
$452$	0	0
$453$	1.50838e6	0.345354
$454$	0	0
$455$	4.02255e6	0.910905
$456$	0	0
$457$	−3.72380e6	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$457$	−3.72380e6	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$458$	0	0
$459$	−513572.	−0.113781
$460$	0	0
$461$	4.74202e6	1.03923	0.519615	−	0.854401i	$-0.326076\pi$
$461$	4.74202e6	1.03923	0.519615	+	0.854401i	$0.326076\pi$
$462$	0	0
$463$	4.82803e6	1.04669	0.523344	−	0.852121i	$-0.324684\pi$
$463$	4.82803e6	1.04669	0.523344	+	0.852121i	$0.324684\pi$
$464$	0	0
$465$	−334743.	−0.0717925
$466$	0	0
$467$	−7.56254e6	−1.60463	−0.802316	−	0.596900i	$-0.796399\pi$
$467$	−7.56254e6	−1.60463	−0.802316	+	0.596900i	$0.796399\pi$
$468$	0	0
$469$	−1.42487e7	−2.99119
$470$	0	0
$471$	569337.	0.118254
$472$	0	0
$473$	490599.	0.100826
$474$	0	0
$475$	592045.	0.120398
$476$	0	0
$477$	−2.14536e6	−0.431722
$478$	0	0
$479$	1.71897e6	0.342319	0.171159	−	0.985243i	$-0.445249\pi$
$479$	1.71897e6	0.342319	0.171159	+	0.985243i	$0.445249\pi$
$480$	0	0
$481$	2.50969e6	0.494604
$482$	0	0
$483$	2.41080e6	0.470212
$484$	0	0
$485$	−5.32663e6	−1.02825
$486$	0	0
$487$	3.08125e6	0.588715	0.294357	−	0.955695i	$-0.404894\pi$
$487$	3.08125e6	0.588715	0.294357	+	0.955695i	$0.404894\pi$
$488$	0	0
$489$	2.13954e6	0.404621
$490$	0	0
$491$	1.05909e6	0.198258	0.0991290	−	0.995075i	$-0.468394\pi$
$491$	1.05909e6	0.198258	0.0991290	+	0.995075i	$0.468394\pi$
$492$	0	0
$493$	−2.77388e6	−0.514009
$494$	0	0
$495$	565741.	0.103778
$496$	0	0
$497$	800218.	0.145317
$498$	0	0
$499$	−2.26415e6	−0.407056	−0.203528	−	0.979069i	$-0.565241\pi$
$499$	−2.26415e6	−0.407056	−0.203528	+	0.979069i	$0.565241\pi$
$500$	0	0
$501$	4.39092e6	0.781558
$502$	0	0
$503$	7.65222e6	1.34855	0.674276	−	0.738480i	$-0.264456\pi$
$503$	7.65222e6	1.34855	0.674276	+	0.738480i	$0.264456\pi$
$504$	0	0
$505$	−2.05182e6	−0.358023
$506$	0	0
$507$	2.64834e6	0.457566
$508$	0	0
$509$	489107.	0.0836777	0.0418388	−	0.999124i	$-0.486678\pi$
$509$	489107.	0.0836777	0.0418388	+	0.999124i	$0.486678\pi$
$510$	0	0
$511$	1.85728e6	0.314649
$512$	0	0
$513$	−2.08580e6	−0.349929
$514$	0	0
$515$	3.45626e6	0.574233
$516$	0	0
$517$	−2.51085e6	−0.413137
$518$	0	0
$519$	−2.93348e6	−0.478041
$520$	0	0
$521$	1.36556e6	0.220402	0.110201	−	0.993909i	$-0.464851\pi$
$521$	1.36556e6	0.220402	0.110201	+	0.993909i	$0.464851\pi$
$522$	0	0
$523$	5.63994e6	0.901613	0.450807	−	0.892622i	$-0.351136\pi$
$523$	5.63994e6	0.901613	0.450807	+	0.892622i	$0.351136\pi$
$524$	0	0
$525$	467591.	0.0740403
$526$	0	0
$527$	453937.	0.0711982
$528$	0	0
$529$	−5.29817e6	−0.823164
$530$	0	0
$531$	−347742.	−0.0535207
$532$	0	0
$533$	−5.05667e6	−0.770986
$534$	0	0
$535$	−5.15198e6	−0.778197
$536$	0	0
$537$	4.16615e6	0.623447
$538$	0	0
$539$	5.59442e6	0.829437
$540$	0	0
$541$	8.24934e6	1.21179	0.605893	−	0.795546i	$-0.292816\pi$
$541$	8.24934e6	1.21179	0.605893	+	0.795546i	$0.292816\pi$
$542$	0	0
$543$	−215111.	−0.0313086
$544$	0	0
$545$	1.31585e6	0.189764
$546$	0	0
$547$	4.74537e6	0.678112	0.339056	−	0.940766i	$-0.389892\pi$
$547$	4.74537e6	0.678112	0.339056	+	0.940766i	$0.389892\pi$
$548$	0	0
$549$	−553372.	−0.0783586
$550$	0	0
$551$	−1.12657e7	−1.58082
$552$	0	0
$553$	6.12480e6	0.851685
$554$	0	0
$555$	4.69755e6	0.647349
$556$	0	0
$557$	2.99690e6	0.409293	0.204647	−	0.978836i	$-0.434395\pi$
$557$	2.99690e6	0.409293	0.204647	+	0.978836i	$0.434395\pi$
$558$	0	0
$559$	−1.12533e6	−0.152318
$560$	0	0
$561$	−767188.	−0.102919
$562$	0	0
$563$	3.89491e6	0.517876	0.258938	−	0.965894i	$-0.416627\pi$
$563$	3.89491e6	0.517876	0.258938	+	0.965894i	$0.416627\pi$
$564$	0	0
$565$	−9.58571e6	−1.26329
$566$	0	0
$567$	−1.64735e6	−0.215192
$568$	0	0
$569$	2.08127e6	0.269493	0.134747	−	0.990880i	$-0.456978\pi$
$569$	2.08127e6	0.269493	0.134747	+	0.990880i	$0.456978\pi$
$570$	0	0
$571$	−1.28958e7	−1.65523	−0.827615	−	0.561297i	$-0.810303\pi$
$571$	−1.28958e7	−1.65523	−0.827615	+	0.561297i	$0.810303\pi$
$572$	0	0
$573$	−5.09387e6	−0.648129
$574$	0	0
$575$	220757.	0.0278448
$576$	0	0
$577$	1.26163e6	0.157759	0.0788795	−	0.996884i	$-0.474866\pi$
$577$	1.26163e6	0.157759	0.0788795	+	0.996884i	$0.474866\pi$
$578$	0	0
$579$	−825861.	−0.102379
$580$	0	0
$581$	2.58100e7	3.17210
$582$	0	0
$583$	−3.20480e6	−0.390508
$584$	0	0
$585$	−1.29769e6	−0.156777
$586$	0	0
$587$	1.25673e7	1.50538	0.752689	−	0.658376i	$-0.228756\pi$
$587$	1.25673e7	1.50538	0.752689	+	0.658376i	$0.228756\pi$
$588$	0	0
$589$	1.84360e6	0.218967
$590$	0	0
$591$	−4.36723e6	−0.514324
$592$	0	0
$593$	−4.32620e6	−0.505207	−0.252604	−	0.967570i	$-0.581287\pi$
$593$	−4.32620e6	−0.505207	−0.252604	+	0.967570i	$0.581287\pi$
$594$	0	0
$595$	−1.02102e7	−1.18234
$596$	0	0
$597$	6.23369e6	0.715829
$598$	0	0
$599$	−1.45262e7	−1.65419	−0.827097	−	0.562060i	$-0.810009\pi$
$599$	−1.45262e7	−1.65419	−0.827097	+	0.562060i	$0.810009\pi$
$600$	0	0
$601$	−380688.	−0.0429915	−0.0214958	−	0.999769i	$-0.506843\pi$
$601$	−380688.	−0.0429915	−0.0214958	+	0.999769i	$0.506843\pi$
$602$	0	0
$603$	4.59671e6	0.514818
$604$	0	0
$605$	845120.	0.0938706
$606$	0	0
$607$	4.99233e6	0.549961	0.274980	−	0.961450i	$-0.411329\pi$
$607$	4.99233e6	0.549961	0.274980	+	0.961450i	$0.411329\pi$
$608$	0	0
$609$	−8.89757e6	−0.972139
$610$	0	0
$611$	5.75936e6	0.624125
$612$	0	0
$613$	2.99353e6	0.321760	0.160880	−	0.986974i	$-0.448567\pi$
$613$	2.99353e6	0.321760	0.160880	+	0.986974i	$0.448567\pi$
$614$	0	0
$615$	−9.46488e6	−1.00908
$616$	0	0
$617$	6.27781e6	0.663889	0.331945	−	0.943299i	$-0.392295\pi$
$617$	6.27781e6	0.663889	0.331945	+	0.943299i	$0.392295\pi$
$618$	0	0
$619$	−3.96477e6	−0.415902	−0.207951	−	0.978139i	$-0.566680\pi$
$619$	−3.96477e6	−0.415902	−0.207951	+	0.978139i	$0.566680\pi$
$620$	0	0
$621$	−777736.	−0.0809288
$622$	0	0
$623$	−1.24535e7	−1.28550
$624$	0	0
$625$	−1.03694e7	−1.06183
$626$	0	0
$627$	−3.11583e6	−0.316523
$628$	0	0
$629$	−6.37023e6	−0.641990
$630$	0	0
$631$	9.62587e6	0.962425	0.481212	−	0.876604i	$-0.340197\pi$
$631$	9.62587e6	0.962425	0.481212	+	0.876604i	$0.340197\pi$
$632$	0	0
$633$	−3.96918e6	−0.393724
$634$	0	0
$635$	−9.18888e6	−0.904333
$636$	0	0
$637$	−1.28324e7	−1.25303
$638$	0	0
$639$	−258154.	−0.0250108
$640$	0	0
$641$	−2.20090e6	−0.211571	−0.105785	−	0.994389i	$-0.533736\pi$
$641$	−2.20090e6	−0.211571	−0.105785	+	0.994389i	$0.533736\pi$
$642$	0	0
$643$	−2.00555e7	−1.91296	−0.956482	−	0.291791i	$-0.905749\pi$
$643$	−2.00555e7	−1.91296	−0.956482	+	0.291791i	$0.905749\pi$
$644$	0	0
$645$	−2.10635e6	−0.199357
$646$	0	0
$647$	−128900.	−0.0121057	−0.00605287	−	0.999982i	$-0.501927\pi$
$647$	−128900.	−0.0121057	−0.00605287	+	0.999982i	$0.501927\pi$
$648$	0	0
$649$	−519467.	−0.0484113
$650$	0	0
$651$	1.45606e6	0.134656
$652$	0	0
$653$	3.03250e6	0.278303	0.139151	−	0.990271i	$-0.455563\pi$
$653$	3.03250e6	0.278303	0.139151	+	0.990271i	$0.455563\pi$
$654$	0	0
$655$	−1.34017e7	−1.22056
$656$	0	0
$657$	−599168.	−0.0541546
$658$	0	0
$659$	1.59132e7	1.42740	0.713698	−	0.700454i	$-0.247019\pi$
$659$	1.59132e7	1.42740	0.713698	+	0.700454i	$0.247019\pi$
$660$	0	0
$661$	6.50892e6	0.579436	0.289718	−	0.957112i	$-0.406439\pi$
$661$	6.50892e6	0.579436	0.289718	+	0.957112i	$0.406439\pi$
$662$	0	0
$663$	1.75977e6	0.155479
$664$	0	0
$665$	−4.14675e7	−3.63625
$666$	0	0
$667$	−4.20067e6	−0.365598
$668$	0	0
$669$	1.01820e7	0.879565
$670$	0	0
$671$	−826643.	−0.0708780
$672$	0	0
$673$	−5.66035e6	−0.481732	−0.240866	−	0.970558i	$-0.577431\pi$
$673$	−5.66035e6	−0.481732	−0.240866	+	0.970558i	$0.577431\pi$
$674$	0	0
$675$	−150847.	−0.0127432
$676$	0	0
$677$	−1.53601e7	−1.28802	−0.644012	−	0.765016i	$-0.722731\pi$
$677$	−1.53601e7	−1.28802	−0.644012	+	0.765016i	$0.722731\pi$
$678$	0	0
$679$	2.31697e7	1.92861
$680$	0	0
$681$	−7.38319e6	−0.610065
$682$	0	0
$683$	−1.47069e6	−0.120634	−0.0603170	−	0.998179i	$-0.519211\pi$
$683$	−1.47069e6	−0.120634	−0.0603170	+	0.998179i	$0.519211\pi$
$684$	0	0
$685$	3.93701e6	0.320583
$686$	0	0
$687$	9.02712e6	0.729722
$688$	0	0
$689$	7.35114e6	0.589939
$690$	0	0
$691$	3.65617e6	0.291294	0.145647	−	0.989337i	$-0.453474\pi$
$691$	3.65617e6	0.291294	0.145647	+	0.989337i	$0.453474\pi$
$692$	0	0
$693$	−2.46085e6	−0.194649
$694$	0	0
$695$	−1.72108e7	−1.35157
$696$	0	0
$697$	1.28351e7	1.00073
$698$	0	0
$699$	−6611.12	−0.000511779 0
$700$	0	0
$701$	−5.21232e6	−0.400623	−0.200312	−	0.979732i	$-0.564195\pi$
$701$	−5.21232e6	−0.400623	−0.200312	+	0.979732i	$0.564195\pi$
$702$	0	0
$703$	−2.58718e7	−1.97442
$704$	0	0
$705$	1.07802e7	0.816869
$706$	0	0
$707$	8.92498e6	0.671519
$708$	0	0
$709$	−1.48388e7	−1.10862	−0.554311	−	0.832310i	$-0.687018\pi$
$709$	−1.48388e7	−1.10862	−0.554311	+	0.832310i	$0.687018\pi$
$710$	0	0
$711$	−1.97589e6	−0.146585
$712$	0	0
$713$	687427.	0.0506410
$714$	0	0
$715$	−1.93853e6	−0.141810
$716$	0	0
$717$	−9.57625e6	−0.695661
$718$	0	0
$719$	−1.37497e7	−0.991906	−0.495953	−	0.868349i	$-0.665181\pi$
$719$	−1.37497e7	−0.991906	−0.495953	+	0.868349i	$0.665181\pi$
$720$	0	0
$721$	−1.50339e7	−1.07705
$722$	0	0
$723$	5.95473e6	0.423659
$724$	0	0
$725$	−814748.	−0.0575676
$726$	0	0
$727$	8.20549e6	0.575796	0.287898	−	0.957661i	$-0.407044\pi$
$727$	8.20549e6	0.575796	0.287898	+	0.957661i	$0.407044\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	2.85638e6	0.197707
$732$	0	0
$733$	1.20636e7	0.829312	0.414656	−	0.909978i	$-0.363902\pi$
$733$	1.20636e7	0.829312	0.414656	+	0.909978i	$0.363902\pi$
$734$	0	0
$735$	−2.40193e7	−1.63999
$736$	0	0
$737$	6.86668e6	0.465670
$738$	0	0
$739$	1.02319e7	0.689203	0.344602	−	0.938749i	$-0.388014\pi$
$739$	1.02319e7	0.689203	0.344602	+	0.938749i	$0.388014\pi$
$740$	0	0
$741$	7.14706e6	0.478170
$742$	0	0
$743$	−2.41980e7	−1.60808	−0.804039	−	0.594577i	$-0.797320\pi$
$743$	−2.41980e7	−1.60808	−0.804039	+	0.594577i	$0.797320\pi$
$744$	0	0
$745$	4.79858e6	0.316754
$746$	0	0
$747$	−8.32642e6	−0.545955
$748$	0	0
$749$	2.24100e7	1.45961
$750$	0	0
$751$	−1.10121e7	−0.712476	−0.356238	−	0.934395i	$-0.615941\pi$
$751$	−1.10121e7	−0.712476	−0.356238	+	0.934395i	$0.615941\pi$
$752$	0	0
$753$	1.21378e7	0.780104
$754$	0	0
$755$	−9.67420e6	−0.617657
$756$	0	0
$757$	−2.08747e7	−1.32398	−0.661989	−	0.749513i	$-0.730288\pi$
$757$	−2.08747e7	−1.32398	−0.661989	+	0.749513i	$0.730288\pi$
$758$	0	0
$759$	−1.16180e6	−0.0732028
$760$	0	0
$761$	2.85723e7	1.78848	0.894240	−	0.447587i	$-0.147716\pi$
$761$	2.85723e7	1.78848	0.894240	+	0.447587i	$0.147716\pi$
$762$	0	0
$763$	−5.72365e6	−0.355928
$764$	0	0
$765$	3.29387e6	0.203495
$766$	0	0
$767$	1.19155e6	0.0731347
$768$	0	0
$769$	−3.07246e6	−0.187357	−0.0936787	−	0.995602i	$-0.529863\pi$
$769$	−3.07246e6	−0.187357	−0.0936787	+	0.995602i	$0.529863\pi$
$770$	0	0
$771$	1.48513e7	0.899765
$772$	0	0
$773$	−3.02076e7	−1.81831	−0.909153	−	0.416461i	$-0.863270\pi$
$773$	−3.02076e7	−1.81831	−0.909153	+	0.416461i	$0.863270\pi$
$774$	0	0
$775$	133331.	0.00797401
$776$	0	0
$777$	−2.04333e7	−1.21419
$778$	0	0
$779$	5.21280e7	3.07771
$780$	0	0
$781$	−385638.	−0.0226231
$782$	0	0
$783$	2.87040e6	0.167316
$784$	0	0
$785$	−3.65153e6	−0.211495
$786$	0	0
$787$	2.07854e7	1.19625	0.598126	−	0.801402i	$-0.295912\pi$
$787$	2.07854e7	1.19625	0.598126	+	0.801402i	$0.295912\pi$
$788$	0	0
$789$	−7.81346e6	−0.446839
$790$	0	0
$791$	4.16957e7	2.36946
$792$	0	0
$793$	1.89615e6	0.107075
$794$	0	0
$795$	1.37596e7	0.772125
$796$	0	0
$797$	−7.95535e6	−0.443622	−0.221811	−	0.975090i	$-0.571197\pi$
$797$	−7.95535e6	−0.443622	−0.221811	+	0.975090i	$0.571197\pi$
$798$	0	0
$799$	−1.46187e7	−0.810106
$800$	0	0
$801$	4.01755e6	0.221248
$802$	0	0
$803$	−895054.	−0.0489847
$804$	0	0
$805$	−1.54620e7	−0.840963
$806$	0	0
$807$	−2.44392e6	−0.132100
$808$	0	0
$809$	−3.04660e7	−1.63661	−0.818303	−	0.574787i	$-0.805085\pi$
$809$	−3.04660e7	−1.63661	−0.818303	+	0.574787i	$0.805085\pi$
$810$	0	0
$811$	−1.28041e7	−0.683594	−0.341797	−	0.939774i	$-0.611036\pi$
$811$	−1.28041e7	−0.683594	−0.341797	+	0.939774i	$0.611036\pi$
$812$	0	0
$813$	−6.77493e6	−0.359483
$814$	0	0
$815$	−1.37223e7	−0.723655
$816$	0	0
$817$	1.16008e7	0.608040
$818$	0	0
$819$	5.64468e6	0.294055
$820$	0	0
$821$	1.13635e7	0.588373	0.294186	−	0.955748i	$-0.404951\pi$
$821$	1.13635e7	0.588373	0.294186	+	0.955748i	$0.404951\pi$
$822$	0	0
$823$	−8.23741e6	−0.423927	−0.211964	−	0.977278i	$-0.567986\pi$
$823$	−8.23741e6	−0.423927	−0.211964	+	0.977278i	$0.567986\pi$
$824$	0	0
$825$	−225339.	−0.0115266
$826$	0	0
$827$	−2.21035e7	−1.12382	−0.561909	−	0.827199i	$-0.689933\pi$
$827$	−2.21035e7	−1.12382	−0.561909	+	0.827199i	$0.689933\pi$
$828$	0	0
$829$	−5.57875e6	−0.281936	−0.140968	−	0.990014i	$-0.545021\pi$
$829$	−5.57875e6	−0.281936	−0.140968	+	0.990014i	$0.545021\pi$
$830$	0	0
$831$	−7.30300e6	−0.366859
$832$	0	0
$833$	3.25720e7	1.62641
$834$	0	0
$835$	−2.81618e7	−1.39780
$836$	0	0
$837$	−469731.	−0.0231758
$838$	0	0
$839$	2.33576e7	1.14558	0.572788	−	0.819704i	$-0.305862\pi$
$839$	2.33576e7	1.14558	0.572788	+	0.819704i	$0.305862\pi$
$840$	0	0
$841$	−5.00769e6	−0.244145
$842$	0	0
$843$	1.55155e7	0.751965
$844$	0	0
$845$	−1.69855e7	−0.818345
$846$	0	0
$847$	−3.67608e6	−0.176067
$848$	0	0
$849$	2.45681e6	0.116977
$850$	0	0
$851$	−9.64685e6	−0.456627
$852$	0	0
$853$	8.94855e6	0.421095	0.210548	−	0.977584i	$-0.432475\pi$
$853$	8.94855e6	0.421095	0.210548	+	0.977584i	$0.432475\pi$
$854$	0	0
$855$	1.33776e7	0.625839
$856$	0	0
$857$	−2.03190e6	−0.0945039	−0.0472520	−	0.998883i	$-0.515046\pi$
$857$	−2.03190e6	−0.0945039	−0.0472520	+	0.998883i	$0.515046\pi$
$858$	0	0
$859$	1.92359e7	0.889467	0.444734	−	0.895663i	$-0.353298\pi$
$859$	1.92359e7	0.889467	0.444734	+	0.895663i	$0.353298\pi$
$860$	0	0
$861$	4.11701e7	1.89267
$862$	0	0
$863$	−2.00380e7	−0.915856	−0.457928	−	0.888989i	$-0.651408\pi$
$863$	−2.00380e7	−0.915856	−0.457928	+	0.888989i	$0.651408\pi$
$864$	0	0
$865$	1.88143e7	0.854965
$866$	0	0
$867$	8.31197e6	0.375540
$868$	0	0
$869$	−2.95164e6	−0.132591
$870$	0	0
$871$	−1.57507e7	−0.703486
$872$	0	0
$873$	−7.47464e6	−0.331936
$874$	0	0
$875$	4.22921e7	1.86741
$876$	0	0
$877$	1.98487e7	0.871430	0.435715	−	0.900085i	$-0.356496\pi$
$877$	1.98487e7	0.871430	0.435715	+	0.900085i	$0.356496\pi$
$878$	0	0
$879$	6.42097e6	0.280304
$880$	0	0
$881$	−2.93546e7	−1.27420	−0.637099	−	0.770782i	$-0.719866\pi$
$881$	−2.93546e7	−1.27420	−0.637099	+	0.770782i	$0.719866\pi$
$882$	0	0
$883$	−3.85199e7	−1.66258	−0.831291	−	0.555838i	$-0.812398\pi$
$883$	−3.85199e7	−1.66258	−0.831291	+	0.555838i	$0.812398\pi$
$884$	0	0
$885$	2.23030e6	0.0957204
$886$	0	0
$887$	2.68797e6	0.114714	0.0573569	−	0.998354i	$-0.481733\pi$
$887$	2.68797e6	0.114714	0.0573569	+	0.998354i	$0.481733\pi$
$888$	0	0
$889$	3.99696e7	1.69619
$890$	0	0
$891$	793881.	0.0335013
$892$	0	0
$893$	−5.93719e7	−2.49145
$894$	0	0
$895$	−2.67202e7	−1.11502
$896$	0	0
$897$	2.66493e6	0.110587
$898$	0	0
$899$	−2.53709e6	−0.104698
$900$	0	0
$901$	−1.86590e7	−0.765733
$902$	0	0
$903$	9.16218e6	0.373921
$904$	0	0
$905$	1.37965e6	0.0559947
$906$	0	0
$907$	7.78340e6	0.314160	0.157080	−	0.987586i	$-0.449792\pi$
$907$	7.78340e6	0.314160	0.157080	+	0.987586i	$0.449792\pi$
$908$	0	0
$909$	−2.87924e6	−0.115576
$910$	0	0
$911$	−3.15361e7	−1.25896	−0.629481	−	0.777016i	$-0.716732\pi$
$911$	−3.15361e7	−1.25896	−0.629481	+	0.777016i	$0.716732\pi$
$912$	0	0
$913$	−1.24382e7	−0.493835
$914$	0	0
$915$	3.54913e6	0.140142
$916$	0	0
$917$	5.82946e7	2.28931
$918$	0	0
$919$	−4.21184e7	−1.64507	−0.822533	−	0.568717i	$-0.807440\pi$
$919$	−4.21184e7	−1.64507	−0.822533	+	0.568717i	$0.807440\pi$
$920$	0	0
$921$	−2.40961e7	−0.936047
$922$	0	0
$923$	884572.	0.0341766
$924$	0	0
$925$	−1.87107e6	−0.0719011
$926$	0	0
$927$	4.85002e6	0.185372
$928$	0	0
$929$	9.43595e6	0.358712	0.179356	−	0.983784i	$-0.442599\pi$
$929$	9.43595e6	0.358712	0.179356	+	0.983784i	$0.442599\pi$
$930$	0	0
$931$	1.32287e8	5.00197
$932$	0	0
$933$	−1.15270e7	−0.433524
$934$	0	0
$935$	4.92047e6	0.184068
$936$	0	0
$937$	2.17869e7	0.810674	0.405337	−	0.914167i	$-0.367154\pi$
$937$	2.17869e7	0.810674	0.405337	+	0.914167i	$0.367154\pi$
$938$	0	0
$939$	−1.23127e7	−0.455710
$940$	0	0
$941$	2.18678e7	0.805065	0.402532	−	0.915406i	$-0.368130\pi$
$941$	2.18678e7	0.805065	0.402532	+	0.915406i	$0.368130\pi$
$942$	0	0
$943$	1.94370e7	0.711787
$944$	0	0
$945$	1.05655e7	0.384867
$946$	0	0
$947$	−2.42335e7	−0.878094	−0.439047	−	0.898464i	$-0.644684\pi$
$947$	−2.42335e7	−0.878094	−0.439047	+	0.898464i	$0.644684\pi$
$948$	0	0
$949$	2.05307e6	0.0740010
$950$	0	0
$951$	69176.5	0.00248032
$952$	0	0
$953$	5.56092e7	1.98342	0.991710	−	0.128499i	$-0.0410159\pi$
$953$	5.56092e7	1.98342	0.991710	+	0.128499i	$0.0410159\pi$
$954$	0	0
$955$	3.26703e7	1.15916
$956$	0	0
$957$	4.28788e6	0.151343
$958$	0	0
$959$	−1.71251e7	−0.601294
$960$	0	0
$961$	−2.82140e7	−0.985498
$962$	0	0
$963$	−7.22956e6	−0.251215
$964$	0	0
$965$	5.29678e6	0.183102
$966$	0	0
$967$	2.09122e7	0.719173	0.359586	−	0.933112i	$-0.382918\pi$
$967$	2.09122e7	0.719173	0.359586	+	0.933112i	$0.382918\pi$
$968$	0	0
$969$	−1.81410e7	−0.620659
$970$	0	0
$971$	−1.68184e7	−0.572448	−0.286224	−	0.958163i	$-0.592400\pi$
$971$	−1.68184e7	−0.572448	−0.286224	+	0.958163i	$0.592400\pi$
$972$	0	0
$973$	7.48630e7	2.53504
$974$	0	0
$975$	516882.	0.0174132
$976$	0	0
$977$	−3.45472e7	−1.15792	−0.578958	−	0.815358i	$-0.696540\pi$
$977$	−3.45472e7	−1.15792	−0.578958	+	0.815358i	$0.696540\pi$
$978$	0	0
$979$	6.00153e6	0.200127
$980$	0	0
$981$	1.84648e6	0.0612592
$982$	0	0
$983$	3.40232e7	1.12303	0.561516	−	0.827466i	$-0.310218\pi$
$983$	3.40232e7	1.12303	0.561516	+	0.827466i	$0.310218\pi$
$984$	0	0
$985$	2.80099e7	0.919857
$986$	0	0
$987$	−4.68913e7	−1.53214
$988$	0	0
$989$	4.32560e6	0.140623
$990$	0	0
$991$	4.47458e7	1.44733	0.723665	−	0.690151i	$-0.242456\pi$
$991$	4.47458e7	1.44733	0.723665	+	0.690151i	$0.242456\pi$
$992$	0	0
$993$	8.62512e6	0.277583
$994$	0	0
$995$	−3.99807e7	−1.28024
$996$	0	0
$997$	−5.86494e6	−0.186864	−0.0934320	−	0.995626i	$-0.529784\pi$
$997$	−5.86494e6	−0.186864	−0.0934320	+	0.995626i	$0.529784\pi$
$998$	0	0
$999$	6.59187e6	0.208975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.a.o.1.2		2
4.3	odd	2		33.6.a.e.1.1	✓	2
12.11	even	2		99.6.a.d.1.2		2
20.19	odd	2		825.6.a.c.1.2		2
44.43	even	2		363.6.a.f.1.2		2
132.131	odd	2		1089.6.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.e.1.1	✓	2	4.3	odd	2
99.6.a.d.1.2		2	12.11	even	2
363.6.a.f.1.2		2	44.43	even	2
528.6.a.o.1.2		2	1.1	even	1	trivial
825.6.a.c.1.2		2	20.19	odd	2
1089.6.a.p.1.1		2	132.131	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:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	528.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +57.7228 q^{5} -251.081 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +57.7228 q^{5} -251.081 q^{7} +81.0000 q^{9} +121.000 q^{11} -277.549 q^{13} -519.505 q^{15} +704.489 q^{17} +2861.18 q^{19} +2259.73 q^{21} +1066.85 q^{23} +206.923 q^{25} -729.000 q^{27} -3937.44 q^{29} +644.350 q^{31} -1089.00 q^{33} -14493.1 q^{35} -9042.34 q^{37} +2497.94 q^{39} +18219.0 q^{41} +4054.54 q^{43} +4675.55 q^{45} -20750.8 q^{47} +46234.9 q^{49} -6340.40 q^{51} -26485.9 q^{53} +6984.46 q^{55} -25750.7 q^{57} -4293.12 q^{59} -6831.76 q^{61} -20337.6 q^{63} -16020.9 q^{65} +56749.5 q^{67} -9601.68 q^{69} -3187.09 q^{71} -7397.14 q^{73} -1862.31 q^{75} -30380.9 q^{77} -24393.7 q^{79} +6561.00 q^{81} -102795. q^{83} +40665.1 q^{85} +35437.0 q^{87} +49599.4 q^{89} +69687.4 q^{91} -5799.15 q^{93} +165156. q^{95} -92279.5 q^{97} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{3} + 58 q^{5} - 146 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q - 18 * q^3 + 58 * q^5 - 146 * q^7 + 162 * q^9	\( 2 q - 18 q^{3} + 58 q^{5} - 146 q^{7} + 162 q^{9} + 242 q^{11} - 130 q^{13} - 522 q^{15} - 728 q^{17} + 828 q^{19} + 1314 q^{21} + 238 q^{23} - 2918 q^{25} - 1458 q^{27} + 696 q^{29} + 10480 q^{31} - 2178 q^{33} - 14464 q^{35} - 1908 q^{37} + 1170 q^{39} + 36484 q^{41} - 9768 q^{43} + 4698 q^{45} - 43742 q^{47} + 40470 q^{49} + 6552 q^{51} - 12174 q^{53} + 7018 q^{55} - 7452 q^{57} + 2788 q^{59} - 25302 q^{61} - 11826 q^{63} - 15980 q^{65} + 40520 q^{67} - 2142 q^{69} - 31386 q^{71} - 46780 q^{73} + 26262 q^{75} - 17666 q^{77} + 16850 q^{79} + 13122 q^{81} - 79440 q^{83} + 40268 q^{85} - 6264 q^{87} - 54204 q^{89} + 85192 q^{91} - 94320 q^{93} + 164592 q^{95} - 241568 q^{97} + 19602 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 + 58 * q^5 - 146 * q^7 + 162 * q^9 + 242 * q^11 - 130 * q^13 - 522 * q^15 - 728 * q^17 + 828 * q^19 + 1314 * q^21 + 238 * q^23 - 2918 * q^25 - 1458 * q^27 + 696 * q^29 + 10480 * q^31 - 2178 * q^33 - 14464 * q^35 - 1908 * q^37 + 1170 * q^39 + 36484 * q^41 - 9768 * q^43 + 4698 * q^45 - 43742 * q^47 + 40470 * q^49 + 6552 * q^51 - 12174 * q^53 + 7018 * q^55 - 7452 * q^57 + 2788 * q^59 - 25302 * q^61 - 11826 * q^63 - 15980 * q^65 + 40520 * q^67 - 2142 * q^69 - 31386 * q^71 - 46780 * q^73 + 26262 * q^75 - 17666 * q^77 + 16850 * q^79 + 13122 * q^81 - 79440 * q^83 + 40268 * q^85 - 6264 * q^87 - 54204 * q^89 + 85192 * q^91 - 94320 * q^93 + 164592 * q^95 - 241568 * q^97 + 19602 * q^99

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